{"id":300,"date":"2017-10-27T16:29:25","date_gmt":"2017-10-27T16:29:25","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/elasticity-stress-and-strain\/"},"modified":"2017-11-08T03:24:14","modified_gmt":"2017-11-08T03:24:14","slug":"elasticity-stress-and-strain","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/elasticity-stress-and-strain\/","title":{"raw":"Elasticity: Stress and Strain","rendered":"Elasticity: Stress and Strain"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>State Hooke\u2019s law.<\/li>\n<li>Explain Hooke\u2019s law using graphical representation between deformation and applied force.<\/li>\n<li>Discuss the three types of deformations such as changes in length, sideways shear and changes in volume.<\/li>\n<li>Describe with examples the young\u2019s modulus, shear modulus and bulk modulus.<\/li>\n<li>Determine the change in length given mass, length and radius.<\/li>\n<\/ul>\n<\/div>\n<p id=\"import-auto-id1165298928075\">We now move from consideration of forces that affect the motion of an object (such as friction and drag) to those that affect an object\u2019s shape. If a bulldozer pushes a car into a wall, the car will not move but it will noticeably change shape. A change in shape due to the application of a force is a <span data-type=\"term\">deformation<\/span>. Even very small forces are known to cause some deformation. For small deformations, two important characteristics are observed. First, the object returns to its original shape when the force is removed\u2014that is, the deformation is elastic for small deformations. Second, the size of the deformation is proportional to the force\u2014that is, for small deformations, Hooke\u2019s law is obeyed. In equation form, <span data-type=\"term\">Hooke\u2019s law<\/span> is given by      <\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]F=k\\Delta L,[\/latex]<\/div>\n<p id=\"import-auto-id1165296525939\">where [latex]\\Delta L[\/latex] is the amount of deformation (the change in length, for example) produced by the force [latex]F[\/latex]<em data-effect=\"italics\">, and [latex]k[\/latex] is a proportionality constant that depends on the shape and composition of the object and the direction of the force. Note that this force is a function of the deformation [latex]\\Delta L[\/latex] \u2014it is not constant as a kinetic friction force is. Rearranging this to<\/em><\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-72\">[latex]\\Delta L=\\frac{F}{k}[\/latex]<\/div>\n<p id=\"import-auto-id1165298765826\">makes it clear that the deformation is proportional to the applied force. <a href=\"#import-auto-id1165298678464\" class=\"autogenerated-content\">(Figure)<\/a> shows the Hooke\u2019s law relationship between the extension [latex]\\Delta L[\/latex] of a spring or of a human bone. For metals or springs, the straight line region in which Hooke\u2019s law pertains is much larger. Bones are brittle and the elastic region is small and the fracture abrupt. Eventually a large enough stress to the material will cause it to break or fracture. <span data-type=\"term\">Tensile strength<\/span> is the breaking stress that will cause permanent deformation or fracture of a material.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1165298979850\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Hooke\u2019s Law<\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-319\">[latex]F=\\mathrm{k\\Delta L},[\/latex]<\/div>\n<p id=\"import-auto-id1165298539788\">where [latex]\\Delta L[\/latex] is the amount of deformation (the change in length, for example) produced by the force [latex]F[\/latex]<em data-effect=\"italics\">, and [latex]k[\/latex] is a proportionality constant that depends on the shape and composition of the object and the direction of the force.<\/em><\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-516\">[latex]\\Delta L=\\frac{F}{k}[\/latex]<\/div>\n<\/div>\n<div class=\"bc-figure figure\">\n<div class=\"bc-figcaption figcaption\">A graph of deformation [latex]\\Delta L[\/latex] versus applied force [latex]F[\/latex]. The straight segment is the linear region where Hooke\u2019s law is obeyed. The slope of the straight region is [latex]\\frac{1}{k}[\/latex]. For larger forces, the graph is curved but the deformation is still elastic\u2014 [latex]\\Delta L[\/latex] will return to zero if the force is removed. Still greater forces permanently deform the object until it finally fractures. The shape of the curve near fracture depends on several factors, including how the force [latex]F[\/latex] is applied. Note that in this graph the slope increases just before fracture, indicating that a small increase in [latex]F[\/latex] is producing a large increase in [latex]L[\/latex] near the fracture.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165298555486\" data-alt=\"Line graph of change in length versus applied force. The line has a constant positive slope from the origin in the region where Hooke\u2019s law is obeyed. The slope then decreases, with a lower, still positive slope until the end of the elastic region. The slope then increases dramatically in the region of permanent deformation until fracturing occurs.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_06_03_01a.jpg\" data-media-type=\"image\/jpg\" alt=\"Line graph of change in length versus applied force. The line has a constant positive slope from the origin in the region where Hooke\u2019s law is obeyed. The slope then decreases, with a lower, still positive slope until the end of the elastic region. The slope then increases dramatically in the region of permanent deformation until fracturing occurs.\" width=\"225\"><\/span><\/p><\/div>\n<p id=\"import-auto-id1165296220522\">The proportionality constant [latex]k[\/latex] depends upon a number of factors for the material. For example, a guitar string made of nylon stretches when it is tightened, and the elongation [latex]\\Delta L[\/latex] is proportional to the force applied (at least for small deformations). Thicker nylon strings and ones made of steel stretch less for the same applied force, implying they have a larger [latex]k[\/latex] (see <a href=\"#import-auto-id1165296232089\" class=\"autogenerated-content\">(Figure)<\/a>). Finally, all three strings return to their normal lengths when the force is removed, provided the deformation is small. Most materials will behave in this manner if the deformation is less than about 0.1% or about 1 part in [latex]{\\text{10}}^{3}[\/latex]<sup>.<\/sup><\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165296232089\">\n<div class=\"bc-figcaption figcaption\">The same force, in this case a weight ([latex]w[\/latex]), applied to three different guitar strings of identical length produces the three different deformations shown as shaded segments. The string on the left is thin nylon, the one in the middle is thicker nylon, and the one on the right is steel.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165296306833\" data-alt=\"Diagram of weight w attached to each of three guitar strings of initial length L zero hanging vertically from a ceiling. The weight pulls down on the strings with force w. The ceiling pulls up on the strings with force w. The first string of thin nylon has a deformation of delta L due to the force of the weight pulling down. The middle string of thicker nylon has a smaller deformation. The third string of thin steel has the smallest deformation.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_06_03_02a.jpg\" data-media-type=\"image\/jpg\" alt=\"Diagram of weight w attached to each of three guitar strings of initial length L zero hanging vertically from a ceiling. The weight pulls down on the strings with force w. The ceiling pulls up on the strings with force w. The first string of thin nylon has a deformation of delta L due to the force of the weight pulling down. The middle string of thicker nylon has a smaller deformation. The third string of thin steel has the smallest deformation.\" width=\"175\"><\/span><\/p><\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1165298669085\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Stretch Yourself a Little<\/div>\n<p id=\"import-auto-id1165298704565\">How would you go about measuring the proportionality constant [latex]k[\/latex] of a rubber band? If a rubber band stretched 3 cm when a 100-g mass was attached to it, then how much would it stretch if two similar rubber bands were attached to the same mass\u2014even if put together in parallel or alternatively if tied together in series?<\/p>\n<\/div>\n<p id=\"import-auto-id1165296217693\">We now consider three specific types of deformations: changes in length (tension and compression), sideways shear (stress), and changes in volume. All deformations are assumed to be small unless otherwise stated.<\/p>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1165298672473\">\n<h1 data-type=\"title\">Changes in Length\u2014Tension and Compression: Elastic Modulus<\/h1>\n<p id=\"import-auto-id1165296231320\">A change in length [latex]\\Delta L[\/latex] is produced when a force is applied to a wire or rod parallel to its length [latex]{L}_{0}[\/latex], either stretching it (a tension) or compressing it. (See <a href=\"#import-auto-id1165298469619\" class=\"autogenerated-content\">(Figure)<\/a>.)<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298469619\">\n<div class=\"bc-figcaption figcaption\">(a) Tension. The rod is stretched a length [latex]\\Delta L[\/latex] when a force is applied parallel to its length. (b) Compression. The same rod is compressed by forces with the same magnitude in the opposite direction. For very small deformations and uniform materials, [latex]\\Delta L[\/latex] is approximately the same for the same magnitude of tension or compression. For larger deformations, the cross-sectional area changes as the rod is compressed or stretched.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165298774917\" data-alt=\"Figure a is a cylindrical rod standing on its end with a height of L sub zero. Two vectors labeled F extend away from each end. A dotted outline indicates that the rod is stretched by a length of delta L. Figure b is a similar rod of identical height L sub zero, but two vectors labeled F exert a force toward the ends of the rod. A dotted line indicates that the rod is compressed by a length of delta L.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_06_03_03a.jpg\" data-media-type=\"image\/jpg\" alt=\"Figure a is a cylindrical rod standing on its end with a height of L sub zero. Two vectors labeled F extend away from each end. A dotted outline indicates that the rod is stretched by a length of delta L. Figure b is a similar rod of identical height L sub zero, but two vectors labeled F exert a force toward the ends of the rod. A dotted line indicates that the rod is compressed by a length of delta L.\" width=\"175\"><\/span><\/p><\/div>\n<p id=\"import-auto-id1165298853431\">Experiments have shown that the change in length ([latex]\\Delta L[\/latex]) depends on only a few variables. As already noted, [latex]\\Delta L[\/latex] is proportional to the force [latex]F[\/latex] and depends on the substance from which the object is made. Additionally, the change in length is proportional to the original length [latex]{L}_{0}[\/latex] and inversely proportional to the cross-sectional area of the wire or rod. For example, a long guitar string will stretch more than a short one, and a thick string will stretch less than a thin one. We can combine all these factors into one equation for [latex]\\Delta L[\/latex]:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-511\">[latex]\\Delta L=\\frac{1}{Y}\\frac{F}{A}{L}_{0},[\/latex]<\/div>\n<p id=\"import-auto-id1165298765805\">where [latex]\\Delta L[\/latex] is the change in length, [latex]F[\/latex] the applied force, <em data-effect=\"italics\">[latex]Y[\/latex] is a factor, called the elastic modulus or Young\u2019s modulus, that depends on the substance, <em data-effect=\"italics\">[latex]A[\/latex] is the cross-sectional area, and <em data-effect=\"italics\">[latex]{L}_{0}[\/latex] is the original length. <a href=\"#import-auto-id1165298671576\" class=\"autogenerated-content\">(Figure)<\/a> lists values of <em data-effect=\"italics\">[latex]Y[\/latex]<\/em> for several materials\u2014those with a large [latex]Y[\/latex]  are said to have a large tensile stifness because they deform less for a given tension or compression.<\/em><\/em><\/em><\/p>\n<p id=\"import-auto-id1165298596914\">\n<\/p><table id=\"import-auto-id1165298671576\" summary=\"Table showing Young\u2019s modulus Y, Shear modulus S, and bulk modulus B for a variety of materials.\">\n<caption><span data-type=\"title\">Elastic Moduli<a data-type=\"footnote-number\" href=\"#footnote1\"><sup>1<\/sup><\/a><\/span><\/caption>\n<thead>\n<tr>\n<th data-align=\"center\">Material<\/th>\n<th data-align=\"center\">Young\u2019s modulus (tension\u2013compression)<em data-effect=\"italics\">Y <\/em>[latex]\\left({\\text{10}}^{\\text{9}}\\phantom{\\rule{0.25em}{0ex}}{\\text{N\/m}}^{\\text{2}}\\right)[\/latex]<\/th>\n<th data-align=\"center\">Shear modulus <em data-effect=\"italics\">S<\/em>[latex]\\left({\\text{10}}^{\\text{9}}\\phantom{\\rule{0.25em}{0ex}}{\\text{N\/m}}^{\\text{2}}\\right)[\/latex]<\/th>\n<th data-align=\"center\">Bulk modulus <em data-effect=\"italics\">B<\/em> [latex]\\left({\\text{10}}^{\\text{9}}\\phantom{\\rule{0.25em}{0ex}}{\\text{N\/m}}^{\\text{2}}\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Aluminum<\/td>\n<td data-align=\"center\">70<\/td>\n<td data-align=\"center\">25<\/td>\n<td data-align=\"center\">75<\/td>\n<\/tr>\n<tr>\n<td>Bone \u2013 tension<\/td>\n<td data-align=\"center\">16<\/td>\n<td data-align=\"center\">80<\/td>\n<td data-align=\"center\">8<\/td>\n<\/tr>\n<tr>\n<td>Bone \u2013 compression<\/td>\n<td data-align=\"center\">9<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Brass<\/td>\n<td data-align=\"center\">90<\/td>\n<td data-align=\"center\">35<\/td>\n<td data-align=\"center\">75<\/td>\n<\/tr>\n<tr>\n<td>Brick<\/td>\n<td data-align=\"center\">15<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Concrete<\/td>\n<td data-align=\"center\">20<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Glass<\/td>\n<td data-align=\"center\">70<\/td>\n<td data-align=\"center\">20<\/td>\n<td data-align=\"center\">30<\/td>\n<\/tr>\n<tr>\n<td>Granite<\/td>\n<td data-align=\"center\">45<\/td>\n<td data-align=\"center\">20<\/td>\n<td data-align=\"center\">45<\/td>\n<\/tr>\n<tr>\n<td>Hair (human)<\/td>\n<td data-align=\"center\">10<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Hardwood<\/td>\n<td data-align=\"center\">15<\/td>\n<td data-align=\"center\">10<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Iron, cast<\/td>\n<td data-align=\"center\">100<\/td>\n<td data-align=\"center\">40<\/td>\n<td data-align=\"center\">90<\/td>\n<\/tr>\n<tr>\n<td>Lead<\/td>\n<td data-align=\"center\">16<\/td>\n<td data-align=\"center\">5<\/td>\n<td data-align=\"center\">50<\/td>\n<\/tr>\n<tr>\n<td>Marble<\/td>\n<td data-align=\"center\">60<\/td>\n<td data-align=\"center\">20<\/td>\n<td data-align=\"center\">70<\/td>\n<\/tr>\n<tr>\n<td>Nylon<\/td>\n<td data-align=\"center\">5<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Polystyrene<\/td>\n<td data-align=\"center\">3<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Silk<\/td>\n<td data-align=\"center\">6<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Spider thread<\/td>\n<td data-align=\"center\">3<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Steel<\/td>\n<td data-align=\"center\">210<\/td>\n<td data-align=\"center\">80<\/td>\n<td data-align=\"center\">130<\/td>\n<\/tr>\n<tr>\n<td>Tendon<\/td>\n<td data-align=\"center\">1<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Acetone<\/td>\n<td><\/td>\n<td><\/td>\n<td data-align=\"center\">0.7<\/td>\n<\/tr>\n<tr>\n<td>Ethanol<\/td>\n<td><\/td>\n<td><\/td>\n<td data-align=\"center\">0.9<\/td>\n<\/tr>\n<tr>\n<td>Glycerin<\/td>\n<td><\/td>\n<td><\/td>\n<td data-align=\"center\">4.5<\/td>\n<\/tr>\n<tr>\n<td>Mercury<\/td>\n<td><\/td>\n<td><\/td>\n<td data-align=\"center\">25<\/td>\n<\/tr>\n<tr>\n<td>Water<\/td>\n<td><\/td>\n<td><\/td>\n<td data-align=\"center\">2.2<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"import-auto-id1165296304602\">Young\u2019s moduli are not listed for liquids and gases in <a href=\"#import-auto-id1165298671576\" class=\"autogenerated-content\">(Figure)<\/a> because they cannot be stretched or compressed in only one direction. Note that there is an assumption that the object does not accelerate, so that there are actually two applied forces of magnitude [latex]F[\/latex] acting in opposite directions. For example, the strings in <a href=\"#import-auto-id1165298469619\" class=\"autogenerated-content\">(Figure)<\/a> are being pulled down by a force of magnitude [latex]w[\/latex] and held up by the ceiling, which also exerts a force of magnitude [latex]w[\/latex].<\/p>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1165298667158\">\n<div data-type=\"title\" class=\"title\">The Stretch of a Long Cable<\/div>\n<p id=\"import-auto-id1165298912883\">Suspension cables are used to carry gondolas at ski resorts. (See <a href=\"#import-auto-id1165298535463\" class=\"autogenerated-content\">(Figure)<\/a>) Consider a suspension cable that includes an unsupported span of 3 km. Calculate the amount of stretch in the steel cable. Assume that the cable has a diameter of 5.6 cm and the maximum tension it can withstand is [latex]3\\text{.}0\u00d7{\\text{10}}^{6}\\phantom{\\rule{0.25em}{0ex}}\\text{N}[\/latex].<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298535463\">\n<div class=\"bc-figcaption figcaption\">Gondolas travel along suspension cables at the Gala Yuzawa ski resort in Japan. (credit: Rudy Herman, Flickr)<\/div>\n<p><span data-type=\"media\" id=\"eip-id2589235\" data-alt=\"Ski gondolas travel along suspension cables. A vast forest and snowy mountain peaks can be seen in the background.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_06_03_04a.jpg\" data-media-type=\"image\/png\" alt=\"Ski gondolas travel along suspension cables. A vast forest and snowy mountain peaks can be seen in the background.\" width=\"200\"><\/span><\/p><\/div>\n<p id=\"eip-911\"><strong>Strategy<\/strong><\/p>\n<p id=\"eip-148\">The force is equal to the maximum tension, or [latex]F=3\\text{.}0\u00d7{\\text{10}}^{6}\\phantom{\\rule{0.25em}{0ex}}\\text{N}[\/latex]. The cross-sectional area is [latex]{\\mathrm{\\pi r}}^{2}=2\\text{.}\\text{46}\u00d7{\\text{10}}^{-3}\\phantom{\\rule{0.25em}{0ex}}{\\text{m}}^{2}[\/latex]. The equation [latex]\\Delta L=\\frac{1}{Y}\\frac{F}{A}{L}_{0}[\/latex] can be used to find the change in length.<\/p>\n<p id=\"import-auto-id1165296320123\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id1165296320124\">All quantities are known. Thus,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-657\">[latex]\\begin{array}{lll}\\Delta L&amp; =&amp; \\left(\\frac{1}{\\text{210}\u00d7{\\text{10}}^{9}\\phantom{\\rule{0.25em}{0ex}}{\\text{N\/m}}^{2}}\\right)\\left(\\frac{3\\text{.}0\u00d7{\\text{10}}^{6}\\phantom{\\rule{0.25em}{0ex}}\\text{N}}{2.46\u00d7{10}^{\u20133}\\phantom{\\rule{0.25em}{0ex}}{\\text{m}}^{2}}\\right)\\left(\\text{3020 m}\\right)\\\\ &amp; =&amp; \\text{18 m}.\\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id1165296524911\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id1165296320125\">This is quite a stretch, but only about 0.6% of the unsupported length. Effects of temperature upon length might be important in these environments.<\/p>\n<\/div>\n<p id=\"import-auto-id1165298463671\">Bones, on the whole, do not fracture due to tension or compression. Rather they generally fracture due to sideways impact or bending, resulting in the bone shearing or snapping. The behavior of bones under tension and compression is important because it determines the load the bones can carry. Bones are classified as weight-bearing structures such as columns in buildings and trees. Weight-bearing structures have special features; columns in building have steel-reinforcing rods while trees and bones are fibrous. The bones in different parts of the body serve different structural functions and are prone to different stresses. Thus the bone in the top of the femur is arranged in thin sheets separated by marrow while in other places the bones can be cylindrical and filled with marrow or just solid. Overweight people have a tendency toward bone damage due to sustained compressions in bone joints and tendons.<\/p>\n<p id=\"import-auto-id1165296301659\">Another biological example of Hooke\u2019s law occurs in tendons. Functionally, the tendon (the tissue connecting muscle to bone) must stretch easily at first when a force is applied, but offer a much greater restoring force for a greater strain. <a href=\"#import-auto-id1165296334455\" class=\"autogenerated-content\">(Figure)<\/a> shows a stress-strain relationship for a human tendon. Some tendons have a high collagen content so there is relatively little strain, or length change; others, like support tendons (as in the leg) can change length up to 10%. Note that this stress-strain curve is nonlinear, since the slope of the line changes in different regions. In the first part of the stretch called the toe region, the fibers in the tendon begin to align in the direction of the stress\u2014this is called <em data-effect=\"italics\">uncrimping<\/em>. In the linear region, the fibrils will be stretched, and in the failure region individual fibers begin to break. A simple model of this relationship can be illustrated by springs in parallel: different springs are activated at different lengths of stretch. Examples of this are given in the problems at end of this chapter. Ligaments (tissue connecting bone to bone) behave in a similar way.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165296334455\">\n<div class=\"bc-figcaption figcaption\">Typical stress-strain curve for mammalian tendon. Three regions are shown: (1) toe region (2) linear region, and (3) failure region.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165298859883\" data-alt=\"The strain on mammalian tendon is shown by a graph, with strain along the x axis and tensile stress along the y axis. The stress strain curve obtained has three regions, namely, toe region at the bottom, linear region between, and failure region at the top.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_06_03_05_xa.jpg\" data-media-type=\"image\/png\" alt=\"The strain on mammalian tendon is shown by a graph, with strain along the x axis and tensile stress along the y axis. The stress strain curve obtained has three regions, namely, toe region at the bottom, linear region between, and failure region at the top.\" width=\"225\"><\/span><\/p><\/div>\n<p id=\"import-auto-id1165298741089\">Unlike bones and tendons, which need to be strong as well as elastic, the arteries and lungs need to be very stretchable. The elastic properties of the arteries are essential for blood flow. The pressure in the arteries increases and arterial walls stretch when the blood is pumped out of the heart. When the aortic valve shuts, the pressure in the arteries drops and the arterial walls relax to maintain the blood flow. When you feel your pulse, you are feeling exactly this\u2014the elastic behavior of the arteries as the blood gushes through with each pump of the heart. If the arteries were rigid, you would not feel a pulse. The heart is also an organ with special elastic properties. The lungs expand with muscular effort when we breathe in but relax freely and elastically when we breathe out. Our skins are particularly elastic, especially for the young. A young person can go from 100 kg to 60 kg with no visible sag in their skins. The elasticity of all organs reduces with age. Gradual physiological aging through reduction in elasticity starts in the early 20s.<\/p>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1165296715479\">\n<div data-type=\"title\" class=\"title\">Calculating Deformation: How Much Does Your Leg Shorten When You Stand on It?<\/div>\n<p id=\"import-auto-id1165298740956\">Calculate the change in length of the upper leg bone (the femur) when a 70.0 kg man supports 62.0 kg of his mass on it, assuming the bone to be equivalent to a uniform rod that is 40.0 cm long and 2.00 cm in radius.<\/p>\n<p id=\"import-auto-id1165298656924\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id1165298656925\">The force is equal to the weight supported, or<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-843\">[latex]F=\\text{mg}=\\left(\\text{62}\\text{.}0 kg\\right)\\left(9\\text{.}\\text{80 m}{\\text{\/s}}^{2}\\right)=\\text{607}\\text{.}6 N,[\/latex]<\/div>\n<p id=\"import-auto-id1165296383090\">and the cross-sectional area is [latex]{\\mathrm{\\pi r}}^{2}=1\\text{.}\\text{257}\u00d7{\\text{10}}^{-3}\\phantom{\\rule{0.25em}{0ex}}{\\text{m}}^{2}[\/latex]. The equation  [latex]\\Delta L=\\frac{1}{Y}\\frac{F}{A}{L}_{0}[\/latex] can be used to find the change in length.<\/p>\n<p id=\"import-auto-id1165298539029\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id1165298656926\">All quantities except [latex]\\Delta L[\/latex] are known. Note that the compression value for Young\u2019s modulus for bone must be used here. Thus,<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\begin{array}{lll}\\Delta L&amp; =&amp; \\left(\\frac{1}{9\u00d7{\\text{10}}^{9}\\phantom{\\rule{0.25em}{0ex}}{\\text{N\/m}}^{2}}\\right)\\left(\\frac{\\text{607}\\text{.}\\text{6 N}}{1.\\text{257}\u00d7{\\text{10}}^{-3}\\phantom{\\rule{0.25em}{0ex}}{\\text{m}}^{2}}\\right)\\left(0\\text{.}\\text{400 m}\\right)\\\\ &amp; =&amp; 2\u00d7{\\text{10}}^{-5}\\phantom{\\rule{0.25em}{0ex}}\\text{m.}\\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id1165298996193\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id1165298656928\">This small change in length seems reasonable, consistent with our experience that bones are rigid. In fact, even the rather large forces encountered during strenuous physical activity do not compress or bend bones by large amounts. Although bone is rigid compared with fat or muscle, several of the substances listed in <a href=\"#import-auto-id1165298671576\" class=\"autogenerated-content\">(Figure)<\/a> have larger values of Young\u2019s modulus [latex]Y[\/latex]. In other words, they are more rigid.<\/p>\n<\/div>\n<p id=\"import-auto-id1165298709547\">The equation for change in length is traditionally rearranged and written in the following form:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-725\">[latex]\\frac{F}{A}=Y\\frac{\\Delta L}{{L}_{0}}.[\/latex]<\/div>\n<p id=\"import-auto-id1165298781488\">The ratio of force to area, <em data-effect=\"italics\">[latex]\\frac{F}{A}[\/latex]<\/em>, is defined as <span data-type=\"term\" id=\"import-auto-id1165298867113\">stress<\/span> (measured in<br>\n[latex]{\\text{N\/m}}^{\\text{2}}[\/latex]), and the ratio of the change in length to length, [latex]\\frac{\\Delta L}{{L}_{0}}[\/latex], is defined as <span data-type=\"term\" id=\"import-auto-id1165298883978\">strain<\/span> (a unitless quantity). In other words,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-760\">[latex]\\text{stress}=Y\u00d7\\text{strain}.[\/latex]<\/div>\n<p id=\"import-auto-id1165298542943\">In this form, the equation is analogous to Hooke\u2019s law, with stress analogous to force and strain analogous to deformation. If we again rearrange this equation to the form<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-77\">[latex]F=\\text{YA}\\frac{\\Delta L}{{L}_{0}},[\/latex]<\/div>\n<p id=\"import-auto-id1165298782834\">we see that it is the same as Hooke\u2019s law with a proportionality constant<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]k=\\frac{\\text{YA}}{{L}_{0}}.[\/latex]<\/div>\n<p id=\"import-auto-id1165298868469\">This general idea\u2014that force and the deformation it causes are proportional for small deformations\u2014applies to changes in length, sideways bending, and changes in volume.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1165296614220\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Stress<\/div>\n<p id=\"import-auto-id1165296580285\">The ratio of force to area, <em data-effect=\"italics\">[latex]\\frac{F}{A}[\/latex]<\/em>, is defined as stress measured in N\/m<sup>2<\/sup>.<\/p>\n<\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Strain<\/div>\n<p id=\"import-auto-id1165298740967\">The ratio of the change in length to length, [latex]\\frac{\\Delta L}{{L}_{0}}[\/latex], is defined as strain (a unitless quantity). In other words,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-282\">[latex]\\text{stress}=Y\u00d7\\text{strain}.[\/latex]<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1165298648094\">\n<h1 data-type=\"title\">Sideways Stress: Shear Modulus<\/h1>\n<p id=\"import-auto-id1165296298657\"><a href=\"#import-auto-id1165296375245\" class=\"autogenerated-content\">(Figure)<\/a> illustrates what is meant by a sideways stress or a <em data-effect=\"italics\">shearing force<\/em>. Here the deformation is called [latex]\\Delta x[\/latex] and it is perpendicular to [latex]{L}_{0}[\/latex], rather than parallel as with tension and compression. Shear deformation behaves similarly to tension and compression and can be described with similar equations. The expression for <span data-type=\"term\" id=\"import-auto-id1165296259204\">shear deformation<\/span> is<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-178\">[latex]\\Delta x=\\frac{1}{S}\\frac{F}{A}{L}_{0},[\/latex]<\/div>\n<p id=\"import-auto-id1165296400329\">where [latex]S[\/latex] is the shear modulus (see <a href=\"#import-auto-id1165298671576\" class=\"autogenerated-content\">(Figure)<\/a>) and [latex]F[\/latex] is the force applied perpendicular to [latex]{L}_{0}[\/latex] and parallel to the cross-sectional area [latex]A[\/latex]. Again, to keep the object from accelerating, there are actually two equal and opposite forces [latex]F[\/latex] applied across opposite faces, as illustrated in <a href=\"#import-auto-id1165296375245\" class=\"autogenerated-content\">(Figure)<\/a>. The equation is logical\u2014for example, it is easier to bend a long thin pencil (small [latex]A[\/latex]) than a short thick one, and both are more easily bent than similar steel rods (large [latex]S[\/latex]).<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1165298835468\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Shear Deformation<\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-894\">[latex]\\Delta x=\\frac{1}{S}\\frac{F}{A}{L}_{0},[\/latex]<\/div>\n<p id=\"import-auto-id1165298776830\">where [latex]S[\/latex] is the shear modulus and [latex]F[\/latex] is the force applied perpendicular to [latex]{L}_{0}[\/latex] and parallel to the cross-sectional area [latex]A[\/latex].<\/p>\n<\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165296375245\">\n<div class=\"bc-figcaption figcaption\">Shearing forces are applied perpendicular to the length [latex]{L}_{\\text{0}}[\/latex] and parallel to the area<br>\n[latex]A[\/latex], producing a deformation<br>\n[latex]\\text{\u0394x}[\/latex]. Vertical forces are not shown, but it should be kept in mind that in addition to the two shearing forces, [latex]\\mathbf{\\text{F}}[\/latex], there must be supporting forces to keep the object from rotating. The distorting effects of these supporting forces are ignored in this treatment. The weight of the object also is not shown, since it is usually negligible compared with forces large enough to cause significant deformations.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165296375246\" data-alt=\"Bookcase sheared by a force applied at the bottom right toward the bottom left, and at the top left toward the top right.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_06_03_06a.jpg\" data-media-type=\"image\/jpg\" alt=\"Bookcase sheared by a force applied at the bottom right toward the bottom left, and at the top left toward the top right.\" width=\"200\"><\/span><\/p><\/div>\n<p id=\"import-auto-id1165296357394\">Examination of the shear moduli in <a href=\"#import-auto-id1165298671576\" class=\"autogenerated-content\">(Figure)<\/a> reveals some telling patterns. For example, shear moduli are less than Young\u2019s moduli for most materials. Bone is a remarkable exception. Its shear modulus is not only greater than its Young\u2019s modulus, but it is as large as that of steel. This is why bones are so rigid.<\/p>\n<p id=\"import-auto-id1165298927874\">The spinal column (consisting of 26 vertebral segments separated by discs) provides the main support for the head and upper part of the body. The spinal column has normal curvature for stability, but this curvature can be increased, leading to increased shearing forces on the lower vertebrae. Discs are better at withstanding compressional forces than shear forces. Because the spine is not vertical, the weight of the upper body exerts some of both. Pregnant women and people that are overweight (with large abdomens) need to move their shoulders back to maintain balance, thereby increasing the curvature in their spine and so increasing the shear component of the stress. An increased angle due to more curvature increases the shear forces along the plane. These higher shear forces increase the risk of back injury through ruptured discs. The lumbosacral disc (the wedge shaped disc below the last vertebrae) is particularly at risk because of its location.<\/p>\n<p id=\"import-auto-id1165298760956\">The shear moduli for concrete and brick are very small; they are too highly variable to be listed. Concrete used in buildings can withstand compression, as in pillars and arches, but is very poor against shear, as might be encountered in heavily loaded floors or during earthquakes. Modern structures were made possible by the use of steel and steel-reinforced concrete. Almost by definition, liquids and gases have shear moduli near zero, because they flow in response to shearing forces.<\/p>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1165296288315\">\n<div data-type=\"title\" class=\"title\">Calculating Force Required to Deform: That Nail Does Not Bend Much Under a Load<\/div>\n<p id=\"import-auto-id1165296580196\">Find the mass of the picture hanging from a steel nail as shown in <a href=\"#import-auto-id1165298644588\" class=\"autogenerated-content\">(Figure)<\/a>, given that the nail bends only [latex]\\text{1.80 \u00b5m}[\/latex]. (Assume the shear modulus is known to two significant figures.)<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298644588\">\n<div class=\"bc-figcaption figcaption\">Side view of a nail with a picture hung from it. The nail flexes very slightly (shown much larger than actual) because of the shearing effect of the supported weight. Also shown is the upward force of the wall on the nail, illustrating that there are equal and opposite forces applied across opposite cross sections of the nail. See <a href=\"#fs-id1165296288315\" class=\"autogenerated-content\">(Figure)<\/a> for a calculation of the mass of the picture.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165298644589\" data-alt=\"Diagram showing the side view a nail in a wall, deformed by the weight of a picture hanging from it. The weight w of the picture is downward. There is an equal force w upward on the nail from the wall. The nail is 1 point five zero millimeters thick. The length of the nail that is outside the wall is five point zero zero millimeters. The deformation delta x of the nail as a result of the picture is 1 point eight zero micrometers.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_06_03_08a.jpg\" data-media-type=\"image\/jpg\" alt=\"Diagram showing the side view a nail in a wall, deformed by the weight of a picture hanging from it. The weight w of the picture is downward. There is an equal force w upward on the nail from the wall. The nail is 1 point five zero millimeters thick. The length of the nail that is outside the wall is five point zero zero millimeters. The deformation delta x of the nail as a result of the picture is 1 point eight zero micrometers.\" width=\"200\"><\/span><\/p><\/div>\n<p id=\"eip-576\"><strong>Strategy<\/strong><\/p>\n<p id=\"eip-286\">The force [latex]F[\/latex] on the nail (neglecting the nail\u2019s own weight) is the weight of the picture [latex]w[\/latex]. If we can find [latex]w[\/latex], then the mass of the picture is just [latex]\\frac{w}{g}[\/latex] . The equation [latex]\\Delta x=\\frac{1}{S}\\frac{F}{A}{L}_{0}[\/latex] can be solved for [latex]F[\/latex].<\/p>\n<p id=\"import-auto-id1165296267425\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id1165296267426\">Solving the equation [latex]\\Delta x=\\frac{1}{S}\\frac{F}{A}{L}_{0}[\/latex] for [latex]F[\/latex], we see that all other quantities can be found:<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]F=\\frac{\\mathrm{SA}}{{L}_{0}}\\Delta x.[\/latex]<\/div>\n<p id=\"import-auto-id1165298475700\"><em data-effect=\"italics\">S<\/em> is found in <a href=\"#import-auto-id1165298671576\" class=\"autogenerated-content\">(Figure)<\/a> and is [latex]S=\\text{80}\u00d7{\\text{10}}^{9}\\phantom{\\rule{0.25em}{0ex}}{\\text{N\/m}}^{2}[\/latex]. The radius [latex]r[\/latex] is 0.750 mm (as seen in the figure), so the cross-sectional area is<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-365\">[latex]A={\\mathrm{\\pi r}}^{2}=1\\text{.}\\text{77}\u00d7{\\text{10}}^{-6}\\phantom{\\rule{0.25em}{0ex}}{\\text{m}}^{2}.[\/latex]<\/div>\n<p id=\"import-auto-id1165298992929\">The value for [latex]{L}_{0}[\/latex] is also shown in the figure. Thus,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-736\">[latex]F=\\frac{\\left(\\text{80}\u00d7{\\text{10}}^{9}\\phantom{\\rule{0.25em}{0ex}}{\\text{N\/m}}^{2}\\right)\\left(1\\text{.}\\text{77}\u00d7{\\text{10}}^{-6}\\phantom{\\rule{0.25em}{0ex}}{\\text{m}}^{2}\\right)}{\\left(5\\text{.}\\text{00}\u00d7{\\text{10}}^{-3}\\phantom{\\rule{0.25em}{0ex}}\\text{m}\\right)}\\left(1\\text{.}\\text{80}\u00d7{\\text{10}}^{-6}\\phantom{\\rule{0.25em}{0ex}}\\text{m}\\right)=\\text{51 N.}[\/latex]<\/div>\n<p id=\"import-auto-id1165296337027\">This 51 N force is the weight [latex]w[\/latex] of the picture, so the picture\u2019s mass is<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-147\">[latex]m=\\frac{w}{g}=\\frac{F}{g}=5\\text{.2 kg}.[\/latex]<\/div>\n<p id=\"import-auto-id1165298935625\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id1165296267432\">This is a fairly massive picture, and it is impressive that the nail flexes only<br>\n[latex]\\text{1.80 \u00b5m}[\/latex]\u2014an amount undetectable to the unaided eye.<\/p>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1165298483858\">\n<h1 data-type=\"title\">Changes in Volume: Bulk Modulus<\/h1>\n<p id=\"import-auto-id1165298993069\">An object will be compressed in all directions if inward forces are applied evenly on all its surfaces as in <a href=\"#import-auto-id1165296230954\" class=\"autogenerated-content\">(Figure)<\/a>. It is relatively easy to compress gases and extremely difficult to compress liquids and solids. For example, air in a wine bottle is compressed when it is corked. But if you try corking a brim-full bottle, you cannot compress the wine\u2014some must be removed if the cork is to be inserted. The reason for these different compressibilities is that atoms and molecules are separated by large empty spaces in gases but packed close together in liquids and solids. To compress a gas, you must force its atoms and molecules closer together. To compress liquids and solids, you must actually compress their atoms and molecules, and very strong electromagnetic forces in them oppose this compression.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165296230954\">\n<div class=\"bc-figcaption figcaption\">An inward force on all surfaces compresses this cube. Its change in volume is proportional to the force per unit area and its original volume, and is related to the compressibility of the substance.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165296230955\" data-alt=\"A cube with area of cross section A and volume V zero is compressed by an inward force F acting on all surfaces. The compression causes a change in volume delta V, which is proportional to the force per unit area and its original volume. This change in volume is related to the compressibility of the substance.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_06_03_09a.jpg\" data-media-type=\"image\/jpg\" alt=\"A cube with area of cross section A and volume V zero is compressed by an inward force F acting on all surfaces. The compression causes a change in volume delta V, which is proportional to the force per unit area and its original volume. This change in volume is related to the compressibility of the substance.\" width=\"200\"><\/span><\/p><\/div>\n<p id=\"import-auto-id1165296371126\">We can describe the compression or volume deformation of an object with an equation. First, we note that a force \u201capplied evenly\u201d is defined to have the same stress, or ratio of force to area [latex]\\frac{F}{A}[\/latex] on all surfaces. The deformation produced is a change in volume [latex]\\Delta V[\/latex], which is found to behave very similarly to the shear, tension, and compression previously discussed. (This is not surprising, since a compression of the entire object is equivalent to compressing each of its three dimensions.) The relationship of the change in volume to other physical quantities is given by<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-63\">[latex]\\Delta V=\\frac{1}{B}\\frac{F}{A}{V}_{0},[\/latex]<\/div>\n<p id=\"import-auto-id1165298863613\">where [latex]B[\/latex] is the bulk modulus (see <a href=\"#import-auto-id1165298671576\" class=\"autogenerated-content\">(Figure)<\/a>), [latex]{V}_{0}[\/latex] is the original volume, and <em data-effect=\"italics\">[latex]\\frac{F}{A}[\/latex]<\/em> is the force per unit area applied uniformly inward on all surfaces. Note that no bulk moduli are given for gases.<\/p>\n<p id=\"import-auto-id1165298862692\">What are some examples of bulk compression of solids and liquids? One practical example is the manufacture of industrial-grade diamonds by compressing carbon with an extremely large force per unit area. The carbon atoms rearrange their crystalline structure into the more tightly packed pattern of diamonds. In nature, a similar process occurs deep underground, where extremely large forces result from the weight of overlying material. Another natural source of large compressive forces is the pressure created by the weight of water, especially in deep parts of the oceans. Water exerts an inward force on all surfaces of a submerged object, and even on the water itself. At great depths, water is measurably compressed, as the following example illustrates.<\/p>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1165298751418\">\n<div data-type=\"title\" class=\"title\">Calculating Change in Volume with Deformation: How Much Is Water Compressed at Great Ocean Depths?<\/div>\n<p id=\"import-auto-id1165296310879\">Calculate the fractional decrease in volume ([latex]\\frac{\\Delta V}{{V}_{0}}[\/latex]) for seawater at 5.00 km depth, where the force per unit area is [latex]5\\text{.}\\text{00}\u00d7{\\text{10}}^{7}\\phantom{\\rule{0.25em}{0ex}}N\/{m}^{2}[\/latex]<br>\n    .<\/p>\n<p id=\"import-auto-id1165298620485\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id1165296267436\">Equation [latex]\\Delta V=\\frac{1}{B}\\frac{F}{A}{V}_{0}[\/latex] is the correct physical relationship. All quantities in the equation except [latex]\\frac{\\Delta V}{{V}_{0}}[\/latex]<br>\n are known.<\/p>\n<p id=\"import-auto-id1165298726312\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id1165296267427\">Solving for the unknown <\/p>\n[latex]\\frac{\\Delta V}{{V}_{0}}[\/latex]\n<p> gives<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-646\">[latex]\\frac{\\Delta V}{{V}_{0}}=\\frac{1}{B}\\frac{F}{A}.[\/latex]<\/div>\n<p id=\"import-auto-id1165298672501\">Substituting known values with the value for the bulk modulus [latex]B[\/latex] from <a href=\"#import-auto-id1165298671576\" class=\"autogenerated-content\">(Figure)<\/a>,<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\begin{array}{lll}\\frac{\\Delta V}{{V}_{0}}&amp; =&amp; \\frac{5.00\u00d7{\\text{10}}^{7}\\phantom{\\rule{0.25em}{0ex}}{\\text{N\/m}}^{2}}{2\\text{.}2\u00d7{\\text{10}}^{9}\\phantom{\\rule{0.25em}{0ex}}{\\text{N\/m}}^{2}}\\\\ &amp; =&amp; 0.023=2.3%.\\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id1165298579258\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id1165298656929\">Although measurable, this is not a significant decrease in volume considering that the force per unit area is about 500 atmospheres (1 million pounds per square foot). Liquids and solids are extraordinarily difficult to compress.<\/p>\n<\/div>\n<p id=\"import-auto-id1165299000208\">Conversely, very large forces are created by liquids and solids when they try to expand but are constrained from doing so\u2014which is equivalent to compressing them to less than their normal volume. This often occurs when a contained material warms up, since most materials expand when their temperature increases. If the materials are tightly constrained, they deform or break their container. Another very common example occurs when water freezes. Water, unlike most materials, expands when it freezes, and it can easily fracture a boulder, rupture a biological cell, or crack an engine block that gets in its way.<\/p>\n<p id=\"import-auto-id1165298745248\">Other types of deformations, such as torsion or twisting, behave analogously to the tension, shear, and bulk deformations considered here.<\/p>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id1165298595602\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul id=\"fs-id1165296391796\">\n<li id=\"import-auto-id1165298476009\">Hooke\u2019s law is given by\n<div data-type=\"equation\" class=\"equation\">[latex]F=k\\text{\u0394}L,[\/latex]<\/div>\n<p id=\"import-auto-id1165296333496\">where [latex]\\Delta L[\/latex] is the amount of deformation (the change in length), [latex]F[\/latex] is the applied force, and [latex]k[\/latex] is a proportionality constant that depends on the shape and composition of the object and the direction of the force. The relationship between the deformation and the applied force can also be written as<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\Delta L=\\frac{1}{Y}\\frac{F}{A}{L}_{0},[\/latex]<\/div>\n<p id=\"import-auto-id1165298686192\">where <em data-effect=\"italics\">[latex]Y\\phantom{\\rule{0.25em}{0ex}}[\/latex]<\/em><em data-effect=\"italics\"> is <em data-effect=\"italics\">Young\u2019s modulus<\/em>, which depends on the substance, <em data-effect=\"italics\">[latex]A[\/latex]<\/em> is the cross-sectional area, and [latex]{L}_{0}[\/latex] is the original length.<\/em><\/p>\n<\/li>\n<li id=\"import-auto-id1165298686194\">The ratio of force to area, <em data-effect=\"italics\">[latex]\\frac{F}{A}[\/latex]<\/em>, is defined as <em data-effect=\"italics\">stress<\/em>, measured in N\/m<sup>2<\/sup>.<\/li>\n<li id=\"import-auto-id1165298726978\">The ratio of the change in length to length, [latex]\\frac{\\Delta L}{{L}_{0}}[\/latex], is defined as <em data-effect=\"italics\">strain<\/em> (a unitless quantity). In other words,\n<div data-type=\"equation\" class=\"equation\">[latex]\\text{stress}=Y\u00d7\\text{strain}.[\/latex]<\/div>\n<\/li>\n<li id=\"import-auto-id1165298586473\">The expression for shear deformation is\n<div data-type=\"equation\" class=\"equation\">[latex]\\Delta x=\\frac{1}{S}\\frac{F}{A}{L}_{0},[\/latex]<\/div>\n<p id=\"import-auto-id1165298779146\">where [latex]S[\/latex] is the shear modulus and [latex]F[\/latex] is the force applied perpendicular to [latex]{L}_{\\text{0}}[\/latex] and parallel to the cross-sectional area [latex]A[\/latex].<\/p>\n<\/li>\n<li id=\"import-auto-id1165298548492\">The relationship of the change in volume to other physical quantities is given by\n<div data-type=\"equation\" class=\"equation\" id=\"eip-492\">[latex]\\Delta V=\\frac{1}{B}\\frac{F}{A}{V}_{0},[\/latex]<\/div>\n<p id=\"import-auto-id1165298931793\">where [latex]B[\/latex] is the bulk modulus, [latex]{V}_{\\text{0}}[\/latex] is the original volume, and [latex]\\frac{F}{A}[\/latex] is the force per unit area applied uniformly inward on all surfaces.<\/p>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" id=\"fs-id1165298807240\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298797418\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298797419\">\n<p id=\"import-auto-id1165298783801\">The elastic properties of the arteries are essential for blood flow. Explain the importance of this in terms of the characteristics of the flow of blood (pulsating or continuous).<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298867810\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298867811\">\n<p id=\"import-auto-id1165298555604\">What are you feeling when you feel your pulse? Measure your pulse rate for 10 s and for 1 min. Is there a factor of 6 difference?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165296252981\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298517998\">\n<p id=\"import-auto-id1165298727446\">Examine different types of shoes, including sports shoes and thongs. In terms of physics, why are the bottom surfaces designed as they are? What differences will dry and wet conditions make for these surfaces?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298840083\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298840084\">\n<p id=\"import-auto-id1165298770860\">Would you expect your height to be different depending upon the time of day? Why or why not?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298650751\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298619439\">\n<p id=\"import-auto-id1165298776145\">Why can a squirrel jump from a tree branch to the ground and run away undamaged, while a human could break a bone in such a fall?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298550516\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298550517\">\n<p id=\"import-auto-id1165298597904\">Explain why pregnant women often suffer from back strain late in their pregnancy.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298545438\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298545440\">\n<p id=\"import-auto-id1165298453194\">An old carpenter\u2019s trick to keep nails from bending when they are pounded into hard materials is to grip the center of the nail firmly with pliers. Why does this help?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298771893\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298771894\">\n<p id=\"import-auto-id1165296268391\">When a glass bottle full of vinegar warms up, both the vinegar and the glass expand, but vinegar expands significantly more with temperature than glass. The bottle will break if it was filled to its tightly capped lid. Explain why, and also explain how a pocket of air above the vinegar would prevent the break. (This is the function of the air above liquids in glass containers.)<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id1165298820012\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298795593\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298795594\">\n<p id=\"import-auto-id1165298766000\">During a circus act, one performer swings upside down hanging from a trapeze holding another, also upside-down, performer by the legs. If the upward force on the lower performer is three times her weight, how much do the bones (the femurs) in her upper legs stretch? You may assume each is equivalent to a uniform rod 35.0 cm long and 1.80 cm in radius. Her mass is 60.0 kg.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1165298841662\">\n<div data-type=\"equation\" class=\"equation\" id=\"eip-id1565399\">[latex]1\\text{.}\\text{90}\u00d7{\\text{10}}^{-3}\\phantom{\\rule{0.25em}{0ex}}\\text{cm}[\/latex]<\/div>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298595705\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298595706\">\n<p id=\"import-auto-id1165298992604\">During a wrestling match, a 150 kg wrestler briefly stands on one hand during a maneuver designed to perplex his already moribund adversary. By how much does the upper arm bone shorten in length? The bone can be represented by a uniform rod 38.0 cm in length and 2.10 cm in radius.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298778825\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298778826\">\n<p id=\"import-auto-id1165296307050\">(a) The \u201clead\u201d in pencils is a graphite composition with a Young\u2019s modulus of about [latex]1\u00d7{\\text{10}}^{9}\\phantom{\\rule{0.25em}{0ex}}\\text{N}\/{\\text{m}}^{2}[\/latex]<br>\n    . Calculate the change in length of the lead in an automatic pencil if you tap it straight into the pencil with a force of 4.0 N. The lead is 0.50 mm in diameter and 60 mm long. (b) Is the answer reasonable? That is, does it seem to be consistent with what you have observed when using pencils? <\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1165296220516\">\n<p id=\"import-auto-id1165298622237\">(a)1 mm <\/p>\n<div data-type=\"newline\"><\/div>\n<p>(b) This does seem reasonable, since the lead does seem to shrink a little when you push on it.<\/p><\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298941254\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298941255\">\n<p id=\"import-auto-id1165298476018\">TV broadcast antennas are the tallest artificial structures on Earth. In 1987, a 72.0-kg physicist placed himself and 400 kg of equipment at the top of one 610-m high antenna to perform gravity experiments. By how much was the antenna compressed, if we consider it to be equivalent to a steel cylinder 0.150 m in radius?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298598606\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298598607\">\n<p id=\"import-auto-id1165296376441\">(a) By how much does a 65.0-kg mountain climber stretch her 0.800-cm diameter nylon rope when she hangs 35.0 m below a rock outcropping? (b) Does the answer seem to be consistent with what you have observed for nylon ropes? Would it make sense if the rope were actually a bungee cord? <\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1165298944074\">\n<p id=\"import-auto-id1165298547983\">(a)9 cm\n<\/p>\n<div data-type=\"newline\"><\/div>\n<p> (b)This seems reasonable for nylon climbing rope, since it is not supposed to stretch that much.<\/p><\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298937277\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298937278\">\n<p id=\"import-auto-id1165298946498\">A 20.0-m tall hollow aluminum flagpole is equivalent in stiffness to a solid cylinder 4.00 cm in diameter. A strong wind bends the pole much as a horizontal force of 900 N exerted at the top would. How far to the side does the top of the pole flex?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165296239036\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165296239037\">\n<p id=\"import-auto-id1165296317212\">As an oil well is drilled, each new section of drill pipe supports its own weight and that of the pipe and drill bit beneath it. Calculate the stretch in a new 6.00 m length of steel pipe that supports 3.00 km of pipe having a mass of 20.0 kg\/m and a 100-kg drill bit. The pipe is equivalent in stiffness to a solid cylinder 5.00 cm in diameter. <\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1165298608719\">\n<p id=\"import-auto-id1165298778980\">8.59 mm<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298786214\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298786215\">\n<p id=\"import-auto-id1165298744049\">Calculate the force a piano tuner applies to stretch a steel piano wire 8.00 mm, if the wire is originally 0.850 mm in diameter and 1.35 m long.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298982360\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298982361\">\n<p id=\"import-auto-id1165296228592\">A vertebra is subjected to a shearing force of 500 N. Find the shear deformation, taking the vertebra to be a cylinder 3.00 cm high and 4.00 cm in diameter. <\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1165298858039\">\n<div data-type=\"equation\" class=\"equation\" id=\"eip-id1373472\">[latex]1\\text{.}\\text{49}\u00d7{\\text{10}}^{-7}\\phantom{\\rule{0.25em}{0ex}}\\text{m}[\/latex]<\/div>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298797863\">\n<p id=\"import-auto-id1165298547950\">A disk between vertebrae in the spine is subjected to a shearing force of 600 N. Find its shear deformation, taking it to have the shear modulus of  [latex]1\u00d7{\\text{10}}^{9}\\phantom{\\rule{0.25em}{0ex}}\\text{N}\/{\\text{m}}^{2}[\/latex]<br>\n    . The disk is equivalent to a solid cylinder 0.700 cm high and 4.00 cm in diameter.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165296255735\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165296255736\">\n<p id=\"import-auto-id1165298708711\">When using a pencil eraser, you exert a vertical force of 6.00 N at a distance of 2.00 cm from the hardwood-eraser joint. The pencil is 6.00 mm in diameter and is held at an angle of [latex]\\text{20.0\u00ba}[\/latex] to the horizontal. (a) By how much does the wood flex perpendicular to its length? (b) How much is it compressed lengthwise? <\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1165298654339\">\n<p id=\"import-auto-id1165298597988\">(a) [latex]3\\text{.}\\text{99}\u00d7{\\text{10}}^{-7}\\phantom{\\rule{0.25em}{0ex}}m[\/latex]<\/p>\n<p id=\"import-auto-id1165299001845\">(b) [latex]9\\text{.}\\text{67}\u00d7{\\text{10}}^{-8}\\phantom{\\rule{0.25em}{0ex}}m[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298783085\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298783086\">\n<p id=\"import-auto-id1165296580296\">To consider the effect of wires hung on poles, we take data from <a href=\"\/contents\/c372561e-08e8-41a1-ba72-9b4c00394c2d@8#fs-id1348788\" class=\"autogenerated-content\">(Figure)<\/a>, in which tensions in wires supporting a traffic light were calculated. The left wire made an angle [latex]\\text{30.0\u00ba}[\/latex] below the horizontal with the top of its pole and carried a tension of 108 N. The 12.0 m tall hollow aluminum pole is equivalent in stiffness to a 4.50 cm diameter solid cylinder. (a) How far is it bent to the side? (b) By how much is it compressed?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298713046\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298713047\">\n<p id=\"import-auto-id1165298597595\">A farmer making grape juice fills a glass bottle to the brim and caps it tightly. The juice expands more than the glass when it warms up, in such a way that the volume increases by 0.2% (that is, [latex]\\Delta V\/{V}_{0}=2\u00d7{\\text{10}}^{-3}[\/latex]) relative to the space available. Calculate the magnitude of the normal force exerted by the juice per square centimeter if its bulk modulus is [latex]1\\text{.}8\u00d7{\\text{10}}^{9}\\phantom{\\rule{0.25em}{0ex}}{\\text{N\/m}}^{2}[\/latex], assuming the bottle does not break. In view of your answer, do you think the bottle will survive? <\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1165298481228\">\n<p id=\"import-auto-id1165298998551\">[latex]4\u00d7{\\text{10}}^{6}\\phantom{\\rule{0.25em}{0ex}}{\\text{N\/m}}^{2}[\/latex]. This is about 36 atm, greater than a typical jar can withstand.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298913397\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298913398\">\n<p id=\"import-auto-id1165296218446\">(a) When water freezes, its volume increases by 9.05% (that is, [latex]\\Delta V\/{V}_{0}=9\\text{.}\\text{05}\u00d7{\\text{10}}^{-2}[\/latex]<br>\n    ). What force per unit area is water capable of exerting on a container when it freezes? (It is acceptable to use the bulk modulus of water in this problem.) (b) Is it surprising that such forces can fracture engine blocks, boulders, and the like?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298740744\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298520715\">\n<p id=\"import-auto-id1165298995984\">This problem returns to the tightrope walker studied in <a href=\"\/contents\/5a51b5c5-1756-4d6c-b9be-84a9627ef003@8#fs-id986136\" class=\"autogenerated-content\">(Figure)<\/a>, who created a tension of [latex]3\\text{.}\\text{94}\u00d7{\\text{10}}^{3}\\phantom{\\rule{0.25em}{0ex}}N[\/latex]<br>\n     in a wire making an angle [latex]\\text{5.0\u00ba}[\/latex] below the horizontal with each supporting pole. Calculate how much this tension stretches the steel wire if it was originally 15 m long and 0.50 cm in diameter. <\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1165298866753\">\n<p id=\"import-auto-id1165298919311\">1.4 cm<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165296233210\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298620362\">\n<p id=\"import-auto-id1165296232572\">The pole in <a href=\"#import-auto-id1165296309614\" class=\"autogenerated-content\">(Figure)<\/a> is at a [latex]\\text{90.0\u00ba}[\/latex] bend in a power line and is therefore subjected to more shear force than poles in straight parts of the line. The tension in each line is [latex]4\\text{.}\\text{00}\u00d7{\\text{10}}^{4}\\phantom{\\rule{0.25em}{0ex}}N[\/latex], at the angles shown. The pole is 15.0 m tall, has an 18.0 cm diameter, and can be considered to have half the stiffness of hardwood. (a) Calculate the compression of the pole. (b) Find how much it bends and in what direction. (c) Find the tension in a guy wire used to keep the pole straight if it is attached to the top of the pole at an angle of [latex]\\text{30.0\u00ba}[\/latex] with the vertical. (Clearly, the guy wire must be in the opposite direction of the bend.)<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165296309614\">\n<div class=\"bc-figcaption figcaption\">This telephone pole is at a [latex]\\text{90}\\text{\u00ba}[\/latex] bend in a power line. A guy wire is attached to the top of the pole at an angle of [latex]\\text{30}\\text{\u00ba}[\/latex] with the vertical.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165298916431\" data-alt=\"A telephone pole is located at a ninety degree bend in a power line. Each part of the line is at an angle of eighty degrees with the pole and has a tension labeled T. A guy wire is attached to the top of the pole at an angle of thirty degrees with the vertical.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_06_03_10a.jpg\" data-media-type=\"image\/jpg\" alt=\"A telephone pole is located at a ninety degree bend in a power line. Each part of the line is at an angle of eighty degrees with the pole and has a tension labeled T. A guy wire is attached to the top of the pole at an angle of thirty degrees with the vertical.\" width=\"185\"><\/span><\/p><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"footnote-refs\">\n<h2 data-type=\"footnote-title\">Footnotes<\/h2>\n<ol>\n<li><a data-type=\"footnote-ref\" href=\"#footnote-ref1\">1<\/a> Approximate and average values. Young\u2019s moduli [latex]Y[\/latex]      for tension and compression sometimes differ but are averaged here. Bone has significantly different Young\u2019s moduli for tension and compression.<\/li>\n<\/ol>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\n<dl class=\"definition\" id=\"import-auto-id1165298828407\">\n<dt>deformation<\/dt>\n<dd id=\"fs-id1165298696821\">change in shape due to the application of force<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"fs-id1987830\">\n<dt>Hooke\u2019s law<\/dt>\n<dd id=\"fs-id1987834\">proportional relationship between the force [latex]F[\/latex] on a material and the deformation [latex]\\Delta L[\/latex] it causes, [latex]F=k\\Delta L[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1165298828410\">\n<dt>tensile strength<\/dt>\n<dd id=\"fs-id1165298787406\">the breaking stress that will cause permanent deformation or fraction of a material<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1165298696762\">\n<dt>stress<\/dt>\n<dd id=\"fs-id1165298600243\">ratio of force to area<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1165296377254\">\n<dt>strain<\/dt>\n<dd id=\"fs-id1165298840132\">ratio of change in length to original length<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"fs-id1268428\">\n<dt>shear deformation<\/dt>\n<dd id=\"fs-id1268431\">deformation perpendicular to the original length of an object<\/dd>\n<\/dl>\n<\/div>\n\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>State Hooke\u2019s law.<\/li>\n<li>Explain Hooke\u2019s law using graphical representation between deformation and applied force.<\/li>\n<li>Discuss the three types of deformations such as changes in length, sideways shear and changes in volume.<\/li>\n<li>Describe with examples the young\u2019s modulus, shear modulus and bulk modulus.<\/li>\n<li>Determine the change in length given mass, length and radius.<\/li>\n<\/ul>\n<\/div>\n<p id=\"import-auto-id1165298928075\">We now move from consideration of forces that affect the motion of an object (such as friction and drag) to those that affect an object\u2019s shape. If a bulldozer pushes a car into a wall, the car will not move but it will noticeably change shape. A change in shape due to the application of a force is a <span data-type=\"term\">deformation<\/span>. Even very small forces are known to cause some deformation. For small deformations, two important characteristics are observed. First, the object returns to its original shape when the force is removed\u2014that is, the deformation is elastic for small deformations. Second, the size of the deformation is proportional to the force\u2014that is, for small deformations, Hooke\u2019s law is obeyed. In equation form, <span data-type=\"term\">Hooke\u2019s law<\/span> is given by      <\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-06fd37866d885d20102d453b8b427b4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;&#61;&#107;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#76;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"78\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"import-auto-id1165296525939\">where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1c907f3a1bf93e7f9930b1deb8554e51_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/> is the amount of deformation (the change in length, for example) produced by the force <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2510519bbe1660dfdffb4195c7287343_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/><em data-effect=\"italics\">, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3422b6bb5c160593658b7c39425d9880_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"9\" style=\"vertical-align: 0px;\" \/> is a proportionality constant that depends on the shape and composition of the object and the direction of the force. Note that this force is a function of the deformation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1c907f3a1bf93e7f9930b1deb8554e51_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/> \u2014it is not constant as a kinetic friction force is. Rearranging this to<\/em><\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-72\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-01524ae8d939c40ba47e0ae5c4bd9988_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#76;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#70;&#125;&#123;&#107;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"63\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id1165298765826\">makes it clear that the deformation is proportional to the applied force. <a href=\"#import-auto-id1165298678464\" class=\"autogenerated-content\">(Figure)<\/a> shows the Hooke\u2019s law relationship between the extension <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1c907f3a1bf93e7f9930b1deb8554e51_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/> of a spring or of a human bone. For metals or springs, the straight line region in which Hooke\u2019s law pertains is much larger. Bones are brittle and the elastic region is small and the fracture abrupt. Eventually a large enough stress to the material will cause it to break or fracture. <span data-type=\"term\">Tensile strength<\/span> is the breaking stress that will cause permanent deformation or fracture of a material.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1165298979850\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Hooke\u2019s Law<\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-319\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6f1fd866958c1d49e02a6316882b561c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;&#61;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#107;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#76;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"77\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"import-auto-id1165298539788\">where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1c907f3a1bf93e7f9930b1deb8554e51_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/> is the amount of deformation (the change in length, for example) produced by the force <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2510519bbe1660dfdffb4195c7287343_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/><em data-effect=\"italics\">, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3422b6bb5c160593658b7c39425d9880_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"9\" style=\"vertical-align: 0px;\" \/> is a proportionality constant that depends on the shape and composition of the object and the direction of the force.<\/em><\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-516\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-01524ae8d939c40ba47e0ae5c4bd9988_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#76;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#70;&#125;&#123;&#107;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"63\" style=\"vertical-align: -6px;\" \/><\/div>\n<\/div>\n<div class=\"bc-figure figure\">\n<div class=\"bc-figcaption figcaption\">A graph of deformation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1c907f3a1bf93e7f9930b1deb8554e51_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/> versus applied force <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2510519bbe1660dfdffb4195c7287343_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/>. The straight segment is the linear region where Hooke\u2019s law is obeyed. The slope of the straight region is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5fcee93106a81b86972787e97fc27f8a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#107;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"8\" style=\"vertical-align: -6px;\" \/>. For larger forces, the graph is curved but the deformation is still elastic\u2014 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1c907f3a1bf93e7f9930b1deb8554e51_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/> will return to zero if the force is removed. Still greater forces permanently deform the object until it finally fractures. The shape of the curve near fracture depends on several factors, including how the force <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2510519bbe1660dfdffb4195c7287343_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> is applied. Note that in this graph the slope increases just before fracture, indicating that a small increase in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2510519bbe1660dfdffb4195c7287343_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> is producing a large increase in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-66a9f474fc3c52efdfb0ba6a70199ee8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/> near the fracture.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165298555486\" data-alt=\"Line graph of change in length versus applied force. The line has a constant positive slope from the origin in the region where Hooke\u2019s law is obeyed. The slope then decreases, with a lower, still positive slope until the end of the elastic region. The slope then increases dramatically in the region of permanent deformation until fracturing occurs.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_06_03_01a.jpg\" data-media-type=\"image\/jpg\" alt=\"Line graph of change in length versus applied force. The line has a constant positive slope from the origin in the region where Hooke\u2019s law is obeyed. The slope then decreases, with a lower, still positive slope until the end of the elastic region. The slope then increases dramatically in the region of permanent deformation until fracturing occurs.\" width=\"225\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id1165296220522\">The proportionality constant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3422b6bb5c160593658b7c39425d9880_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"9\" style=\"vertical-align: 0px;\" \/> depends upon a number of factors for the material. For example, a guitar string made of nylon stretches when it is tightened, and the elongation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1c907f3a1bf93e7f9930b1deb8554e51_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/> is proportional to the force applied (at least for small deformations). Thicker nylon strings and ones made of steel stretch less for the same applied force, implying they have a larger <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3422b6bb5c160593658b7c39425d9880_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"9\" style=\"vertical-align: 0px;\" \/> (see <a href=\"#import-auto-id1165296232089\" class=\"autogenerated-content\">(Figure)<\/a>). Finally, all three strings return to their normal lengths when the force is removed, provided the deformation is small. Most materials will behave in this manner if the deformation is less than about 0.1% or about 1 part in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2785a351d5c590d8be7d440e0bed118e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"24\" style=\"vertical-align: -1px;\" \/><sup>.<\/sup><\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165296232089\">\n<div class=\"bc-figcaption figcaption\">The same force, in this case a weight (<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dfee5c980777976ae8cf6541893fb572_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#119;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"13\" style=\"vertical-align: 0px;\" \/>), applied to three different guitar strings of identical length produces the three different deformations shown as shaded segments. The string on the left is thin nylon, the one in the middle is thicker nylon, and the one on the right is steel.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165296306833\" data-alt=\"Diagram of weight w attached to each of three guitar strings of initial length L zero hanging vertically from a ceiling. The weight pulls down on the strings with force w. The ceiling pulls up on the strings with force w. The first string of thin nylon has a deformation of delta L due to the force of the weight pulling down. The middle string of thicker nylon has a smaller deformation. The third string of thin steel has the smallest deformation.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_06_03_02a.jpg\" data-media-type=\"image\/jpg\" alt=\"Diagram of weight w attached to each of three guitar strings of initial length L zero hanging vertically from a ceiling. The weight pulls down on the strings with force w. The ceiling pulls up on the strings with force w. The first string of thin nylon has a deformation of delta L due to the force of the weight pulling down. The middle string of thicker nylon has a smaller deformation. The third string of thin steel has the smallest deformation.\" width=\"175\" \/><\/span><\/p>\n<\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1165298669085\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Stretch Yourself a Little<\/div>\n<p id=\"import-auto-id1165298704565\">How would you go about measuring the proportionality constant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3422b6bb5c160593658b7c39425d9880_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"9\" style=\"vertical-align: 0px;\" \/> of a rubber band? If a rubber band stretched 3 cm when a 100-g mass was attached to it, then how much would it stretch if two similar rubber bands were attached to the same mass\u2014even if put together in parallel or alternatively if tied together in series?<\/p>\n<\/div>\n<p id=\"import-auto-id1165296217693\">We now consider three specific types of deformations: changes in length (tension and compression), sideways shear (stress), and changes in volume. All deformations are assumed to be small unless otherwise stated.<\/p>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1165298672473\">\n<h1 data-type=\"title\">Changes in Length\u2014Tension and Compression: Elastic Modulus<\/h1>\n<p id=\"import-auto-id1165296231320\">A change in length <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1c907f3a1bf93e7f9930b1deb8554e51_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/> is produced when a force is applied to a wire or rod parallel to its length <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4b534bdf9b43c38daf787ca868a98823_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#76;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"19\" style=\"vertical-align: -3px;\" \/>, either stretching it (a tension) or compressing it. (See <a href=\"#import-auto-id1165298469619\" class=\"autogenerated-content\">(Figure)<\/a>.)<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298469619\">\n<div class=\"bc-figcaption figcaption\">(a) Tension. The rod is stretched a length <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1c907f3a1bf93e7f9930b1deb8554e51_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/> when a force is applied parallel to its length. (b) Compression. The same rod is compressed by forces with the same magnitude in the opposite direction. For very small deformations and uniform materials, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1c907f3a1bf93e7f9930b1deb8554e51_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/> is approximately the same for the same magnitude of tension or compression. For larger deformations, the cross-sectional area changes as the rod is compressed or stretched.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165298774917\" data-alt=\"Figure a is a cylindrical rod standing on its end with a height of L sub zero. Two vectors labeled F extend away from each end. A dotted outline indicates that the rod is stretched by a length of delta L. Figure b is a similar rod of identical height L sub zero, but two vectors labeled F exert a force toward the ends of the rod. A dotted line indicates that the rod is compressed by a length of delta L.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_06_03_03a.jpg\" data-media-type=\"image\/jpg\" alt=\"Figure a is a cylindrical rod standing on its end with a height of L sub zero. Two vectors labeled F extend away from each end. A dotted outline indicates that the rod is stretched by a length of delta L. Figure b is a similar rod of identical height L sub zero, but two vectors labeled F exert a force toward the ends of the rod. A dotted line indicates that the rod is compressed by a length of delta L.\" width=\"175\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id1165298853431\">Experiments have shown that the change in length (<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1c907f3a1bf93e7f9930b1deb8554e51_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/>) depends on only a few variables. As already noted, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1c907f3a1bf93e7f9930b1deb8554e51_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/> is proportional to the force <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2510519bbe1660dfdffb4195c7287343_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> and depends on the substance from which the object is made. Additionally, the change in length is proportional to the original length <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4b534bdf9b43c38daf787ca868a98823_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#76;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"19\" style=\"vertical-align: -3px;\" \/> and inversely proportional to the cross-sectional area of the wire or rod. For example, a long guitar string will stretch more than a short one, and a thick string will stretch less than a thin one. We can combine all these factors into one equation for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1c907f3a1bf93e7f9930b1deb8554e51_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/>:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-511\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d52365995dc6a1fc5337b33c9350c9a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#76;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#89;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#70;&#125;&#123;&#65;&#125;&#123;&#76;&#125;&#95;&#123;&#48;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"104\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id1165298765805\">where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1c907f3a1bf93e7f9930b1deb8554e51_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/> is the change in length, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2510519bbe1660dfdffb4195c7287343_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> the applied force, <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-82606c3098bb09002088b0f6f9ffbb2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#89;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> is a factor, called the elastic modulus or Young\u2019s modulus, that depends on the substance, <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/> is the cross-sectional area, and <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4b534bdf9b43c38daf787ca868a98823_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#76;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"19\" style=\"vertical-align: -3px;\" \/> is the original length. <a href=\"#import-auto-id1165298671576\" class=\"autogenerated-content\">(Figure)<\/a> lists values of <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-82606c3098bb09002088b0f6f9ffbb2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#89;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/><\/em> for several materials\u2014those with a large <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-82606c3098bb09002088b0f6f9ffbb2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#89;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/>  are said to have a large tensile stifness because they deform less for a given tension or compression.<\/em><\/em><\/em><\/p>\n<p id=\"import-auto-id1165298596914\">\n<table id=\"import-auto-id1165298671576\" summary=\"Table showing Young\u2019s modulus Y, Shear modulus S, and bulk modulus B for a variety of materials.\">\n<caption><span data-type=\"title\">Elastic Moduli<a data-type=\"footnote-number\" href=\"#footnote1\"><sup>1<\/sup><\/a><\/span><\/caption>\n<thead>\n<tr>\n<th data-align=\"center\">Material<\/th>\n<th data-align=\"center\">Young\u2019s modulus (tension\u2013compression)<em data-effect=\"italics\">Y <\/em><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6afa197aecad024fef05c85ae96ea87d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#47;&#109;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"90\" style=\"vertical-align: -12px;\" \/><\/th>\n<th data-align=\"center\">Shear modulus <em data-effect=\"italics\">S<\/em><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6afa197aecad024fef05c85ae96ea87d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#47;&#109;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"90\" style=\"vertical-align: -12px;\" \/><\/th>\n<th data-align=\"center\">Bulk modulus <em data-effect=\"italics\">B<\/em> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6afa197aecad024fef05c85ae96ea87d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#47;&#109;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"90\" style=\"vertical-align: -12px;\" \/><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Aluminum<\/td>\n<td data-align=\"center\">70<\/td>\n<td data-align=\"center\">25<\/td>\n<td data-align=\"center\">75<\/td>\n<\/tr>\n<tr>\n<td>Bone \u2013 tension<\/td>\n<td data-align=\"center\">16<\/td>\n<td data-align=\"center\">80<\/td>\n<td data-align=\"center\">8<\/td>\n<\/tr>\n<tr>\n<td>Bone \u2013 compression<\/td>\n<td data-align=\"center\">9<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Brass<\/td>\n<td data-align=\"center\">90<\/td>\n<td data-align=\"center\">35<\/td>\n<td data-align=\"center\">75<\/td>\n<\/tr>\n<tr>\n<td>Brick<\/td>\n<td data-align=\"center\">15<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Concrete<\/td>\n<td data-align=\"center\">20<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Glass<\/td>\n<td data-align=\"center\">70<\/td>\n<td data-align=\"center\">20<\/td>\n<td data-align=\"center\">30<\/td>\n<\/tr>\n<tr>\n<td>Granite<\/td>\n<td data-align=\"center\">45<\/td>\n<td data-align=\"center\">20<\/td>\n<td data-align=\"center\">45<\/td>\n<\/tr>\n<tr>\n<td>Hair (human)<\/td>\n<td data-align=\"center\">10<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Hardwood<\/td>\n<td data-align=\"center\">15<\/td>\n<td data-align=\"center\">10<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Iron, cast<\/td>\n<td data-align=\"center\">100<\/td>\n<td data-align=\"center\">40<\/td>\n<td data-align=\"center\">90<\/td>\n<\/tr>\n<tr>\n<td>Lead<\/td>\n<td data-align=\"center\">16<\/td>\n<td data-align=\"center\">5<\/td>\n<td data-align=\"center\">50<\/td>\n<\/tr>\n<tr>\n<td>Marble<\/td>\n<td data-align=\"center\">60<\/td>\n<td data-align=\"center\">20<\/td>\n<td data-align=\"center\">70<\/td>\n<\/tr>\n<tr>\n<td>Nylon<\/td>\n<td data-align=\"center\">5<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Polystyrene<\/td>\n<td data-align=\"center\">3<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Silk<\/td>\n<td data-align=\"center\">6<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Spider thread<\/td>\n<td data-align=\"center\">3<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Steel<\/td>\n<td data-align=\"center\">210<\/td>\n<td data-align=\"center\">80<\/td>\n<td data-align=\"center\">130<\/td>\n<\/tr>\n<tr>\n<td>Tendon<\/td>\n<td data-align=\"center\">1<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Acetone<\/td>\n<td><\/td>\n<td><\/td>\n<td data-align=\"center\">0.7<\/td>\n<\/tr>\n<tr>\n<td>Ethanol<\/td>\n<td><\/td>\n<td><\/td>\n<td data-align=\"center\">0.9<\/td>\n<\/tr>\n<tr>\n<td>Glycerin<\/td>\n<td><\/td>\n<td><\/td>\n<td data-align=\"center\">4.5<\/td>\n<\/tr>\n<tr>\n<td>Mercury<\/td>\n<td><\/td>\n<td><\/td>\n<td data-align=\"center\">25<\/td>\n<\/tr>\n<tr>\n<td>Water<\/td>\n<td><\/td>\n<td><\/td>\n<td data-align=\"center\">2.2<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"import-auto-id1165296304602\">Young\u2019s moduli are not listed for liquids and gases in <a href=\"#import-auto-id1165298671576\" class=\"autogenerated-content\">(Figure)<\/a> because they cannot be stretched or compressed in only one direction. Note that there is an assumption that the object does not accelerate, so that there are actually two applied forces of magnitude <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2510519bbe1660dfdffb4195c7287343_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> acting in opposite directions. For example, the strings in <a href=\"#import-auto-id1165298469619\" class=\"autogenerated-content\">(Figure)<\/a> are being pulled down by a force of magnitude <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dfee5c980777976ae8cf6541893fb572_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#119;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"13\" style=\"vertical-align: 0px;\" \/> and held up by the ceiling, which also exerts a force of magnitude <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dfee5c980777976ae8cf6541893fb572_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#119;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"13\" style=\"vertical-align: 0px;\" \/>.<\/p>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1165298667158\">\n<div data-type=\"title\" class=\"title\">The Stretch of a Long Cable<\/div>\n<p id=\"import-auto-id1165298912883\">Suspension cables are used to carry gondolas at ski resorts. (See <a href=\"#import-auto-id1165298535463\" class=\"autogenerated-content\">(Figure)<\/a>) Consider a suspension cable that includes an unsupported span of 3 km. Calculate the amount of stretch in the steel cable. Assume that the cable has a diameter of 5.6 cm and the maximum tension it can withstand is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-14be5a169fc9f1380de5e38ce1090efd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#54;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"66\" style=\"vertical-align: -1px;\" \/>.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298535463\">\n<div class=\"bc-figcaption figcaption\">Gondolas travel along suspension cables at the Gala Yuzawa ski resort in Japan. (credit: Rudy Herman, Flickr)<\/div>\n<p><span data-type=\"media\" id=\"eip-id2589235\" data-alt=\"Ski gondolas travel along suspension cables. A vast forest and snowy mountain peaks can be seen in the background.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_06_03_04a.jpg\" data-media-type=\"image\/png\" alt=\"Ski gondolas travel along suspension cables. A vast forest and snowy mountain peaks can be seen in the background.\" width=\"200\" \/><\/span><\/p>\n<\/div>\n<p id=\"eip-911\"><strong>Strategy<\/strong><\/p>\n<p id=\"eip-148\">The force is equal to the maximum tension, or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e04d96d88b4be96e08e398be926d3729_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;&#61;&#51;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#54;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"103\" style=\"vertical-align: -1px;\" \/>. The cross-sectional area is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e9a13c6d5e4a0155625f6a46716f3c79_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#92;&#112;&#105;&#32;&#114;&#125;&#125;&#94;&#123;&#50;&#125;&#61;&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#54;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"143\" style=\"vertical-align: -1px;\" \/>. The equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1c27ab0c42e3733cc053de319b32d9ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#76;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#89;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#70;&#125;&#123;&#65;&#125;&#123;&#76;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"99\" style=\"vertical-align: -6px;\" \/> can be used to find the change in length.<\/p>\n<p id=\"import-auto-id1165296320123\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id1165296320124\">All quantities are known. Thus,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-657\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-460fa39313089eeb2635f95d4845e484_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#108;&#108;&#125;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#76;&#38;&#32;&#61;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#49;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#57;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#47;&#109;&#125;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#54;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#125;&#125;&#123;&#50;&#46;&#52;&#54;&times;&#123;&#49;&#48;&#125;&#94;&#123;&#45;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#48;&#50;&#48;&#32;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#56;&#32;&#109;&#125;&#46;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"356\" style=\"vertical-align: -17px;\" \/><\/div>\n<p id=\"import-auto-id1165296524911\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id1165296320125\">This is quite a stretch, but only about 0.6% of the unsupported length. Effects of temperature upon length might be important in these environments.<\/p>\n<\/div>\n<p id=\"import-auto-id1165298463671\">Bones, on the whole, do not fracture due to tension or compression. Rather they generally fracture due to sideways impact or bending, resulting in the bone shearing or snapping. The behavior of bones under tension and compression is important because it determines the load the bones can carry. Bones are classified as weight-bearing structures such as columns in buildings and trees. Weight-bearing structures have special features; columns in building have steel-reinforcing rods while trees and bones are fibrous. The bones in different parts of the body serve different structural functions and are prone to different stresses. Thus the bone in the top of the femur is arranged in thin sheets separated by marrow while in other places the bones can be cylindrical and filled with marrow or just solid. Overweight people have a tendency toward bone damage due to sustained compressions in bone joints and tendons.<\/p>\n<p id=\"import-auto-id1165296301659\">Another biological example of Hooke\u2019s law occurs in tendons. Functionally, the tendon (the tissue connecting muscle to bone) must stretch easily at first when a force is applied, but offer a much greater restoring force for a greater strain. <a href=\"#import-auto-id1165296334455\" class=\"autogenerated-content\">(Figure)<\/a> shows a stress-strain relationship for a human tendon. Some tendons have a high collagen content so there is relatively little strain, or length change; others, like support tendons (as in the leg) can change length up to 10%. Note that this stress-strain curve is nonlinear, since the slope of the line changes in different regions. In the first part of the stretch called the toe region, the fibers in the tendon begin to align in the direction of the stress\u2014this is called <em data-effect=\"italics\">uncrimping<\/em>. In the linear region, the fibrils will be stretched, and in the failure region individual fibers begin to break. A simple model of this relationship can be illustrated by springs in parallel: different springs are activated at different lengths of stretch. Examples of this are given in the problems at end of this chapter. Ligaments (tissue connecting bone to bone) behave in a similar way.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165296334455\">\n<div class=\"bc-figcaption figcaption\">Typical stress-strain curve for mammalian tendon. Three regions are shown: (1) toe region (2) linear region, and (3) failure region.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165298859883\" data-alt=\"The strain on mammalian tendon is shown by a graph, with strain along the x axis and tensile stress along the y axis. The stress strain curve obtained has three regions, namely, toe region at the bottom, linear region between, and failure region at the top.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_06_03_05_xa.jpg\" data-media-type=\"image\/png\" alt=\"The strain on mammalian tendon is shown by a graph, with strain along the x axis and tensile stress along the y axis. The stress strain curve obtained has three regions, namely, toe region at the bottom, linear region between, and failure region at the top.\" width=\"225\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id1165298741089\">Unlike bones and tendons, which need to be strong as well as elastic, the arteries and lungs need to be very stretchable. The elastic properties of the arteries are essential for blood flow. The pressure in the arteries increases and arterial walls stretch when the blood is pumped out of the heart. When the aortic valve shuts, the pressure in the arteries drops and the arterial walls relax to maintain the blood flow. When you feel your pulse, you are feeling exactly this\u2014the elastic behavior of the arteries as the blood gushes through with each pump of the heart. If the arteries were rigid, you would not feel a pulse. The heart is also an organ with special elastic properties. The lungs expand with muscular effort when we breathe in but relax freely and elastically when we breathe out. Our skins are particularly elastic, especially for the young. A young person can go from 100 kg to 60 kg with no visible sag in their skins. The elasticity of all organs reduces with age. Gradual physiological aging through reduction in elasticity starts in the early 20s.<\/p>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1165296715479\">\n<div data-type=\"title\" class=\"title\">Calculating Deformation: How Much Does Your Leg Shorten When You Stand on It?<\/div>\n<p id=\"import-auto-id1165298740956\">Calculate the change in length of the upper leg bone (the femur) when a 70.0 kg man supports 62.0 kg of his mass on it, assuming the bone to be equivalent to a uniform rod that is 40.0 cm long and 2.00 cm in radius.<\/p>\n<p id=\"import-auto-id1165298656924\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id1165298656925\">The force is equal to the weight supported, or<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-843\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b55cc9d602781977917e4d7ef4a05f09_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#103;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#32;&#107;&#103;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#48;&#32;&#109;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#48;&#55;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#54;&#32;&#78;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"332\" style=\"vertical-align: -12px;\" \/><\/div>\n<p id=\"import-auto-id1165296383090\">and the cross-sectional area is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-50d9a89b7dbfb53d8dad79372bf98a02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#92;&#112;&#105;&#32;&#114;&#125;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#53;&#55;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"152\" style=\"vertical-align: -1px;\" \/>. The equation  <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1c27ab0c42e3733cc053de319b32d9ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#76;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#89;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#70;&#125;&#123;&#65;&#125;&#123;&#76;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"99\" style=\"vertical-align: -6px;\" \/> can be used to find the change in length.<\/p>\n<p id=\"import-auto-id1165298539029\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id1165298656926\">All quantities except <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1c907f3a1bf93e7f9930b1deb8554e51_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/> are known. Note that the compression value for Young\u2019s modulus for bone must be used here. Thus,<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c28dd72e8e747d76af39ee32644a7c5b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#108;&#108;&#125;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#76;&#38;&#32;&#61;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#57;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#57;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#47;&#109;&#125;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#48;&#55;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#32;&#78;&#125;&#125;&#123;&#49;&#46;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#53;&#55;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#48;&#48;&#32;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#50;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#53;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#46;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"349\" style=\"vertical-align: -17px;\" \/><\/div>\n<p id=\"import-auto-id1165298996193\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id1165298656928\">This small change in length seems reasonable, consistent with our experience that bones are rigid. In fact, even the rather large forces encountered during strenuous physical activity do not compress or bend bones by large amounts. Although bone is rigid compared with fat or muscle, several of the substances listed in <a href=\"#import-auto-id1165298671576\" class=\"autogenerated-content\">(Figure)<\/a> have larger values of Young\u2019s modulus <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-82606c3098bb09002088b0f6f9ffbb2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#89;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/>. In other words, they are more rigid.<\/p>\n<\/div>\n<p id=\"import-auto-id1165298709547\">The equation for change in length is traditionally rearranged and written in the following form:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-725\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b34d2ab7913828328576dcab699211bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#70;&#125;&#123;&#65;&#125;&#61;&#89;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#76;&#125;&#123;&#123;&#76;&#125;&#95;&#123;&#48;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"79\" style=\"vertical-align: -8px;\" \/><\/div>\n<p id=\"import-auto-id1165298781488\">The ratio of force to area, <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d839367af0a3b5c4f9215f914d45d496_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#70;&#125;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"11\" style=\"vertical-align: -6px;\" \/><\/em>, is defined as <span data-type=\"term\" id=\"import-auto-id1165298867113\">stress<\/span> (measured in<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ceb8ee30f59173b1d5416edfdede5419_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#47;&#109;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"44\" style=\"vertical-align: -4px;\" \/>), and the ratio of the change in length to length, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-35dc59d3e472b2c3e025593d161151d1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#76;&#125;&#123;&#123;&#76;&#125;&#95;&#123;&#48;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"21\" style=\"vertical-align: -8px;\" \/>, is defined as <span data-type=\"term\" id=\"import-auto-id1165298883978\">strain<\/span> (a unitless quantity). In other words,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-760\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-22f3d936f04bbf57b30d416efe1860e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#116;&#114;&#101;&#115;&#115;&#125;&#61;&#89;&times;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#116;&#114;&#97;&#105;&#110;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"129\" style=\"vertical-align: -1px;\" \/><\/div>\n<p id=\"import-auto-id1165298542943\">In this form, the equation is analogous to Hooke\u2019s law, with stress analogous to force and strain analogous to deformation. If we again rearrange this equation to the form<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-77\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-fdced4189dc59d606f075a9a4cf6d4df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#89;&#65;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#76;&#125;&#123;&#123;&#76;&#125;&#95;&#123;&#48;&#125;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"91\" style=\"vertical-align: -8px;\" \/><\/div>\n<p id=\"import-auto-id1165298782834\">we see that it is the same as Hooke\u2019s law with a proportionality constant<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d15512793bb817bec421c4bbb4dabff2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#107;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#89;&#65;&#125;&#125;&#123;&#123;&#76;&#125;&#95;&#123;&#48;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"61\" style=\"vertical-align: -8px;\" \/><\/div>\n<p id=\"import-auto-id1165298868469\">This general idea\u2014that force and the deformation it causes are proportional for small deformations\u2014applies to changes in length, sideways bending, and changes in volume.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1165296614220\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Stress<\/div>\n<p id=\"import-auto-id1165296580285\">The ratio of force to area, <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d839367af0a3b5c4f9215f914d45d496_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#70;&#125;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"11\" style=\"vertical-align: -6px;\" \/><\/em>, is defined as stress measured in N\/m<sup>2<\/sup>.<\/p>\n<\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Strain<\/div>\n<p id=\"import-auto-id1165298740967\">The ratio of the change in length to length, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-35dc59d3e472b2c3e025593d161151d1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#76;&#125;&#123;&#123;&#76;&#125;&#95;&#123;&#48;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"21\" style=\"vertical-align: -8px;\" \/>, is defined as strain (a unitless quantity). In other words,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-282\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-22f3d936f04bbf57b30d416efe1860e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#116;&#114;&#101;&#115;&#115;&#125;&#61;&#89;&times;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#116;&#114;&#97;&#105;&#110;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"129\" style=\"vertical-align: -1px;\" \/><\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1165298648094\">\n<h1 data-type=\"title\">Sideways Stress: Shear Modulus<\/h1>\n<p id=\"import-auto-id1165296298657\"><a href=\"#import-auto-id1165296375245\" class=\"autogenerated-content\">(Figure)<\/a> illustrates what is meant by a sideways stress or a <em data-effect=\"italics\">shearing force<\/em>. Here the deformation is called <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b7398eb19d4a7b42e0948c84ca560c29_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"25\" style=\"vertical-align: 0px;\" \/> and it is perpendicular to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4b534bdf9b43c38daf787ca868a98823_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#76;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"19\" style=\"vertical-align: -3px;\" \/>, rather than parallel as with tension and compression. Shear deformation behaves similarly to tension and compression and can be described with similar equations. The expression for <span data-type=\"term\" id=\"import-auto-id1165296259204\">shear deformation<\/span> is<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-178\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9fbf4d302043ef0bc4aa318256d681c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#83;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#70;&#125;&#123;&#65;&#125;&#123;&#76;&#125;&#95;&#123;&#48;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"100\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id1165296400329\">where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-520cb534cd5b6bed768a61515b57cb7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#83;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/> is the shear modulus (see <a href=\"#import-auto-id1165298671576\" class=\"autogenerated-content\">(Figure)<\/a>) and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2510519bbe1660dfdffb4195c7287343_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> is the force applied perpendicular to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4b534bdf9b43c38daf787ca868a98823_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#76;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"19\" style=\"vertical-align: -3px;\" \/> and parallel to the cross-sectional area <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/>. Again, to keep the object from accelerating, there are actually two equal and opposite forces <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2510519bbe1660dfdffb4195c7287343_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> applied across opposite faces, as illustrated in <a href=\"#import-auto-id1165296375245\" class=\"autogenerated-content\">(Figure)<\/a>. The equation is logical\u2014for example, it is easier to bend a long thin pencil (small <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/>) than a short thick one, and both are more easily bent than similar steel rods (large <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-520cb534cd5b6bed768a61515b57cb7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#83;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/>).<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1165298835468\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Shear Deformation<\/div>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-894\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9fbf4d302043ef0bc4aa318256d681c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#83;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#70;&#125;&#123;&#65;&#125;&#123;&#76;&#125;&#95;&#123;&#48;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"100\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id1165298776830\">where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-520cb534cd5b6bed768a61515b57cb7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#83;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/> is the shear modulus and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2510519bbe1660dfdffb4195c7287343_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> is the force applied perpendicular to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4b534bdf9b43c38daf787ca868a98823_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#76;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"19\" style=\"vertical-align: -3px;\" \/> and parallel to the cross-sectional area <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/>.<\/p>\n<\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165296375245\">\n<div class=\"bc-figcaption figcaption\">Shearing forces are applied perpendicular to the length <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6ea9195001e8ac5d80555195a5c918f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#76;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"19\" style=\"vertical-align: -3px;\" \/> and parallel to the area<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/>, producing a deformation<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8fcc2e6cec22ff8b375c37620cbbaf78_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"10\" width=\"10\" style=\"vertical-align: -1px;\" \/>. Vertical forces are not shown, but it should be kept in mind that in addition to the two shearing forces, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c5cd16916557096d431ff3a1af0c9119_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"11\" style=\"vertical-align: -1px;\" \/>, there must be supporting forces to keep the object from rotating. The distorting effects of these supporting forces are ignored in this treatment. The weight of the object also is not shown, since it is usually negligible compared with forces large enough to cause significant deformations.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165296375246\" data-alt=\"Bookcase sheared by a force applied at the bottom right toward the bottom left, and at the top left toward the top right.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_06_03_06a.jpg\" data-media-type=\"image\/jpg\" alt=\"Bookcase sheared by a force applied at the bottom right toward the bottom left, and at the top left toward the top right.\" width=\"200\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id1165296357394\">Examination of the shear moduli in <a href=\"#import-auto-id1165298671576\" class=\"autogenerated-content\">(Figure)<\/a> reveals some telling patterns. For example, shear moduli are less than Young\u2019s moduli for most materials. Bone is a remarkable exception. Its shear modulus is not only greater than its Young\u2019s modulus, but it is as large as that of steel. This is why bones are so rigid.<\/p>\n<p id=\"import-auto-id1165298927874\">The spinal column (consisting of 26 vertebral segments separated by discs) provides the main support for the head and upper part of the body. The spinal column has normal curvature for stability, but this curvature can be increased, leading to increased shearing forces on the lower vertebrae. Discs are better at withstanding compressional forces than shear forces. Because the spine is not vertical, the weight of the upper body exerts some of both. Pregnant women and people that are overweight (with large abdomens) need to move their shoulders back to maintain balance, thereby increasing the curvature in their spine and so increasing the shear component of the stress. An increased angle due to more curvature increases the shear forces along the plane. These higher shear forces increase the risk of back injury through ruptured discs. The lumbosacral disc (the wedge shaped disc below the last vertebrae) is particularly at risk because of its location.<\/p>\n<p id=\"import-auto-id1165298760956\">The shear moduli for concrete and brick are very small; they are too highly variable to be listed. Concrete used in buildings can withstand compression, as in pillars and arches, but is very poor against shear, as might be encountered in heavily loaded floors or during earthquakes. Modern structures were made possible by the use of steel and steel-reinforced concrete. Almost by definition, liquids and gases have shear moduli near zero, because they flow in response to shearing forces.<\/p>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1165296288315\">\n<div data-type=\"title\" class=\"title\">Calculating Force Required to Deform: That Nail Does Not Bend Much Under a Load<\/div>\n<p id=\"import-auto-id1165296580196\">Find the mass of the picture hanging from a steel nail as shown in <a href=\"#import-auto-id1165298644588\" class=\"autogenerated-content\">(Figure)<\/a>, given that the nail bends only <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-32c1950eaa815ee325b8bad396c04fb2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#46;&#56;&#48;&#32;&micro;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"51\" style=\"vertical-align: -1px;\" \/>. (Assume the shear modulus is known to two significant figures.)<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165298644588\">\n<div class=\"bc-figcaption figcaption\">Side view of a nail with a picture hung from it. The nail flexes very slightly (shown much larger than actual) because of the shearing effect of the supported weight. Also shown is the upward force of the wall on the nail, illustrating that there are equal and opposite forces applied across opposite cross sections of the nail. See <a href=\"#fs-id1165296288315\" class=\"autogenerated-content\">(Figure)<\/a> for a calculation of the mass of the picture.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165298644589\" data-alt=\"Diagram showing the side view a nail in a wall, deformed by the weight of a picture hanging from it. The weight w of the picture is downward. There is an equal force w upward on the nail from the wall. The nail is 1 point five zero millimeters thick. The length of the nail that is outside the wall is five point zero zero millimeters. The deformation delta x of the nail as a result of the picture is 1 point eight zero micrometers.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_06_03_08a.jpg\" data-media-type=\"image\/jpg\" alt=\"Diagram showing the side view a nail in a wall, deformed by the weight of a picture hanging from it. The weight w of the picture is downward. There is an equal force w upward on the nail from the wall. The nail is 1 point five zero millimeters thick. The length of the nail that is outside the wall is five point zero zero millimeters. The deformation delta x of the nail as a result of the picture is 1 point eight zero micrometers.\" width=\"200\" \/><\/span><\/p>\n<\/div>\n<p id=\"eip-576\"><strong>Strategy<\/strong><\/p>\n<p id=\"eip-286\">The force <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2510519bbe1660dfdffb4195c7287343_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> on the nail (neglecting the nail\u2019s own weight) is the weight of the picture <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dfee5c980777976ae8cf6541893fb572_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#119;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"13\" style=\"vertical-align: 0px;\" \/>. If we can find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dfee5c980777976ae8cf6541893fb572_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#119;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"13\" style=\"vertical-align: 0px;\" \/>, then the mass of the picture is just <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d25f76e90d769d84bff983a4c88043ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#119;&#125;&#123;&#103;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"11\" style=\"vertical-align: -9px;\" \/> . The equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d9287682eb79c79e478261b815ab35f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#83;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#70;&#125;&#123;&#65;&#125;&#123;&#76;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"95\" style=\"vertical-align: -6px;\" \/> can be solved for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2510519bbe1660dfdffb4195c7287343_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/>.<\/p>\n<p id=\"import-auto-id1165296267425\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id1165296267426\">Solving the equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d9287682eb79c79e478261b815ab35f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#83;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#70;&#125;&#123;&#65;&#125;&#123;&#76;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"95\" style=\"vertical-align: -6px;\" \/> for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2510519bbe1660dfdffb4195c7287343_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/>, we see that all other quantities can be found:<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-24a6e7db886540c2baa2f6c73291321c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#83;&#65;&#125;&#125;&#123;&#123;&#76;&#125;&#95;&#123;&#48;&#125;&#125;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"88\" style=\"vertical-align: -8px;\" \/><\/div>\n<p id=\"import-auto-id1165298475700\"><em data-effect=\"italics\">S<\/em> is found in <a href=\"#import-auto-id1165298671576\" class=\"autogenerated-content\">(Figure)<\/a> and is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e1aeff83229310257ab968d31f992278_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#83;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#57;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#47;&#109;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"127\" style=\"vertical-align: -4px;\" \/>. The radius <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c409433a9e2dfcdb83360a974d243f18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/> is 0.750 mm (as seen in the figure), so the cross-sectional area is<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-365\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-03883917a0f9bb1e4750d76fc6a16688_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#61;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#92;&#112;&#105;&#32;&#114;&#125;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#55;&#55;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#54;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"184\" style=\"vertical-align: -1px;\" \/><\/div>\n<p id=\"import-auto-id1165298992929\">The value for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4b534bdf9b43c38daf787ca868a98823_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#76;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"19\" style=\"vertical-align: -3px;\" \/> is also shown in the figure. Thus,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-736\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-217729892683b08f3997c5b7187e1c88_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#57;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#47;&#109;&#125;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#55;&#55;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#54;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#54;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#49;&#32;&#78;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"390\" style=\"vertical-align: -14px;\" \/><\/div>\n<p id=\"import-auto-id1165296337027\">This 51 N force is the weight <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dfee5c980777976ae8cf6541893fb572_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#119;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"13\" style=\"vertical-align: 0px;\" \/> of the picture, so the picture\u2019s mass is<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-147\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8cbb076639ffa2e9c64464d905b01ada_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#119;&#125;&#123;&#103;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#70;&#125;&#123;&#103;&#125;&#61;&#53;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#50;&#32;&#107;&#103;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"166\" style=\"vertical-align: -9px;\" \/><\/div>\n<p id=\"import-auto-id1165298935625\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id1165296267432\">This is a fairly massive picture, and it is impressive that the nail flexes only<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-32c1950eaa815ee325b8bad396c04fb2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#46;&#56;&#48;&#32;&micro;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"51\" style=\"vertical-align: -1px;\" \/>\u2014an amount undetectable to the unaided eye.<\/p>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1165298483858\">\n<h1 data-type=\"title\">Changes in Volume: Bulk Modulus<\/h1>\n<p id=\"import-auto-id1165298993069\">An object will be compressed in all directions if inward forces are applied evenly on all its surfaces as in <a href=\"#import-auto-id1165296230954\" class=\"autogenerated-content\">(Figure)<\/a>. It is relatively easy to compress gases and extremely difficult to compress liquids and solids. For example, air in a wine bottle is compressed when it is corked. But if you try corking a brim-full bottle, you cannot compress the wine\u2014some must be removed if the cork is to be inserted. The reason for these different compressibilities is that atoms and molecules are separated by large empty spaces in gases but packed close together in liquids and solids. To compress a gas, you must force its atoms and molecules closer together. To compress liquids and solids, you must actually compress their atoms and molecules, and very strong electromagnetic forces in them oppose this compression.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165296230954\">\n<div class=\"bc-figcaption figcaption\">An inward force on all surfaces compresses this cube. Its change in volume is proportional to the force per unit area and its original volume, and is related to the compressibility of the substance.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165296230955\" data-alt=\"A cube with area of cross section A and volume V zero is compressed by an inward force F acting on all surfaces. The compression causes a change in volume delta V, which is proportional to the force per unit area and its original volume. This change in volume is related to the compressibility of the substance.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_06_03_09a.jpg\" data-media-type=\"image\/jpg\" alt=\"A cube with area of cross section A and volume V zero is compressed by an inward force F acting on all surfaces. The compression causes a change in volume delta V, which is proportional to the force per unit area and its original volume. This change in volume is related to the compressibility of the substance.\" width=\"200\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id1165296371126\">We can describe the compression or volume deformation of an object with an equation. First, we note that a force \u201capplied evenly\u201d is defined to have the same stress, or ratio of force to area <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d839367af0a3b5c4f9215f914d45d496_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#70;&#125;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"11\" style=\"vertical-align: -6px;\" \/> on all surfaces. The deformation produced is a change in volume <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-92b790b2cbb74074a065721477fbf108_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#86;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"29\" style=\"vertical-align: 0px;\" \/>, which is found to behave very similarly to the shear, tension, and compression previously discussed. (This is not surprising, since a compression of the entire object is equivalent to compressing each of its three dimensions.) The relationship of the change in volume to other physical quantities is given by<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-63\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-10d41f4d1b4931d18e3b24eaaf5b1b2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#86;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#66;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#70;&#125;&#123;&#65;&#125;&#123;&#86;&#125;&#95;&#123;&#48;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"104\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id1165298863613\">where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-770fd1447ccf2fc229801b486b0d8f8a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> is the bulk modulus (see <a href=\"#import-auto-id1165298671576\" class=\"autogenerated-content\">(Figure)<\/a>), <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-16f33352c457c46135b0862350da2bd4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#86;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: -3px;\" \/> is the original volume, and <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d839367af0a3b5c4f9215f914d45d496_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#70;&#125;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"11\" style=\"vertical-align: -6px;\" \/><\/em> is the force per unit area applied uniformly inward on all surfaces. Note that no bulk moduli are given for gases.<\/p>\n<p id=\"import-auto-id1165298862692\">What are some examples of bulk compression of solids and liquids? One practical example is the manufacture of industrial-grade diamonds by compressing carbon with an extremely large force per unit area. The carbon atoms rearrange their crystalline structure into the more tightly packed pattern of diamonds. In nature, a similar process occurs deep underground, where extremely large forces result from the weight of overlying material. Another natural source of large compressive forces is the pressure created by the weight of water, especially in deep parts of the oceans. Water exerts an inward force on all surfaces of a submerged object, and even on the water itself. At great depths, water is measurably compressed, as the following example illustrates.<\/p>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1165298751418\">\n<div data-type=\"title\" class=\"title\">Calculating Change in Volume with Deformation: How Much Is Water Compressed at Great Ocean Depths?<\/div>\n<p id=\"import-auto-id1165296310879\">Calculate the fractional decrease in volume (<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0c328028e7ea3ba94a1b4295bba0a6ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#86;&#125;&#123;&#123;&#86;&#125;&#95;&#123;&#48;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"23\" style=\"vertical-align: -8px;\" \/>) for seawater at 5.00 km depth, where the force per unit area is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ba4960e7932e2dd8fa6f55badc91cc5d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#55;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#78;&#47;&#123;&#109;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"108\" style=\"vertical-align: -5px;\" \/><br \/>\n    .<\/p>\n<p id=\"import-auto-id1165298620485\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id1165296267436\">Equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5b355fc501181b72bb5d8775c4719269_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#86;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#66;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#70;&#125;&#123;&#65;&#125;&#123;&#86;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"99\" style=\"vertical-align: -6px;\" \/> is the correct physical relationship. All quantities in the equation except <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0c328028e7ea3ba94a1b4295bba0a6ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#86;&#125;&#123;&#123;&#86;&#125;&#95;&#123;&#48;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"23\" style=\"vertical-align: -8px;\" \/><br \/>\n are known.<\/p>\n<p id=\"import-auto-id1165298726312\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id1165296267427\">Solving for the unknown <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0c328028e7ea3ba94a1b4295bba0a6ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#86;&#125;&#123;&#123;&#86;&#125;&#95;&#123;&#48;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"23\" style=\"vertical-align: -8px;\" \/><\/p>\n<p> gives<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-646\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1a21e78cfc731ecfd2ab1da2a9d0dbdc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#86;&#125;&#123;&#123;&#86;&#125;&#95;&#123;&#48;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#66;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#70;&#125;&#123;&#65;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"81\" style=\"vertical-align: -8px;\" \/><\/div>\n<p id=\"import-auto-id1165298672501\">Substituting known values with the value for the bulk modulus <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-770fd1447ccf2fc229801b486b0d8f8a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> from <a href=\"#import-auto-id1165298671576\" class=\"autogenerated-content\">(Figure)<\/a>,<\/p>\n<div data-type=\"equation\" class=\"equation\">\n<pre class=\"ql-errors\">*** QuickLaTeX cannot compile formula:\n&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#108;&#108;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#86;&#125;&#123;&#123;&#86;&#125;&#95;&#123;&#48;&#125;&#125;&#38;&#32;&#61;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#46;&#48;&#48;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#55;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#47;&#109;&#125;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#50;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#57;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#47;&#109;&#125;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#48;&#46;&#48;&#50;&#51;&#61;&#50;&#46;&#51;&#37;&#46;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\n\n*** Error 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id=\"import-auto-id1165298579258\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id1165298656929\">Although measurable, this is not a significant decrease in volume considering that the force per unit area is about 500 atmospheres (1 million pounds per square foot). Liquids and solids are extraordinarily difficult to compress.<\/p>\n<\/div>\n<p id=\"import-auto-id1165299000208\">Conversely, very large forces are created by liquids and solids when they try to expand but are constrained from doing so\u2014which is equivalent to compressing them to less than their normal volume. This often occurs when a contained material warms up, since most materials expand when their temperature increases. If the materials are tightly constrained, they deform or break their container. Another very common example occurs when water freezes. Water, unlike most materials, expands when it freezes, and it can easily fracture a boulder, rupture a biological cell, or crack an engine block that gets in its way.<\/p>\n<p id=\"import-auto-id1165298745248\">Other types of deformations, such as torsion or twisting, behave analogously to the tension, shear, and bulk deformations considered here.<\/p>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id1165298595602\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul id=\"fs-id1165296391796\">\n<li id=\"import-auto-id1165298476009\">Hooke\u2019s law is given by\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e0a15bd85d72ac967922383153f0f560_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;&#61;&#107;&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#76;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"63\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"import-auto-id1165296333496\">where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1c907f3a1bf93e7f9930b1deb8554e51_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/> is the amount of deformation (the change in length), <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2510519bbe1660dfdffb4195c7287343_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> is the applied force, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3422b6bb5c160593658b7c39425d9880_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"9\" style=\"vertical-align: 0px;\" \/> is a proportionality constant that depends on the shape and composition of the object and the direction of the force. The relationship between the deformation and the applied force can also be written as<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d52365995dc6a1fc5337b33c9350c9a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#76;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#89;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#70;&#125;&#123;&#65;&#125;&#123;&#76;&#125;&#95;&#123;&#48;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"104\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id1165298686192\">where <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-73157e26905587d7195c94d544a30431_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#89;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/><\/em><em data-effect=\"italics\"> is <em data-effect=\"italics\">Young\u2019s modulus<\/em>, which depends on the substance, <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/><\/em> is the cross-sectional area, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4b534bdf9b43c38daf787ca868a98823_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#76;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"19\" style=\"vertical-align: -3px;\" \/> is the original length.<\/em><\/p>\n<\/li>\n<li id=\"import-auto-id1165298686194\">The ratio of force to area, <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d839367af0a3b5c4f9215f914d45d496_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#70;&#125;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"11\" style=\"vertical-align: -6px;\" \/><\/em>, is defined as <em data-effect=\"italics\">stress<\/em>, measured in N\/m<sup>2<\/sup>.<\/li>\n<li id=\"import-auto-id1165298726978\">The ratio of the change in length to length, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-35dc59d3e472b2c3e025593d161151d1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#76;&#125;&#123;&#123;&#76;&#125;&#95;&#123;&#48;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"21\" style=\"vertical-align: -8px;\" \/>, is defined as <em data-effect=\"italics\">strain<\/em> (a unitless quantity). In other words,\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-22f3d936f04bbf57b30d416efe1860e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#116;&#114;&#101;&#115;&#115;&#125;&#61;&#89;&times;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#116;&#114;&#97;&#105;&#110;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"129\" style=\"vertical-align: -1px;\" \/><\/div>\n<\/li>\n<li id=\"import-auto-id1165298586473\">The expression for shear deformation is\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9fbf4d302043ef0bc4aa318256d681c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#83;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#70;&#125;&#123;&#65;&#125;&#123;&#76;&#125;&#95;&#123;&#48;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"100\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id1165298779146\">where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-520cb534cd5b6bed768a61515b57cb7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#83;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/> is the shear modulus and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2510519bbe1660dfdffb4195c7287343_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> is the force applied perpendicular to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6ea9195001e8ac5d80555195a5c918f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#76;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"19\" style=\"vertical-align: -3px;\" \/> and parallel to the cross-sectional area <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/>.<\/p>\n<\/li>\n<li id=\"import-auto-id1165298548492\">The relationship of the change in volume to other physical quantities is given by\n<div data-type=\"equation\" class=\"equation\" id=\"eip-492\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-10d41f4d1b4931d18e3b24eaaf5b1b2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#86;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#66;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#70;&#125;&#123;&#65;&#125;&#123;&#86;&#125;&#95;&#123;&#48;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"104\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id1165298931793\">where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-770fd1447ccf2fc229801b486b0d8f8a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> is the bulk modulus, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1a261e0ac702f688e003de4e314a5d03_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#86;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: -3px;\" \/> is the original volume, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d839367af0a3b5c4f9215f914d45d496_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#70;&#125;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"11\" style=\"vertical-align: -6px;\" \/> is the force per unit area applied uniformly inward on all surfaces.<\/p>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" id=\"fs-id1165298807240\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298797418\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298797419\">\n<p id=\"import-auto-id1165298783801\">The elastic properties of the arteries are essential for blood flow. Explain the importance of this in terms of the characteristics of the flow of blood (pulsating or continuous).<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298867810\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298867811\">\n<p id=\"import-auto-id1165298555604\">What are you feeling when you feel your pulse? Measure your pulse rate for 10 s and for 1 min. Is there a factor of 6 difference?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165296252981\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298517998\">\n<p id=\"import-auto-id1165298727446\">Examine different types of shoes, including sports shoes and thongs. In terms of physics, why are the bottom surfaces designed as they are? What differences will dry and wet conditions make for these surfaces?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298840083\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298840084\">\n<p id=\"import-auto-id1165298770860\">Would you expect your height to be different depending upon the time of day? Why or why not?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298650751\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298619439\">\n<p id=\"import-auto-id1165298776145\">Why can a squirrel jump from a tree branch to the ground and run away undamaged, while a human could break a bone in such a fall?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298550516\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298550517\">\n<p id=\"import-auto-id1165298597904\">Explain why pregnant women often suffer from back strain late in their pregnancy.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298545438\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298545440\">\n<p id=\"import-auto-id1165298453194\">An old carpenter\u2019s trick to keep nails from bending when they are pounded into hard materials is to grip the center of the nail firmly with pliers. Why does this help?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298771893\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298771894\">\n<p id=\"import-auto-id1165296268391\">When a glass bottle full of vinegar warms up, both the vinegar and the glass expand, but vinegar expands significantly more with temperature than glass. The bottle will break if it was filled to its tightly capped lid. Explain why, and also explain how a pocket of air above the vinegar would prevent the break. (This is the function of the air above liquids in glass containers.)<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id1165298820012\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298795593\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298795594\">\n<p id=\"import-auto-id1165298766000\">During a circus act, one performer swings upside down hanging from a trapeze holding another, also upside-down, performer by the legs. If the upward force on the lower performer is three times her weight, how much do the bones (the femurs) in her upper legs stretch? You may assume each is equivalent to a uniform rod 35.0 cm long and 1.80 cm in radius. Her mass is 60.0 kg.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1165298841662\">\n<div data-type=\"equation\" class=\"equation\" id=\"eip-id1565399\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c9c6efdd274654a3f43eea31c73ffea9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"94\" style=\"vertical-align: -1px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298595705\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298595706\">\n<p id=\"import-auto-id1165298992604\">During a wrestling match, a 150 kg wrestler briefly stands on one hand during a maneuver designed to perplex his already moribund adversary. By how much does the upper arm bone shorten in length? The bone can be represented by a uniform rod 38.0 cm in length and 2.10 cm in radius.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298778825\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298778826\">\n<p id=\"import-auto-id1165296307050\">(a) The \u201clead\u201d in pencils is a graphite composition with a Young\u2019s modulus of about <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-381f93982773f90b7b42d93a60b99d7c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#57;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#125;&#47;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"82\" style=\"vertical-align: -5px;\" \/><br \/>\n    . Calculate the change in length of the lead in an automatic pencil if you tap it straight into the pencil with a force of 4.0 N. The lead is 0.50 mm in diameter and 60 mm long. (b) Is the answer reasonable? That is, does it seem to be consistent with what you have observed when using pencils? <\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1165296220516\">\n<p id=\"import-auto-id1165298622237\">(a)1 mm <\/p>\n<div data-type=\"newline\"><\/div>\n<p>(b) This does seem reasonable, since the lead does seem to shrink a little when you push on it.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298941254\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298941255\">\n<p id=\"import-auto-id1165298476018\">TV broadcast antennas are the tallest artificial structures on Earth. In 1987, a 72.0-kg physicist placed himself and 400 kg of equipment at the top of one 610-m high antenna to perform gravity experiments. By how much was the antenna compressed, if we consider it to be equivalent to a steel cylinder 0.150 m in radius?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298598606\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298598607\">\n<p id=\"import-auto-id1165296376441\">(a) By how much does a 65.0-kg mountain climber stretch her 0.800-cm diameter nylon rope when she hangs 35.0 m below a rock outcropping? (b) Does the answer seem to be consistent with what you have observed for nylon ropes? Would it make sense if the rope were actually a bungee cord? <\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1165298944074\">\n<p id=\"import-auto-id1165298547983\">(a)9 cm\n<\/p>\n<div data-type=\"newline\"><\/div>\n<p> (b)This seems reasonable for nylon climbing rope, since it is not supposed to stretch that much.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298937277\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298937278\">\n<p id=\"import-auto-id1165298946498\">A 20.0-m tall hollow aluminum flagpole is equivalent in stiffness to a solid cylinder 4.00 cm in diameter. A strong wind bends the pole much as a horizontal force of 900 N exerted at the top would. How far to the side does the top of the pole flex?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165296239036\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165296239037\">\n<p id=\"import-auto-id1165296317212\">As an oil well is drilled, each new section of drill pipe supports its own weight and that of the pipe and drill bit beneath it. Calculate the stretch in a new 6.00 m length of steel pipe that supports 3.00 km of pipe having a mass of 20.0 kg\/m and a 100-kg drill bit. The pipe is equivalent in stiffness to a solid cylinder 5.00 cm in diameter. <\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1165298608719\">\n<p id=\"import-auto-id1165298778980\">8.59 mm<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298786214\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298786215\">\n<p id=\"import-auto-id1165298744049\">Calculate the force a piano tuner applies to stretch a steel piano wire 8.00 mm, if the wire is originally 0.850 mm in diameter and 1.35 m long.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298982360\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298982361\">\n<p id=\"import-auto-id1165296228592\">A vertebra is subjected to a shearing force of 500 N. Find the shear deformation, taking the vertebra to be a cylinder 3.00 cm high and 4.00 cm in diameter. <\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1165298858039\">\n<div data-type=\"equation\" class=\"equation\" id=\"eip-id1373472\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-72df83636ca0b928af27ed660937a8f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#57;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#55;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"86\" style=\"vertical-align: -1px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298797863\">\n<p id=\"import-auto-id1165298547950\">A disk between vertebrae in the spine is subjected to a shearing force of 600 N. Find its shear deformation, taking it to have the shear modulus of  <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-381f93982773f90b7b42d93a60b99d7c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#57;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#125;&#47;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"82\" style=\"vertical-align: -5px;\" \/><br \/>\n    . The disk is equivalent to a solid cylinder 0.700 cm high and 4.00 cm in diameter.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165296255735\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165296255736\">\n<p id=\"import-auto-id1165298708711\">When using a pencil eraser, you exert a vertical force of 6.00 N at a distance of 2.00 cm from the hardwood-eraser joint. The pencil is 6.00 mm in diameter and is held at an angle of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c918e80bba36ea1d2e565858d4a55186_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&#46;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"32\" style=\"vertical-align: 0px;\" \/> to the horizontal. (a) By how much does the wood flex perpendicular to its length? (b) How much is it compressed lengthwise? <\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1165298654339\">\n<p id=\"import-auto-id1165298597988\">(a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c0bc85f1b623c881331fa1fe177319b9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#57;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#55;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"87\" style=\"vertical-align: -1px;\" \/><\/p>\n<p id=\"import-auto-id1165299001845\">(b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8e97307e97f9dab3254443c59147d04e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#55;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#56;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"87\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298783085\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298783086\">\n<p id=\"import-auto-id1165296580296\">To consider the effect of wires hung on poles, we take data from <a href=\"\/contents\/c372561e-08e8-41a1-ba72-9b4c00394c2d@8#fs-id1348788\" class=\"autogenerated-content\">(Figure)<\/a>, in which tensions in wires supporting a traffic light were calculated. The left wire made an angle <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ba41f26fc3ce0d18bf23de978fc31b76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#48;&#46;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"32\" style=\"vertical-align: 0px;\" \/> below the horizontal with the top of its pole and carried a tension of 108 N. The 12.0 m tall hollow aluminum pole is equivalent in stiffness to a 4.50 cm diameter solid cylinder. (a) How far is it bent to the side? (b) By how much is it compressed?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298713046\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298713047\">\n<p id=\"import-auto-id1165298597595\">A farmer making grape juice fills a glass bottle to the brim and caps it tightly. The juice expands more than the glass when it warms up, in such a way that the volume increases by 0.2% (that is, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-aa3f45e574ce032edff6ce15defafd61_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#86;&#47;&#123;&#86;&#125;&#95;&#123;&#48;&#125;&#61;&#50;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"122\" style=\"vertical-align: -5px;\" \/>) relative to the space available. Calculate the magnitude of the normal force exerted by the juice per square centimeter if its bulk modulus is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5530848f34fdc8ac289222d0853ef21b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#56;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#57;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#47;&#109;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"95\" style=\"vertical-align: -4px;\" \/>, assuming the bottle does not break. In view of your answer, do you think the bottle will survive? <\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1165298481228\">\n<p id=\"import-auto-id1165298998551\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-28293bb65bbff83e5695e9f98faf00a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#54;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#47;&#109;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"83\" style=\"vertical-align: -4px;\" \/>. This is about 36 atm, greater than a typical jar can withstand.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298913397\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298913398\">\n<p id=\"import-auto-id1165296218446\">(a) When water freezes, its volume increases by 9.05% (that is, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e74577cb2be9e7c1212bc865a507a011_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#86;&#47;&#123;&#86;&#125;&#95;&#123;&#48;&#125;&#61;&#57;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#53;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"145\" style=\"vertical-align: -5px;\" \/><br \/>\n    ). What force per unit area is water capable of exerting on a container when it freezes? (It is acceptable to use the bulk modulus of water in this problem.) (b) Is it surprising that such forces can fracture engine blocks, boulders, and the like?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165298740744\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298520715\">\n<p id=\"import-auto-id1165298995984\">This problem returns to the tightrope walker studied in <a href=\"\/contents\/5a51b5c5-1756-4d6c-b9be-84a9627ef003@8#fs-id986136\" class=\"autogenerated-content\">(Figure)<\/a>, who created a tension of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-905460ed4916a81e5d3a6c27bae29f6c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#52;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#78;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"77\" style=\"vertical-align: -1px;\" \/><br \/>\n     in a wire making an angle <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1b1426aef1f37c25c80da56c74add29b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#46;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"23\" style=\"vertical-align: 0px;\" \/> below the horizontal with each supporting pole. Calculate how much this tension stretches the steel wire if it was originally 15 m long and 0.50 cm in diameter. <\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1165298866753\">\n<p id=\"import-auto-id1165298919311\">1.4 cm<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1165296233210\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1165298620362\">\n<p id=\"import-auto-id1165296232572\">The pole in <a href=\"#import-auto-id1165296309614\" class=\"autogenerated-content\">(Figure)<\/a> is at a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3f5cec5cb5d1b54154d5483e559d13b7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#48;&#46;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"32\" style=\"vertical-align: 0px;\" \/> bend in a power line and is therefore subjected to more shear force than poles in straight parts of the line. The tension in each line is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-168f8f6fa62ad464e242f6a0691ca4ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#52;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#78;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"77\" style=\"vertical-align: -1px;\" \/>, at the angles shown. The pole is 15.0 m tall, has an 18.0 cm diameter, and can be considered to have half the stiffness of hardwood. (a) Calculate the compression of the pole. (b) Find how much it bends and in what direction. (c) Find the tension in a guy wire used to keep the pole straight if it is attached to the top of the pole at an angle of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ba41f26fc3ce0d18bf23de978fc31b76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#48;&#46;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"32\" style=\"vertical-align: 0px;\" \/> with the vertical. (Clearly, the guy wire must be in the opposite direction of the bend.)<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1165296309614\">\n<div class=\"bc-figcaption figcaption\">This telephone pole is at a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d237135e055cb753fca9232c2768ec76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/> bend in a power line. A guy wire is attached to the top of the pole at an angle of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-66108dc539d6fb6e5d5756edc8287139_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/> with the vertical.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1165298916431\" data-alt=\"A telephone pole is located at a ninety degree bend in a power line. Each part of the line is at an angle of eighty degrees with the pole and has a tension labeled T. A guy wire is attached to the top of the pole at an angle of thirty degrees with the vertical.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_06_03_10a.jpg\" data-media-type=\"image\/jpg\" alt=\"A telephone pole is located at a ninety degree bend in a power line. Each part of the line is at an angle of eighty degrees with the pole and has a tension labeled T. A guy wire is attached to the top of the pole at an angle of thirty degrees with the vertical.\" width=\"185\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"footnote-refs\">\n<h2 data-type=\"footnote-title\">Footnotes<\/h2>\n<ol>\n<li><a data-type=\"footnote-ref\" href=\"#footnote-ref1\">1<\/a> Approximate and average values. Young\u2019s moduli <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-82606c3098bb09002088b0f6f9ffbb2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#89;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/>      for tension and compression sometimes differ but are averaged here. Bone has significantly different Young\u2019s moduli for tension and compression.<\/li>\n<\/ol>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\n<dl class=\"definition\" id=\"import-auto-id1165298828407\">\n<dt>deformation<\/dt>\n<dd id=\"fs-id1165298696821\">change in shape due to the application of force<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"fs-id1987830\">\n<dt>Hooke\u2019s law<\/dt>\n<dd id=\"fs-id1987834\">proportional relationship between the force <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2510519bbe1660dfdffb4195c7287343_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> on a material and the deformation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1c907f3a1bf93e7f9930b1deb8554e51_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/> it causes, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c769cd38b6b11448d8239e49b615a05b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;&#61;&#107;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"74\" style=\"vertical-align: 0px;\" \/><\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1165298828410\">\n<dt>tensile strength<\/dt>\n<dd id=\"fs-id1165298787406\">the breaking stress that will cause permanent deformation or fraction of a material<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1165298696762\">\n<dt>stress<\/dt>\n<dd id=\"fs-id1165298600243\">ratio of force to area<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1165296377254\">\n<dt>strain<\/dt>\n<dd id=\"fs-id1165298840132\">ratio of change in length to original length<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"fs-id1268428\">\n<dt>shear deformation<\/dt>\n<dd id=\"fs-id1268431\">deformation perpendicular to the original length of an object<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":211,"menu_order":1,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":"all-rights-reserved"},"chapter-type":[],"contributor":[],"license":[56],"class_list":["post-300","chapter","type-chapter","status-publish","hentry","license-all-rights-reserved"],"part":272,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/300","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/users\/211"}],"version-history":[{"count":1,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/300\/revisions"}],"predecessor-version":[{"id":301,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/300\/revisions\/301"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/parts\/272"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/300\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/media?parent=300"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapter-type?post=300"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/contributor?post=300"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/license?post=300"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}