{"id":477,"date":"2017-09-18T18:04:59","date_gmt":"2017-09-18T22:04:59","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/douglasphys1108\/chapter\/9-1-rotation-angle-and-angular-velocity\/"},"modified":"2020-11-07T09:32:24","modified_gmt":"2020-11-07T14:32:24","slug":"9-1-rotation-angle-and-angular-velocity","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/douglasphys1108\/chapter\/9-1-rotation-angle-and-angular-velocity\/","title":{"raw":"9.1 Rotation Angle and Angular Velocity","rendered":"9.1 Rotation Angle and Angular Velocity"},"content":{"raw":"<div>\n<div class=\"bcc-box bcc-highlight\">\n<h3>Summary<\/h3>\n<div>\n<ul>\n \t<li>Define arc length, rotation angle, radius of curvature and angular velocity.<\/li>\n \t<li>Calculate the angular velocity of a car wheel spin.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"import-auto-id1571938\">In 1D Kinematics, we studied motion along a straight line and introduced such concepts as displacement, velocity, and acceleration. 2D kinematics dealt with motion in two dimensions. Projectile motion is a special case of two-dimensional kinematics in which the object is projected into the air, while being subject to the gravitational force, and lands a distance away. In this chapter, we consider situations where the object does moves in a curve (a special case of this type of motion is <a href=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1108\/part\/chapter-6-uniform-circular-motion-and-gravitation\/\">uniform circular motion<\/a>). We begin the study of rotational motion with rotation kinematics, and defining two angular quantities needed to describe this new type of motion.<\/p>\n\n<section id=\"fs-id1488693\">\n<h1>Rotation Angle<\/h1>\n<p id=\"import-auto-id1917384\">When objects rotate about some axis\u2014for example, when the CD (compact disc) in <a class=\"autogenerated-content\" href=\"#import-auto-id3402904\">Figure 1<\/a> rotates about its center\u2014each point in the object follows a circular arc. Consider a line from the center of the CD to its edge. Each <span id=\"import-auto-id2654027\">pit<\/span> used to record sound along this line moves through the same angle in the same amount of time. The rotation angle is the amount of rotation and is analogous to linear distance. We define the <strong><span id=\"import-auto-id3255842\">rotation angle\u00a0<\/span><span id=\"import-auto-id3255842\">\u0394<em>\u03b8<\/em><\/span><\/strong> to be the ratio of the arc length to the radius of curvature:<\/p>\n\n<div id=\"eip-211\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{\\Delta\\theta=}[\/latex][latex size=\"2\"]\\boldsymbol{\\frac{\\Delta{s}}{r}}.[\/latex]<\/div>\n<figure id=\"import-auto-id3402904\"><figcaption><\/figcaption>\n\n[caption id=\"\" align=\"aligncenter\" width=\"300\"]<img class=\"\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1107\/wp-content\/uploads\/sites\/1184\/2020\/01\/Figure_07_01_01aa-1.jpg\" alt=\"The figure shows the back side of a compact disc. There is a scratched part on the upper right side of the C D, about one-fifth size of the whole area, with inner circular dots clearly visible. Two line segments are drawn enclosing the scratched area from the border of the C D to the middle plastic portion. A curved arrow is drawn between the two line segments near this middle portion and angle delta theta written alongside it.\" width=\"300\" height=\"300\"> <strong>Figure 1.<\/strong> All points on a CD travel in circular arcs. The pits along a line from the center to the edge all move through the same angle <strong>\u0394<em>\u03b8<\/em><\/strong> in a time <strong>\u0394<em>t<\/em><\/strong>.[\/caption]\n<p style=\"text-align: center\"><\/p>\n<\/figure>\n<figure id=\"import-auto-id3418263\"><figcaption><\/figcaption>\n\n[caption id=\"\" align=\"aligncenter\" width=\"300\"]<img class=\"\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1107\/wp-content\/uploads\/sites\/1184\/2020\/11\/Figure_07_01_01ab-1.jpg\" alt=\"A circle of radius r and center O is shown. A radius O-A of the circle is rotated through angle delta theta about the center O to terminate as radius O-B. The arc length A-B is marked as delta s.\" width=\"300\" height=\"250\"> <strong>Figure 2.<\/strong> The radius of a circle is rotated through an angle <strong>\u0394<em>\u03b8<\/em><\/strong>. The arc length <strong>\u0394<em>s<\/em><\/strong> is described on the circumference.[\/caption]<\/figure>\n<p id=\"import-auto-id3025908\">The <span id=\"import-auto-id2679165\"><strong>arc length \u0394<em>s <\/em><\/strong><\/span>is the distance traveled along a circular path as shown in <a class=\"autogenerated-content\" href=\"#import-auto-id3418263\">Figure 2<\/a> Note that <em><strong>r<\/strong><\/em> is the <span id=\"import-auto-id2920709\"><strong>radius of curvature<\/strong><\/span> of the circular path.<\/p>\n<p id=\"import-auto-id1471752\">We know that for one complete revolution, the arc length is the circumference of a circle of radius <em><strong>r<\/strong><\/em>. The circumference of a circle is <strong>2\u03c0<em>r<\/em><\/strong>. Thus for one complete revolution the rotation angle is<\/p>\n\n<div id=\"eip-191\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{\\Delta\\theta\\:=}[\/latex][latex size=\"2\"]\\boldsymbol{\\frac{2\\pi{r}}{r}}[\/latex][latex]\\boldsymbol{=\\:2\\pi}.[\/latex]<\/div>\n<p id=\"import-auto-id1842986\">This result is the basis for defining the units used to measure rotation angles, <strong><span id=\"import-auto-id3255842\">\u0394<em>\u03b8<\/em><\/span><\/strong> to be <strong><span id=\"import-auto-id2625941\">radians<\/span> <\/strong>(rad), defined so that<\/p>\n\n<div id=\"eip-135\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{2\\pi\\textbf{ rad} = 1\\textbf{ revolution}.}[\/latex]<\/div>\n<p id=\"eip-425\">A comparison of some useful angles expressed in both degrees and radians is shown in <a class=\"autogenerated-content\" href=\"#import-auto-id2588905\">Table 1<\/a>.<\/p>\n\n<table id=\"import-auto-id2588905\" summary=\"The table compares various angle measures in degrees (first column) and radians (second colum).\">\n<thead>\n<tr>\n<th>[latex]\\textbf{Degree Measures}[\/latex]<\/th>\n<th>[latex]\\textbf{Radian Measure}[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: center\">[latex]\\boldsymbol{30^0}[\/latex]<\/td>\n<td style=\"text-align: center\">[latex size=\"1\"]\\boldsymbol{\\frac{\\pi}{6}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center\">[latex]\\boldsymbol{60^0}[\/latex]<\/td>\n<td style=\"text-align: center\">[latex size=\"1\"]\\boldsymbol{\\frac{\\pi}{3}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center\">[latex]\\boldsymbol{90^0}[\/latex]<\/td>\n<td style=\"text-align: center\">[latex size=\"1\"]\\boldsymbol{\\frac{\\pi}{2}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center\">[latex]\\boldsymbol{120^0}[\/latex]<\/td>\n<td style=\"text-align: center\">[latex size=\"1\"]\\boldsymbol{\\frac{2\\pi}{3}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center\">[latex]\\boldsymbol{135^0}[\/latex]<\/td>\n<td style=\"text-align: center\">[latex size=\"1\"]\\boldsymbol{\\frac{3\\pi}{4}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center\">[latex]\\boldsymbol{180^0}[\/latex]<\/td>\n<td style=\"text-align: center\">[latex]\\boldsymbol{\\pi}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<tbody>\n<tr>\n<td colspan=\"2\"><strong>Table 1.<\/strong> Comparison of Angular Units.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<figure id=\"import-auto-id2442865\"><figcaption><\/figcaption>\n\n[caption id=\"\" align=\"aligncenter\" width=\"300\"]<img class=\"\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1107\/wp-content\/uploads\/sites\/1184\/2020\/11\/Figure_07_01_02a-1.jpg\" alt=\"A circle is shown. Two radii of the circle, inclined at an acute angle delta theta, are shown. On one of the radii, two points, one and two are marked. The point one is inside the circle through which an arc between the two radii is shown. The point two is on the cirumfenrence of the circle. The two arc lengths are delta s one and delta s two respectively for the two points.\" width=\"300\" height=\"294\"> <strong>Figure 3.<\/strong> Points 1 and 2 rotate through the same angle <strong>(\u0394<em>\u03b8<\/em>)<\/strong>, but point 2 moves through a greater arc length <strong>(\u0394<em>s<\/em>)<\/strong> because it is at a greater distance from the center of rotation <strong>(<em>r<\/em>)<\/strong>.[\/caption]<\/figure>\n<p id=\"import-auto-id1930108\">If <strong><span id=\"import-auto-id3255842\">\u0394\u03b8<\/span> = 2\u03c0 rad<\/strong>, then the CD has made one complete revolution, and every point on the CD is back at its original position. Because there are <strong>360\u00b0<\/strong> in a circle or one revolution, the relationship between radians and degrees is thus<\/p>\n\n<div id=\"eip-808\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{2\\pi\\textbf{rad}=360^0}[\/latex]<\/div>\n<p id=\"import-auto-id2052087\">so that<\/p>\n\n<div class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{1\\textbf{ rad}\\:=}[\/latex][latex size=\"2\"]\\boldsymbol{\\frac{360^0}{2\\pi}}[\/latex][latex]\\boldsymbol{\\approx\\:57.3^0}.[\/latex]<\/div>\n<\/section><section id=\"fs-id3104613\">\n<h1>Angular Velocity<\/h1>\n<p id=\"import-auto-id2681279\">How fast is an object rotating? We define <strong><span id=\"import-auto-id2962847\">angular velocity\u00a0\u03c9<\/span><\/strong> as the rate of change of an angle. In symbols, this is<\/p>\n\n<div id=\"eip-759\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{\\omega\\:=}[\/latex][latex size=\"2\"]\\boldsymbol{\\frac{\\Delta\\theta}{\\Delta{t}}},[\/latex]<\/div>\n<p id=\"import-auto-id2604506\">where an angular rotation <strong><span id=\"import-auto-id3255842\">\u0394<em>\u03b8<\/em><\/span><\/strong> takes place in a time <strong><span id=\"import-auto-id3255842\">\u0394<\/span><em>t<\/em><\/strong>. The greater the rotation angle in a given amount of time, the greater the angular velocity. The units for angular velocity are radians per second (rad\/s). Angular velocity is often expressed in units of <strong>rev\/min<\/strong> (\"rpm\" or \"revolutions per minute\"). You can convert from <strong>rev\/min<\/strong> to <strong>rad\/s<\/strong> using the fact that that [latex]\\boldsymbol{2\\pi~\\textbf{rad} = 1~\\textbf{rev}}[\/latex] and <strong>1 min = 60 sec<\/strong>.<\/p>\n<p id=\"import-auto-id2621168\">Angular velocity <strong><span id=\"import-auto-id2962847\">\u03c9<\/span><\/strong> is analogous to linear velocity <em><strong>v<\/strong><\/em>. To get the precise relationship between angular and linear velocity, we again consider a pit on the rotating CD. This pit moves an arc length <strong><span id=\"import-auto-id3255842\">\u0394<\/span><em>s<\/em><\/strong> in a time <strong><span id=\"import-auto-id3255842\">\u0394<\/span><em>t<\/em><\/strong>, and so it has a linear velocity<\/p>\n\n<div id=\"eip-400\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{v\\:=}[\/latex][latex size=\"2\"]\\boldsymbol{\\frac{\\Delta{s}}{\\Delta{t}}}.[\/latex]<\/div>\n<p id=\"import-auto-id1840944\">From [latex]\\boldsymbol{\\Delta\\theta=\\frac{\\Delta{s}}{r}}[\/latex] we see that <strong><span id=\"import-auto-id3255842\">\u0394<\/span><em>s<\/em> = <em>r<\/em><span id=\"import-auto-id3255842\">\u0394<em>\u03b8<\/em><\/span><\/strong>. Substituting this into the expression for <em><strong>v<\/strong><\/em> gives<\/p>\n\n<div class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{v\\:=}[\/latex][latex size=\"2\"]\\boldsymbol{\\frac{r\\Delta\\theta}{\\Delta{t}}}[\/latex][latex]\\boldsymbol{=\\:r\\omega}.[\/latex]<\/div>\n<p id=\"import-auto-id1549179\">We write this relationship in two different ways and gain two different insights:<\/p>\n\n<div id=\"eip-639\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{v=r\\omega\\textbf{ or }\\omega\\:=}[\/latex][latex size=\"2\"]\\boldsymbol{\\frac{v}{r}}.[\/latex]<\/div>\n<p id=\"import-auto-id2680923\">The first relationship in <strong><em>v<\/em> = <em>r<\/em><span id=\"import-auto-id2962847\">\u03c9<\/span><\/strong>\u00a0 states that the linear velocity <em><strong>v<\/strong><\/em> is proportional to the distance from the center of rotation, thus, it is largest for a point on the rim (largest <em><strong>r<\/strong><\/em>), as you might expect. We can also call this linear speed <em><strong>v<\/strong><\/em> of a point on the rim the <em>tangential speed<\/em>. The second relationship in\u00a0<strong><em>v<\/em> = <em>r<\/em><span id=\"import-auto-id2962847\">\u03c9<\/span><\/strong>\u00a0 can be illustrated by considering the tire of a moving car. Note that the speed of a point on the rim of the tire is the same as the speed <em><strong>v<\/strong> <\/em>of the car. See <a class=\"autogenerated-content\" href=\"#import-auto-id2931190\">Figure 4<\/a>. So the faster the car moves, the faster the tire spins\u2014large <em><strong>v<\/strong><\/em> means a large <strong><span id=\"import-auto-id2962847\">\u03c9<\/span><\/strong>, because <strong><em>v <\/em>= <em>r<\/em><span id=\"import-auto-id2962847\">\u03c9<\/span><\/strong>. Similarly, a larger-radius tire rotating at the same angular velocity (<strong><span id=\"import-auto-id2962847\">\u03c9<\/span><\/strong>) will produce a greater linear speed (<em><strong>v<\/strong><\/em>) for the car.<\/p>\n\n<figure id=\"import-auto-id2931190\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"300\"]<img src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1107\/wp-content\/uploads\/sites\/1184\/2020\/11\/Figure_07_01_03a-1.jpg\" alt=\"The given figure shows the front wheel of a car. The radius of the car wheel, r, is shown as an arrow and the linear velocity, v, is shown with a green horizontal arrow pointing rightward. The angular velocity, omega, is shown with a clockwise-curved arrow over the wheel.\" width=\"300\" height=\"621\"> <strong>Figure 4.<\/strong> A car moving at a velocity <strong><em>v<\/em><\/strong> to the right has a tire rotating with an angular velocity <strong>\u03c9<\/strong>. The speed of the tread of the tire relative to the axle is <em><strong>v<\/strong><\/em>, the same as if the car were jacked up. Thus the car moves forward at linear velocity <strong><em>v<\/em>=<em>r<\/em>\u03c9<\/strong>, where <em><strong>r<\/strong><\/em> is the tire radius. A larger angular velocity for the tire means a greater velocity for the car.[\/caption]<\/figure>\n<div class=\"textbox shaded\">\n<div id=\"fs-id2589253\" class=\"example\">\n<h3 id=\"import-auto-id3402720\">Example 1: How Fast Does a Car Tire Spin?<\/h3>\nCalculate the angular velocity of a 0.300 m radius car tire when the car travels at <strong>15.0 m\/s<\/strong> (about <strong>54 km\/h<\/strong>). See <a class=\"autogenerated-content\" href=\"#import-auto-id2931190\">Figure 4<\/a>.\n<p id=\"import-auto-id2968606\"><strong>Strategy<\/strong><\/p>\nBecause the linear speed of the tire rim is the same as the speed of the car, we have <strong><em>v<\/em> = 15.0 m\/s<\/strong>. The radius of the tire is given to be <strong><em>r<\/em> = 0.300 m<\/strong>. Knowing <em><strong>v<\/strong><\/em> and <em><strong>r<\/strong><\/em>, we can use the second relationship in <strong><em>v<\/em> = <em>r<\/em>\u03c9<\/strong>, [latex]\\boldsymbol{\\omega=\\frac{v}{r}}[\/latex] to calculate the angular velocity.\n<p id=\"import-auto-id2949936\"><strong>Solution<\/strong><\/p>\n<p id=\"eip-105\">To calculate the angular velocity, we will use the following relationship:<\/p>\n\n<div id=\"eip-97\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{\\omega\\:=}[\/latex][latex size=\"2\"]\\boldsymbol{\\frac{v}{r}}.[\/latex]<\/div>\n<p id=\"import-auto-id3199878\">Substituting the knowns,<\/p>\n\n<div id=\"eip-451\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{\\omega\\:=}[\/latex][latex size=\"2\"]\\boldsymbol{\\frac{15.0\\textbf{ m\/s}}{0.300\\textbf{ m}}}[\/latex][latex]\\boldsymbol{=\\:50.0\\textbf{ rad\/s}}.[\/latex]<\/div>\n<p id=\"import-auto-id1889900\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id956895\">When we cancel units in the above calculation, we get 50.0\/s. But the angular velocity must have units of rad\/s. Because radians are actually unitless (radians are defined as a ratio of distance), we can simply insert them into the answer for the angular velocity. Also note that if an earth mover with much larger tires, say 1.20 m in radius, were moving at the same speed of 15.0 m\/s, its tires would rotate more slowly. They would have an angular velocity<\/p>\n\n<div id=\"eip-971\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{\\omega=(15.0\\textbf{ m\/s})\/(1.20\\textbf{ m})=12.5\\textbf{ rad\/s}}.[\/latex]<\/div>\n<\/div>\n<\/div>\n<p id=\"import-auto-id2415283\">Both[latex]~\\boldsymbol{\\omega}~[\/latex]and[latex]~\\boldsymbol{v}~[\/latex]have directions (hence they are angular and linear <em>velocities<\/em>, respectively). Angular velocity has only two directions with respect to the axis of rotation\u2014it is either clockwise or counterclockwise. Linear velocity is tangent to the path, as illustrated in <a class=\"autogenerated-content\" href=\"#import-auto-id1452850\">Figure 5<\/a>.<\/p>\n\n<h2>Connection to Centripetal Acceleration<\/h2>\nWe call the acceleration of an object moving in uniform circular motion (resulting from a net external force) the <strong><span id=\"import-auto-id3108952\">centripetal acceleration <\/span><\/strong>(<strong><em>a<\/em><sub>c<\/sub><\/strong>); centripetal means \u201ctoward the center\u201d or \u201ccenter seeking\". In <a href=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1108\/chapter\/5-2-centripetal-acceleration\/\">Section 5.1<\/a>, we saw that the magnitude of the centripetal acceleration is\n<div id=\"eip-684\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{a_{\\textbf{c}}\\:=}[\/latex][latex size=\"2\"]\\boldsymbol{\\frac{v^2}{r}},[\/latex]<\/div>\nwhich is the acceleration of an object in a circle of radius <em><strong>r<\/strong><\/em> at a speed <em><strong>v<\/strong><\/em>. It is also useful to express <strong><em>a<\/em><sub>c<\/sub><\/strong> in terms of angular velocity. Substituting <strong><em>v<\/em> = <em>r<\/em>\u03c9<\/strong> into the above expression, we find <strong><em>a<\/em><sub>c <\/sub>= (<em>r<\/em>\u03c9)<sup>2<\/sup>\/<em>r<\/em> = <em>r<\/em>\u03c9<sup>2<\/sup><\/strong>. We can express the magnitude of centripetal acceleration using either of two equations:\n<div id=\"eip-740\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{a_{\\textbf{c}}\\:=}[\/latex][latex size=\"2\"]\\boldsymbol{\\frac{v^2}{r}}[\/latex][latex]\\boldsymbol{;\\:a_{\\textbf{c}}=r\\omega^2}.[\/latex]<\/div>\n<div>\n<div class=\"textbox shaded\">\n<div id=\"fs-id2598952\" class=\"example\">\n<h3>Example 2: How Big Is The Centripetal Acceleration in an Ultracentrifuge?<\/h3>\nCalculate the centripetal acceleration of a point 7.50 cm from the axis of an <strong><span id=\"import-auto-id2449410\">ultracentrifuge<\/span> <\/strong>spinning at <strong>7.5 \u00d7 10<sup>4<\/sup> rev\/min<\/strong>. Determine the ratio of this acceleration to that due to gravity. Note that this example is similar to <a href=\"\/douglasphys1108\/chapter\/5-2-centripetal-acceleration\/#ca_ex2\">Example 2<\/a> in the Section 5.2 on centripetal acceleration.\n\n&nbsp;\n<p id=\"import-auto-id3105981\"><strong>Strategy<\/strong><\/p>\nThe term rev\/min stands for revolutions per minute. By converting this to radians per second, we obtain the angular velocity <strong>\u03c9<\/strong>. Because <em><strong>r<\/strong><\/em> is given, we can use the second expression in the equation [latex]\\boldsymbol{a_{\\textbf{c}}=\\frac{v^2}{r};\\:a_{\\textbf{c}}=r\\omega^2}[\/latex] to calculate the centripetal acceleration.\n\n&nbsp;\n<p id=\"import-auto-id1140598\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id2849150\">To convert <strong>7.5 \u00d7 10<sup>4<\/sup> rev\/min<\/strong> to radians per second, we use the facts that one revolution is <strong>2\u03c0 rad<\/strong> and one minute is 60.0 s. Thus,<\/p>\n\n<div class=\"equation\">[latex]\\boldsymbol{\\omega\\:=7.50\\times10^4}[\/latex][latex size=\"2\"]\\boldsymbol{\\frac{\\textbf{rev}}{\\textbf{min}}\\times\\frac{2\\pi\\textbf{ rad}}{1\\textbf{ rev}}\\times\\frac{1\\textbf{ min}}{60.0\\textbf{ s}}}[\/latex][latex]\\boldsymbol{=\\:7854\\textbf{ rad\/s}}.[\/latex]<\/div>\n<p id=\"import-auto-id3191658\">Now the centripetal acceleration is given by the second expression in [latex]\\boldsymbol{a_{\\textbf{c}}=\\frac{v^2}{r};\\:a_{\\textbf{c}}=r\\omega^2}[\/latex] as<\/p>\n\n<div class=\"equation\">[latex]\\boldsymbol{a_{\\textbf{c}}=r\\omega^2}.[\/latex]<\/div>\n<p id=\"import-auto-id1926483\">Converting 7.50 cm to meters and substituting known values gives<\/p>\n\n<div class=\"equation\">[latex]\\boldsymbol{a_{\\textbf{c}}=(0.0750\\textbf{ m})(7854\\textbf{ rad\/s})^2=4.63\\times10^6\\textbf{ m\/s}^2}.[\/latex]<\/div>\n<p id=\"import-auto-id2673973\">Note that the unitless radians are discarded in order to get the correct units for centripetal acceleration. Taking the ratio of <strong><em>a<\/em><sub>c<\/sub><\/strong> to <em><strong>g<\/strong><\/em> yields<\/p>\n\n<div class=\"equation\">[latex size=\"2\"]\\boldsymbol{\\frac{a_{\\textbf{c}}}{g}}[\/latex][latex]\\boldsymbol{=}[\/latex][latex size=\"2\"]\\boldsymbol{\\frac{4.63\\times10^6}{9.80}}[\/latex][latex]\\boldsymbol{=4.72\\times10^5}.[\/latex]<\/div>\n<div><\/div>\n<p id=\"import-auto-id871156\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id2604418\">This last result means that the centripetal acceleration is 472,000 times as strong as <em><strong>g<\/strong><\/em>. It is no wonder that such high <strong>\u03c9<\/strong> centrifuges are called ultracentrifuges. The extremely large accelerations involved greatly decrease the time needed to cause the sedimentation of blood cells or other materials.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2584087\" class=\"note\">\n<div class=\"textbox shaded\">\n<div class=\"note\">\n<h3 class=\"title\">TAKE-HOME EXPERIMENT<span style=\"text-decoration: underline\">\n<\/span><\/h3>\n<p id=\"import-auto-id1986367\">Tie an object to the end of a string and swing it around in a horizontal circle above your head (swing at your wrist). Maintain uniform speed as the object swings and measure the angular velocity of the motion. What is the approximate speed of the object? Identify a point close to your hand and take appropriate measurements to calculate the linear speed at this point. Identify other circular motions and measure their angular velocities.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<figure id=\"import-auto-id1452850\"><figcaption><\/figcaption>\n\n[caption id=\"\" align=\"aligncenter\" width=\"250\"]<img src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1107\/wp-content\/uploads\/sites\/1184\/2020\/11\/Figure_07_01_04a-1.jpg\" alt=\"The given figure shows the top view of an old fashioned vinyl record. Two perpendicular line segments are drawn through the center of the circular record, one vertically upward and one horizontal to the right side. Two flies are shown at the end points of the vertical lines near the borders of the record. Two arrows are also drawn perpendicularly rightward through the end points of these vertical lines depicting linear velocities. A curved arrow is also drawn at the center circular part of the record which shows the angular velocity.\" width=\"250\" height=\"755\"> <strong>Figure 5.<\/strong> As an object moves in a circle, here a fly on the edge of an old-fashioned vinyl record, its instantaneous velocity is always tangent to the circle. The direction of the angular velocity is clockwise in this case.[\/caption]<\/figure>\n<\/section>\n<div id=\"eip-270\" class=\"note\">\n<div class=\"textbox shaded\">\n<div class=\"note\">\n<h3 class=\"title\">PHET EXPLORATIONS: LADYBUG REVOLUTION<\/h3>\n<figure id=\"eip-id1171550\"><figcaption><\/figcaption>\n\n[caption id=\"\" align=\"aligncenter\" width=\"450\"]<a href=\"\/resources\/7c52f36f755df0a4d1bf4e75814c8135735d7055\/rotation_en.jar\"><img src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1107\/wp-content\/uploads\/sites\/1184\/2020\/11\/PhET_Icon-5-1.png\" alt=\"image\" width=\"450\" height=\"147\"><\/a> <strong>Figure 6.<\/strong><a href=\"https:\/\/phet.colorado.edu\/en\/simulation\/rotation\"> Ladybug Revolution<\/a>[\/caption]<\/figure>\n<p id=\"eip-id1169738118030\">Join the ladybug in an exploration of rotational motion. Rotate the merry-go-round to change its angle, or choose a constant angular velocity or angular acceleration. Explore how circular motion relates to the bug's x,y position, velocity, and acceleration using vectors or graphs.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<section id=\"fs-id3399151\" class=\"section-summary\">\n<h1>Section Summary<\/h1>\n<ul id=\"fs-id1104163\">\n \t<li id=\"import-auto-id2603294\">Uniform circular motion is motion in a circle at constant speed. The rotation angle<strong> \u0394<em>\u03b8<\/em><\/strong> is defined as the ratio of the arc length to the radius of curvature:\n<div class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{\\Delta\\theta\\:=}[\/latex][latex size=\"2\"]\\boldsymbol{\\frac{\\Delta{s}}{r}},[\/latex]<\/div>\n<p id=\"import-auto-id3062885\">where arc length <strong>\u0394<em>s <\/em><\/strong>is distance traveled along a circular path and <em><strong>r<\/strong><\/em> is the radius of curvature of the circular path. The quantity <strong>\u0394<em>\u03b8<\/em><\/strong> is measured in units of radians (rad), for which<\/p>\n\n<div id=\"eip-567\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{2\\pi\\textbf{ rad}=360^0=1\\textbf{ revolution}}.[\/latex]<\/div><\/li>\n \t<li id=\"import-auto-id2384053\">The conversion between radians and degrees is <strong>1 rad = 57.3\u00b0<\/strong>.<\/li>\n \t<li id=\"import-auto-id2442213\">Angular velocity <strong>\u03c9<\/strong> is the rate of change of an angle,\n<div id=\"eip-969\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{\\omega\\:=}[\/latex][latex size=\"2\"]\\boldsymbol{\\frac{\\Delta\\theta}{\\Delta{t}}},[\/latex]<\/div>\n<p id=\"import-auto-id1439148\">where a rotation <strong>\u0394<em>\u03b8<\/em><\/strong> takes place in a time <strong>\u0394<em>t<\/em><\/strong>. The units of angular velocity are radians per second (rad\/s). Linear velocity <em><strong>v<\/strong><\/em> and angular velocity <strong>\u03c9<\/strong> are related by<\/p>\n\n<div id=\"eip-513\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{v=r\\omega\\textbf{ or }\\omega\\:=}[\/latex][latex size=\"2\"]\\boldsymbol{\\frac{v}{r}}.[\/latex]<\/div><\/li>\n<\/ul>\n<\/section><section id=\"fs-id3103314\" class=\"conceptual-questions\">\n<div id=\"fs-id3119404\" class=\"exercise\">\n<div id=\"fs-id2969010\" class=\"problem\">\n<div class=\"bcc-box bcc-info\">\n<h3>Conceptual Questions<\/h3>\n<strong>1: <\/strong>There is an analogy between rotational and linear physical quantities. What rotational quantities are analogous to distance and velocity?\n\n<\/div>\n<\/div>\n<\/div>\n<\/section><section id=\"fs-id2381548\" class=\"problems-exercises\">\n<div class=\"bcc-box bcc-info\">\n<h3>Problems &amp; Exercises<\/h3>\n<div id=\"fs-id3004274\" class=\"exercise\">\n<div id=\"fs-id3253787\" class=\"problem\">\n<p id=\"import-auto-id3415350\"><strong>1: <\/strong>Semi-trailer trucks have an odometer on one hub of a trailer wheel. The hub is weighted so that it does not rotate, but it contains gears to count the number of wheel revolutions\u2014it then calculates the distance traveled. If the wheel has a 1.15 m diameter and goes through 200,000 rotations, how many kilometers should the odometer read?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1004074\" class=\"exercise\">\n<div id=\"fs-id2448666\" class=\"problem\">\n<p id=\"fs-id2052623\"><strong>2: <\/strong>Microwave ovens rotate at a rate of about 6 rev\/min. What is this in revolutions per second? What is the angular velocity in radians per second?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1921627\" class=\"exercise\">\n<div id=\"fs-id2953385\" class=\"problem\">\n<p id=\"fs-id1574958\"><strong>3: <\/strong>An automobile with 0.260 m radius tires travels 80,000 km before wearing them out. How many revolutions do the tires make, neglecting any backing up and any change in radius due to wear?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1524972\" class=\"exercise\">\n<div id=\"fs-id3254586\" class=\"problem\">\n<p id=\"fs-id3035846\"><strong>4: <\/strong>(a) What is the period of rotation of Earth in seconds? (b) What is the angular velocity of Earth? (c) Given that Earth has a radius of 6.4 \u00d7 10<sup>6<\/sup> m at its equator, what is the linear velocity at Earth\u2019s surface?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2979194\" class=\"exercise\">\n<div id=\"fs-id1361145\" class=\"problem\">\n<p id=\"fs-id3093713\"><strong>5: <\/strong>A baseball pitcher brings his arm forward during a pitch, rotating the forearm about the elbow. If the velocity of the ball in the pitcher\u2019s hand is 35.0 m\/s and the ball is 0.300 m from the elbow joint, what is the angular velocity of the forearm?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id954942\" class=\"exercise\">\n<div id=\"fs-id3181723\" class=\"problem\">\n<p id=\"fs-id2595491\"><strong>6: <\/strong>In lacrosse, a ball is thrown from a net on the end of a stick by rotating the stick and forearm about the elbow. If the angular velocity of the ball about the elbow joint is 30.0 rad\/s and the ball is 1.30 m from the elbow joint, what is the velocity of the ball?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2678694\" class=\"exercise\">\n<div id=\"fs-id1576562\" class=\"problem\">\n<p id=\"fs-id2979343\"><strong>7: <\/strong>A truck with 0.420-m-radius tires travels at 32.0 m\/s. What is the angular velocity of the rotating tires in radians per second? What is this in rev\/min?<\/p>\n<strong>8: <\/strong>A rotating space station is said to create \u201cartificial gravity\u201d\u2014a loosely-defined term used for an acceleration that would be crudely similar to gravity. The outer wall of the rotating space station would become a floor for the astronauts, and centripetal acceleration supplied by the floor would allow astronauts to exercise and maintain muscle and bone strength more naturally than in non-rotating space environments. If the space station is 200 m in diameter, what angular velocity would produce an \u201cartificial gravity\u201d of 9.80 m\/s<sup>2<\/sup> at the rim?\n\n<\/div>\n<\/div>\n<div id=\"fs-id2017206\" class=\"exercise\">\n<div id=\"fs-id2400941\" class=\"problem\">\n<p id=\"fs-id2953765\"><strong>9: <\/strong>An ordinary workshop grindstone has a radius of 7.50 cm and rotates at 6500 rev\/min.<\/p>\n<p id=\"fs-id1990575\">(a) Calculate the magnitude of the centripetal acceleration at its edge in meters per second squared and convert it to multiples of[latex]\\boldsymbol{g}.[\/latex]<\/p>\n<p id=\"fs-id3122407\">(b) What is the linear speed of a point on its edge?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1474903\" class=\"exercise\">\n<div id=\"fs-id1272517\" class=\"problem\">\n<p id=\"fs-id3424715\"><strong>10: <\/strong>Helicopter blades withstand tremendous stresses. In addition to supporting the weight of a helicopter, they are spun at rapid rates and experience large centripetal accelerations, especially at the tip.<\/p>\n<p id=\"fs-id3054343\">(a) Calculate the magnitude of the centripetal acceleration at the tip of a 4.00 m long helicopter blade that rotates at 300 rev\/min.<\/p>\n<p id=\"fs-id3033654\">(b) Compare the linear speed of the tip with the speed of sound (taken to be 340 m\/s).<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id3224044\" class=\"exercise\">\n<div id=\"fs-id1514610\" class=\"problem\">\n<p id=\"fs-id1004097\"><strong>11: <\/strong>Olympic ice skaters are able to spin at about 5 rev\/s.<\/p>\n<p id=\"fs-id3358931\">(a) What is their angular velocity in radians per second?<\/p>\n<p id=\"fs-id1355573\">(b) What is the centripetal acceleration of the skater\u2019s nose if it is 0.120 m from the axis of rotation?<\/p>\n<p id=\"fs-id1947068\">(c) An exceptional skater named Dick Button was able to spin much faster in the 1950s than anyone since\u2014at about 9 rev\/s. What was the centripetal acceleration of the tip of his nose, assuming it is at 0.120 m radius?<\/p>\n(d) Comment on the magnitudes of the accelerations found. It is reputed that Button ruptured small blood vessels during his spins.\n\n<strong>12: <\/strong>Verify that the linear speed of an ultracentrifuge is about 0.50 km\/s, and Earth in its orbit is about 30 km\/s by calculating:\n<p id=\"fs-id3138086\">(a) The linear speed of a point on an ultracentrifuge 0.100 m from its center, rotating at 50,000 rev\/min.<\/p>\n(b) The linear speed of Earth in its orbit about the Sun (use data from the text on the radius of Earth\u2019s orbit and approximate it as being circular).\n<strong>\n13: <\/strong>At takeoff, a commercial jet has a 60.0 m\/s speed. Its tires have a diameter of 0.850 m.\n<p id=\"fs-id1869365\">(a) At how many rev\/min are the tires rotating?<\/p>\n<p id=\"fs-id1436140\">(b) What is the centripetal acceleration at the edge of the tire?<\/p>\n<p id=\"fs-id1486366\">(c) With what force must a determined 1.00 \u00d7 10<sup>-15<\/sup> kg bacterium cling to the rim?<\/p>\n<p id=\"fs-id2400686\">(d) Take the ratio of this force to the bacterium\u2019s weight.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1429548\" class=\"exercise\">\n<div id=\"fs-id1926043\" class=\"problem\">\n<p id=\"fs-id2001055\"><strong>14: Integrated Concepts<\/strong><\/p>\nWhen kicking a football, the kicker rotates his leg about the hip joint.\n<p id=\"fs-id1997458\">(a) If the velocity of the tip of the kicker\u2019s shoe is 35.0 m\/s and the hip joint is 1.05 m from the tip of the shoe, what is the shoe tip\u2019s angular velocity?<\/p>\n<p id=\"fs-id1934232\">(b) The shoe is in contact with the initially stationary 0.500 kg football for 20.0 ms. What average force is exerted on the football to give it a velocity of 20.0 m\/s?<\/p>\n<p id=\"fs-id2057369\">(c) Find the maximum range of the football, neglecting air resistance.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2578682\" class=\"exercise\">\n<div id=\"fs-id1545891\" class=\"problem\">\n<p id=\"fs-id2654282\"><strong>15: Construct Your Own Problem<\/strong><\/p>\n<p id=\"eip-id2601357\">Consider an amusement park ride in which participants are rotated about a vertical axis in a cylinder with vertical walls. Once the angular velocity reaches its full value, the floor drops away and friction between the walls and the riders prevents them from sliding down. Construct a problem in which you calculate the necessary angular velocity that assures the riders will not slide down the wall. Include a free body diagram of a single rider. Among the variables to consider are the radius of the cylinder and the coefficients of friction between the riders\u2019 clothing and the wall.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<div>\n<h2>Glossary<\/h2>\n<dl id=\"import-auto-id3161313\" class=\"definition\">\n \t<dt>arc length<\/dt>\n \t<dd id=\"fs-id1350787\"><strong>\u0394<em>s<\/em><\/strong>, the distance traveled by an object along a circular path<\/dd>\n<\/dl>\n<dl id=\"import-auto-id1818172\" class=\"definition\">\n \t<dt>pit<\/dt>\n \t<dd id=\"fs-id3191636\">a tiny indentation on the spiral track moulded into the top of the polycarbonate layer of CD<\/dd>\n<\/dl>\n<dl id=\"import-auto-id2010216\" class=\"definition\">\n \t<dt>rotation angle<\/dt>\n \t<dd id=\"fs-id2836750\">the ratio of the arc length to the radius of curvature on a circular path:\n<p id=\"import-auto-id2448776\">[latex]\\boldsymbol{\\Delta\\theta\\:=}[\/latex][latex size=\"2\"]\\boldsymbol{\\frac{\\Delta{s}}{r}}[\/latex]<\/p>\n<\/dd>\n<\/dl>\n<dl id=\"import-auto-id3046840\" class=\"definition\">\n \t<dt>radius of curvature<\/dt>\n \t<dd id=\"fs-id1927592\">radius of a circular path<\/dd>\n<\/dl>\n<dl id=\"import-auto-id2031924\" class=\"definition\">\n \t<dt>radians<\/dt>\n \t<dd id=\"fs-id3400298\">a unit of angle measurement<\/dd>\n<\/dl>\n<dl id=\"import-auto-id1019372\" class=\"definition\">\n \t<dt>angular velocity<\/dt>\n \t<dd id=\"fs-id3010164\"><strong>\u03c9<\/strong>, the rate of change of the angle with which an object moves on a circular path<\/dd>\n<\/dl>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Solutions<\/h3>\n<strong>Problems &amp; Exercises<\/strong>\n\n<strong>1:<\/strong> [latex]\\boldsymbol{723\\textbf{ km}}[\/latex]\n\n<strong>3:<\/strong> [latex]\\boldsymbol{5\\times10^7\\textbf{ rotations}}[\/latex]\n\n<strong>5:<\/strong> [latex]\\boldsymbol{117\\textbf{ rad\/s}}[\/latex]\n<p id=\"eip-id2409910\"><strong>7:<\/strong> [latex]\\boldsymbol{76.2\\textbf{ rad\/s }728\\textbf{ rpm}}[\/latex]<\/p>\n<p id=\"eip-id2587151\"><strong>9: <\/strong>(a) [latex]\\boldsymbol{3.47\\times10^4\\textbf{ m\/s}^2},\\boldsymbol{\\:3.55\\times10^3\\textbf{ g}}[\/latex] (b) [latex]\\boldsymbol{51.1\\textbf{ m\/s}}[\/latex]<\/p>\n<p id=\"fs-id3121982\"><strong>11: <\/strong>(a) [latex]\\boldsymbol{31.4\\textbf{ rad\/s}}[\/latex] (b) [latex]\\boldsymbol{118\\textbf{ m\/s}}[\/latex] (c) [latex]\\boldsymbol{384\\textbf{ m\/s}}[\/latex] (d) The centripetal acceleration felt by Olympic skaters is 12 times larger than the acceleration due to gravity. That\u2019s quite a lot of acceleration in itself. The centripetal acceleration felt by Button\u2019s nose was 39.2 times larger than the acceleration due to gravity. It is no wonder that he ruptured small blood vessels in his spins.<\/p>\n<strong>13: <\/strong>(a) [latex]\\boldsymbol{1.35\\times10^3\\textbf{ rpm}}[\/latex] (b) [latex]\\boldsymbol{8.47\\times10^3\\textbf{ m\/s}^2}[\/latex] (c) [latex]\\boldsymbol{8.47\\times10^{-12}\\textbf{ N}}[\/latex] (d) [latex]\\boldsymbol{865}[\/latex]\n<p id=\"eip-id3252463\"><strong>14:<\/strong> (a) [latex]\\boldsymbol{33.3\\textbf{ rad\/s}}[\/latex] (b) [latex]\\boldsymbol{500\\textbf{ N}}[\/latex] (c) [latex]\\boldsymbol{40.8\\textbf{ m}}[\/latex]<\/p>\n\n<\/div>","rendered":"<div>\n<div class=\"bcc-box bcc-highlight\">\n<h3>Summary<\/h3>\n<div>\n<ul>\n<li>Define arc length, rotation angle, radius of curvature and angular velocity.<\/li>\n<li>Calculate the angular velocity of a car wheel spin.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"import-auto-id1571938\">In 1D Kinematics, we studied motion along a straight line and introduced such concepts as displacement, velocity, and acceleration. 2D kinematics dealt with motion in two dimensions. Projectile motion is a special case of two-dimensional kinematics in which the object is projected into the air, while being subject to the gravitational force, and lands a distance away. In this chapter, we consider situations where the object does moves in a curve (a special case of this type of motion is <a href=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1108\/part\/chapter-6-uniform-circular-motion-and-gravitation\/\">uniform circular motion<\/a>). We begin the study of rotational motion with rotation kinematics, and defining two angular quantities needed to describe this new type of motion.<\/p>\n<section id=\"fs-id1488693\">\n<h1>Rotation Angle<\/h1>\n<p id=\"import-auto-id1917384\">When objects rotate about some axis\u2014for example, when the CD (compact disc) in <a class=\"autogenerated-content\" href=\"#import-auto-id3402904\">Figure 1<\/a> rotates about its center\u2014each point in the object follows a circular arc. Consider a line from the center of the CD to its edge. Each <span id=\"import-auto-id2654027\">pit<\/span> used to record sound along this line moves through the same angle in the same amount of time. The rotation angle is the amount of rotation and is analogous to linear distance. We define the <strong><span id=\"import-auto-id3255842\">rotation angle\u00a0<\/span><span id=\"import-auto-id3255842\">\u0394<em>\u03b8<\/em><\/span><\/strong> to be the ratio of the arc length to the radius of curvature:<\/p>\n<div id=\"eip-211\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{\\Delta\\theta=}[\/latex][latex]\\boldsymbol{\\frac{\\Delta{s}}{r}}.[\/latex]<\/div>\n<figure id=\"import-auto-id3402904\"><figcaption><\/figcaption><figure style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1107\/wp-content\/uploads\/sites\/1184\/2020\/01\/Figure_07_01_01aa-1.jpg\" alt=\"The figure shows the back side of a compact disc. There is a scratched part on the upper right side of the C D, about one-fifth size of the whole area, with inner circular dots clearly visible. Two line segments are drawn enclosing the scratched area from the border of the C D to the middle plastic portion. A curved arrow is drawn between the two line segments near this middle portion and angle delta theta written alongside it.\" width=\"300\" height=\"300\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 1.<\/strong> All points on a CD travel in circular arcs. The pits along a line from the center to the edge all move through the same angle <strong>\u0394<em>\u03b8<\/em><\/strong> in a time <strong>\u0394<em>t<\/em><\/strong>.<\/figcaption><\/figure>\n<p style=\"text-align: center\">\n<\/figure>\n<figure id=\"import-auto-id3418263\"><figcaption><\/figcaption><figure style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1107\/wp-content\/uploads\/sites\/1184\/2020\/11\/Figure_07_01_01ab-1.jpg\" alt=\"A circle of radius r and center O is shown. A radius O-A of the circle is rotated through angle delta theta about the center O to terminate as radius O-B. The arc length A-B is marked as delta s.\" width=\"300\" height=\"250\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 2.<\/strong> The radius of a circle is rotated through an angle <strong>\u0394<em>\u03b8<\/em><\/strong>. The arc length <strong>\u0394<em>s<\/em><\/strong> is described on the circumference.<\/figcaption><\/figure>\n<\/figure>\n<p id=\"import-auto-id3025908\">The <span id=\"import-auto-id2679165\"><strong>arc length \u0394<em>s <\/em><\/strong><\/span>is the distance traveled along a circular path as shown in <a class=\"autogenerated-content\" href=\"#import-auto-id3418263\">Figure 2<\/a> Note that <em><strong>r<\/strong><\/em> is the <span id=\"import-auto-id2920709\"><strong>radius of curvature<\/strong><\/span> of the circular path.<\/p>\n<p id=\"import-auto-id1471752\">We know that for one complete revolution, the arc length is the circumference of a circle of radius <em><strong>r<\/strong><\/em>. The circumference of a circle is <strong>2\u03c0<em>r<\/em><\/strong>. Thus for one complete revolution the rotation angle is<\/p>\n<div id=\"eip-191\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{\\Delta\\theta\\:=}[\/latex][latex]\\boldsymbol{\\frac{2\\pi{r}}{r}}[\/latex][latex]\\boldsymbol{=\\:2\\pi}.[\/latex]<\/div>\n<p id=\"import-auto-id1842986\">This result is the basis for defining the units used to measure rotation angles, <strong><span id=\"import-auto-id3255842\">\u0394<em>\u03b8<\/em><\/span><\/strong> to be <strong><span id=\"import-auto-id2625941\">radians<\/span> <\/strong>(rad), defined so that<\/p>\n<div id=\"eip-135\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{2\\pi\\textbf{ rad} = 1\\textbf{ revolution}.}[\/latex]<\/div>\n<p id=\"eip-425\">A comparison of some useful angles expressed in both degrees and radians is shown in <a class=\"autogenerated-content\" href=\"#import-auto-id2588905\">Table 1<\/a>.<\/p>\n<table id=\"import-auto-id2588905\" summary=\"The table compares various angle measures in degrees (first column) and radians (second colum).\">\n<thead>\n<tr>\n<th>[latex]\\textbf{Degree Measures}[\/latex]<\/th>\n<th>[latex]\\textbf{Radian Measure}[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: center\">[latex]\\boldsymbol{30^0}[\/latex]<\/td>\n<td style=\"text-align: center\">[latex]\\boldsymbol{\\frac{\\pi}{6}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center\">[latex]\\boldsymbol{60^0}[\/latex]<\/td>\n<td style=\"text-align: center\">[latex]\\boldsymbol{\\frac{\\pi}{3}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center\">[latex]\\boldsymbol{90^0}[\/latex]<\/td>\n<td style=\"text-align: center\">[latex]\\boldsymbol{\\frac{\\pi}{2}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center\">[latex]\\boldsymbol{120^0}[\/latex]<\/td>\n<td style=\"text-align: center\">[latex]\\boldsymbol{\\frac{2\\pi}{3}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center\">[latex]\\boldsymbol{135^0}[\/latex]<\/td>\n<td style=\"text-align: center\">[latex]\\boldsymbol{\\frac{3\\pi}{4}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center\">[latex]\\boldsymbol{180^0}[\/latex]<\/td>\n<td style=\"text-align: center\">[latex]\\boldsymbol{\\pi}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<tbody>\n<tr>\n<td colspan=\"2\"><strong>Table 1.<\/strong> Comparison of Angular Units.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<figure id=\"import-auto-id2442865\"><figcaption><\/figcaption><figure style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1107\/wp-content\/uploads\/sites\/1184\/2020\/11\/Figure_07_01_02a-1.jpg\" alt=\"A circle is shown. Two radii of the circle, inclined at an acute angle delta theta, are shown. On one of the radii, two points, one and two are marked. The point one is inside the circle through which an arc between the two radii is shown. The point two is on the cirumfenrence of the circle. The two arc lengths are delta s one and delta s two respectively for the two points.\" width=\"300\" height=\"294\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 3.<\/strong> Points 1 and 2 rotate through the same angle <strong>(\u0394<em>\u03b8<\/em>)<\/strong>, but point 2 moves through a greater arc length <strong>(\u0394<em>s<\/em>)<\/strong> because it is at a greater distance from the center of rotation <strong>(<em>r<\/em>)<\/strong>.<\/figcaption><\/figure>\n<\/figure>\n<p id=\"import-auto-id1930108\">If <strong><span id=\"import-auto-id3255842\">\u0394\u03b8<\/span> = 2\u03c0 rad<\/strong>, then the CD has made one complete revolution, and every point on the CD is back at its original position. Because there are <strong>360\u00b0<\/strong> in a circle or one revolution, the relationship between radians and degrees is thus<\/p>\n<div id=\"eip-808\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{2\\pi\\textbf{rad}=360^0}[\/latex]<\/div>\n<p id=\"import-auto-id2052087\">so that<\/p>\n<div class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{1\\textbf{ rad}\\:=}[\/latex][latex]\\boldsymbol{\\frac{360^0}{2\\pi}}[\/latex][latex]\\boldsymbol{\\approx\\:57.3^0}.[\/latex]<\/div>\n<\/section>\n<section id=\"fs-id3104613\">\n<h1>Angular Velocity<\/h1>\n<p id=\"import-auto-id2681279\">How fast is an object rotating? We define <strong><span id=\"import-auto-id2962847\">angular velocity\u00a0\u03c9<\/span><\/strong> as the rate of change of an angle. In symbols, this is<\/p>\n<div id=\"eip-759\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{\\omega\\:=}[\/latex][latex]\\boldsymbol{\\frac{\\Delta\\theta}{\\Delta{t}}},[\/latex]<\/div>\n<p id=\"import-auto-id2604506\">where an angular rotation <strong><span id=\"import-auto-id3255842\">\u0394<em>\u03b8<\/em><\/span><\/strong> takes place in a time <strong><span id=\"import-auto-id3255842\">\u0394<\/span><em>t<\/em><\/strong>. The greater the rotation angle in a given amount of time, the greater the angular velocity. The units for angular velocity are radians per second (rad\/s). Angular velocity is often expressed in units of <strong>rev\/min<\/strong> (&#8220;rpm&#8221; or &#8220;revolutions per minute&#8221;). You can convert from <strong>rev\/min<\/strong> to <strong>rad\/s<\/strong> using the fact that that [latex]\\boldsymbol{2\\pi~\\textbf{rad} = 1~\\textbf{rev}}[\/latex] and <strong>1 min = 60 sec<\/strong>.<\/p>\n<p id=\"import-auto-id2621168\">Angular velocity <strong><span id=\"import-auto-id2962847\">\u03c9<\/span><\/strong> is analogous to linear velocity <em><strong>v<\/strong><\/em>. To get the precise relationship between angular and linear velocity, we again consider a pit on the rotating CD. This pit moves an arc length <strong><span id=\"import-auto-id3255842\">\u0394<\/span><em>s<\/em><\/strong> in a time <strong><span id=\"import-auto-id3255842\">\u0394<\/span><em>t<\/em><\/strong>, and so it has a linear velocity<\/p>\n<div id=\"eip-400\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{v\\:=}[\/latex][latex]\\boldsymbol{\\frac{\\Delta{s}}{\\Delta{t}}}.[\/latex]<\/div>\n<p id=\"import-auto-id1840944\">From [latex]\\boldsymbol{\\Delta\\theta=\\frac{\\Delta{s}}{r}}[\/latex] we see that <strong><span id=\"import-auto-id3255842\">\u0394<\/span><em>s<\/em> = <em>r<\/em><span id=\"import-auto-id3255842\">\u0394<em>\u03b8<\/em><\/span><\/strong>. Substituting this into the expression for <em><strong>v<\/strong><\/em> gives<\/p>\n<div class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{v\\:=}[\/latex][latex]\\boldsymbol{\\frac{r\\Delta\\theta}{\\Delta{t}}}[\/latex][latex]\\boldsymbol{=\\:r\\omega}.[\/latex]<\/div>\n<p id=\"import-auto-id1549179\">We write this relationship in two different ways and gain two different insights:<\/p>\n<div id=\"eip-639\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{v=r\\omega\\textbf{ or }\\omega\\:=}[\/latex][latex]\\boldsymbol{\\frac{v}{r}}.[\/latex]<\/div>\n<p id=\"import-auto-id2680923\">The first relationship in <strong><em>v<\/em> = <em>r<\/em><span id=\"import-auto-id2962847\">\u03c9<\/span><\/strong>\u00a0 states that the linear velocity <em><strong>v<\/strong><\/em> is proportional to the distance from the center of rotation, thus, it is largest for a point on the rim (largest <em><strong>r<\/strong><\/em>), as you might expect. We can also call this linear speed <em><strong>v<\/strong><\/em> of a point on the rim the <em>tangential speed<\/em>. The second relationship in\u00a0<strong><em>v<\/em> = <em>r<\/em><span id=\"import-auto-id2962847\">\u03c9<\/span><\/strong>\u00a0 can be illustrated by considering the tire of a moving car. Note that the speed of a point on the rim of the tire is the same as the speed <em><strong>v<\/strong> <\/em>of the car. See <a class=\"autogenerated-content\" href=\"#import-auto-id2931190\">Figure 4<\/a>. So the faster the car moves, the faster the tire spins\u2014large <em><strong>v<\/strong><\/em> means a large <strong><span id=\"import-auto-id2962847\">\u03c9<\/span><\/strong>, because <strong><em>v <\/em>= <em>r<\/em><span id=\"import-auto-id2962847\">\u03c9<\/span><\/strong>. Similarly, a larger-radius tire rotating at the same angular velocity (<strong><span id=\"import-auto-id2962847\">\u03c9<\/span><\/strong>) will produce a greater linear speed (<em><strong>v<\/strong><\/em>) for the car.<\/p>\n<figure id=\"import-auto-id2931190\">\n<figure style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1107\/wp-content\/uploads\/sites\/1184\/2020\/11\/Figure_07_01_03a-1.jpg\" alt=\"The given figure shows the front wheel of a car. The radius of the car wheel, r, is shown as an arrow and the linear velocity, v, is shown with a green horizontal arrow pointing rightward. The angular velocity, omega, is shown with a clockwise-curved arrow over the wheel.\" width=\"300\" height=\"621\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 4.<\/strong> A car moving at a velocity <strong><em>v<\/em><\/strong> to the right has a tire rotating with an angular velocity <strong>\u03c9<\/strong>. The speed of the tread of the tire relative to the axle is <em><strong>v<\/strong><\/em>, the same as if the car were jacked up. Thus the car moves forward at linear velocity <strong><em>v<\/em>=<em>r<\/em>\u03c9<\/strong>, where <em><strong>r<\/strong><\/em> is the tire radius. A larger angular velocity for the tire means a greater velocity for the car.<\/figcaption><\/figure>\n<\/figure>\n<div class=\"textbox shaded\">\n<div id=\"fs-id2589253\" class=\"example\">\n<h3 id=\"import-auto-id3402720\">Example 1: How Fast Does a Car Tire Spin?<\/h3>\n<p>Calculate the angular velocity of a 0.300 m radius car tire when the car travels at <strong>15.0 m\/s<\/strong> (about <strong>54 km\/h<\/strong>). See <a class=\"autogenerated-content\" href=\"#import-auto-id2931190\">Figure 4<\/a>.<\/p>\n<p id=\"import-auto-id2968606\"><strong>Strategy<\/strong><\/p>\n<p>Because the linear speed of the tire rim is the same as the speed of the car, we have <strong><em>v<\/em> = 15.0 m\/s<\/strong>. The radius of the tire is given to be <strong><em>r<\/em> = 0.300 m<\/strong>. Knowing <em><strong>v<\/strong><\/em> and <em><strong>r<\/strong><\/em>, we can use the second relationship in <strong><em>v<\/em> = <em>r<\/em>\u03c9<\/strong>, [latex]\\boldsymbol{\\omega=\\frac{v}{r}}[\/latex] to calculate the angular velocity.<\/p>\n<p id=\"import-auto-id2949936\"><strong>Solution<\/strong><\/p>\n<p id=\"eip-105\">To calculate the angular velocity, we will use the following relationship:<\/p>\n<div id=\"eip-97\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{\\omega\\:=}[\/latex][latex]\\boldsymbol{\\frac{v}{r}}.[\/latex]<\/div>\n<p id=\"import-auto-id3199878\">Substituting the knowns,<\/p>\n<div id=\"eip-451\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{\\omega\\:=}[\/latex][latex]\\boldsymbol{\\frac{15.0\\textbf{ m\/s}}{0.300\\textbf{ m}}}[\/latex][latex]\\boldsymbol{=\\:50.0\\textbf{ rad\/s}}.[\/latex]<\/div>\n<p id=\"import-auto-id1889900\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id956895\">When we cancel units in the above calculation, we get 50.0\/s. But the angular velocity must have units of rad\/s. Because radians are actually unitless (radians are defined as a ratio of distance), we can simply insert them into the answer for the angular velocity. Also note that if an earth mover with much larger tires, say 1.20 m in radius, were moving at the same speed of 15.0 m\/s, its tires would rotate more slowly. They would have an angular velocity<\/p>\n<div id=\"eip-971\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{\\omega=(15.0\\textbf{ m\/s})\/(1.20\\textbf{ m})=12.5\\textbf{ rad\/s}}.[\/latex]<\/div>\n<\/div>\n<\/div>\n<p id=\"import-auto-id2415283\">Both[latex]~\\boldsymbol{\\omega}~[\/latex]and[latex]~\\boldsymbol{v}~[\/latex]have directions (hence they are angular and linear <em>velocities<\/em>, respectively). Angular velocity has only two directions with respect to the axis of rotation\u2014it is either clockwise or counterclockwise. Linear velocity is tangent to the path, as illustrated in <a class=\"autogenerated-content\" href=\"#import-auto-id1452850\">Figure 5<\/a>.<\/p>\n<h2>Connection to Centripetal Acceleration<\/h2>\n<p>We call the acceleration of an object moving in uniform circular motion (resulting from a net external force) the <strong><span id=\"import-auto-id3108952\">centripetal acceleration <\/span><\/strong>(<strong><em>a<\/em><sub>c<\/sub><\/strong>); centripetal means \u201ctoward the center\u201d or \u201ccenter seeking&#8221;. In <a href=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1108\/chapter\/5-2-centripetal-acceleration\/\">Section 5.1<\/a>, we saw that the magnitude of the centripetal acceleration is<\/p>\n<div id=\"eip-684\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{a_{\\textbf{c}}\\:=}[\/latex][latex]\\boldsymbol{\\frac{v^2}{r}},[\/latex]<\/div>\n<p>which is the acceleration of an object in a circle of radius <em><strong>r<\/strong><\/em> at a speed <em><strong>v<\/strong><\/em>. It is also useful to express <strong><em>a<\/em><sub>c<\/sub><\/strong> in terms of angular velocity. Substituting <strong><em>v<\/em> = <em>r<\/em>\u03c9<\/strong> into the above expression, we find <strong><em>a<\/em><sub>c <\/sub>= (<em>r<\/em>\u03c9)<sup>2<\/sup>\/<em>r<\/em> = <em>r<\/em>\u03c9<sup>2<\/sup><\/strong>. We can express the magnitude of centripetal acceleration using either of two equations:<\/p>\n<div id=\"eip-740\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{a_{\\textbf{c}}\\:=}[\/latex][latex]\\boldsymbol{\\frac{v^2}{r}}[\/latex][latex]\\boldsymbol{;\\:a_{\\textbf{c}}=r\\omega^2}.[\/latex]<\/div>\n<div>\n<div class=\"textbox shaded\">\n<div id=\"fs-id2598952\" class=\"example\">\n<h3>Example 2: How Big Is The Centripetal Acceleration in an Ultracentrifuge?<\/h3>\n<p>Calculate the centripetal acceleration of a point 7.50 cm from the axis of an <strong><span id=\"import-auto-id2449410\">ultracentrifuge<\/span> <\/strong>spinning at <strong>7.5 \u00d7 10<sup>4<\/sup> rev\/min<\/strong>. Determine the ratio of this acceleration to that due to gravity. Note that this example is similar to <a href=\"\/douglasphys1108\/chapter\/5-2-centripetal-acceleration\/#ca_ex2\">Example 2<\/a> in the Section 5.2 on centripetal acceleration.<\/p>\n<p>&nbsp;<\/p>\n<p id=\"import-auto-id3105981\"><strong>Strategy<\/strong><\/p>\n<p>The term rev\/min stands for revolutions per minute. By converting this to radians per second, we obtain the angular velocity <strong>\u03c9<\/strong>. Because <em><strong>r<\/strong><\/em> is given, we can use the second expression in the equation [latex]\\boldsymbol{a_{\\textbf{c}}=\\frac{v^2}{r};\\:a_{\\textbf{c}}=r\\omega^2}[\/latex] to calculate the centripetal acceleration.<\/p>\n<p>&nbsp;<\/p>\n<p id=\"import-auto-id1140598\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id2849150\">To convert <strong>7.5 \u00d7 10<sup>4<\/sup> rev\/min<\/strong> to radians per second, we use the facts that one revolution is <strong>2\u03c0 rad<\/strong> and one minute is 60.0 s. Thus,<\/p>\n<div class=\"equation\">[latex]\\boldsymbol{\\omega\\:=7.50\\times10^4}[\/latex][latex]\\boldsymbol{\\frac{\\textbf{rev}}{\\textbf{min}}\\times\\frac{2\\pi\\textbf{ rad}}{1\\textbf{ rev}}\\times\\frac{1\\textbf{ min}}{60.0\\textbf{ s}}}[\/latex][latex]\\boldsymbol{=\\:7854\\textbf{ rad\/s}}.[\/latex]<\/div>\n<p id=\"import-auto-id3191658\">Now the centripetal acceleration is given by the second expression in [latex]\\boldsymbol{a_{\\textbf{c}}=\\frac{v^2}{r};\\:a_{\\textbf{c}}=r\\omega^2}[\/latex] as<\/p>\n<div class=\"equation\">[latex]\\boldsymbol{a_{\\textbf{c}}=r\\omega^2}.[\/latex]<\/div>\n<p id=\"import-auto-id1926483\">Converting 7.50 cm to meters and substituting known values gives<\/p>\n<div class=\"equation\">[latex]\\boldsymbol{a_{\\textbf{c}}=(0.0750\\textbf{ m})(7854\\textbf{ rad\/s})^2=4.63\\times10^6\\textbf{ m\/s}^2}.[\/latex]<\/div>\n<p id=\"import-auto-id2673973\">Note that the unitless radians are discarded in order to get the correct units for centripetal acceleration. Taking the ratio of <strong><em>a<\/em><sub>c<\/sub><\/strong> to <em><strong>g<\/strong><\/em> yields<\/p>\n<div class=\"equation\">[latex]\\boldsymbol{\\frac{a_{\\textbf{c}}}{g}}[\/latex][latex]\\boldsymbol{=}[\/latex][latex]\\boldsymbol{\\frac{4.63\\times10^6}{9.80}}[\/latex][latex]\\boldsymbol{=4.72\\times10^5}.[\/latex]<\/div>\n<div><\/div>\n<p id=\"import-auto-id871156\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id2604418\">This last result means that the centripetal acceleration is 472,000 times as strong as <em><strong>g<\/strong><\/em>. It is no wonder that such high <strong>\u03c9<\/strong> centrifuges are called ultracentrifuges. The extremely large accelerations involved greatly decrease the time needed to cause the sedimentation of blood cells or other materials.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2584087\" class=\"note\">\n<div class=\"textbox shaded\">\n<div class=\"note\">\n<h3 class=\"title\">TAKE-HOME EXPERIMENT<span style=\"text-decoration: underline\"><br \/>\n<\/span><\/h3>\n<p id=\"import-auto-id1986367\">Tie an object to the end of a string and swing it around in a horizontal circle above your head (swing at your wrist). Maintain uniform speed as the object swings and measure the angular velocity of the motion. What is the approximate speed of the object? Identify a point close to your hand and take appropriate measurements to calculate the linear speed at this point. Identify other circular motions and measure their angular velocities.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<figure id=\"import-auto-id1452850\"><figcaption><\/figcaption><figure style=\"width: 250px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1107\/wp-content\/uploads\/sites\/1184\/2020\/11\/Figure_07_01_04a-1.jpg\" alt=\"The given figure shows the top view of an old fashioned vinyl record. Two perpendicular line segments are drawn through the center of the circular record, one vertically upward and one horizontal to the right side. Two flies are shown at the end points of the vertical lines near the borders of the record. Two arrows are also drawn perpendicularly rightward through the end points of these vertical lines depicting linear velocities. A curved arrow is also drawn at the center circular part of the record which shows the angular velocity.\" width=\"250\" height=\"755\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 5.<\/strong> As an object moves in a circle, here a fly on the edge of an old-fashioned vinyl record, its instantaneous velocity is always tangent to the circle. The direction of the angular velocity is clockwise in this case.<\/figcaption><\/figure>\n<\/figure>\n<\/section>\n<div id=\"eip-270\" class=\"note\">\n<div class=\"textbox shaded\">\n<div class=\"note\">\n<h3 class=\"title\">PHET EXPLORATIONS: LADYBUG REVOLUTION<\/h3>\n<figure id=\"eip-id1171550\"><figcaption><\/figcaption><figure style=\"width: 450px\" class=\"wp-caption aligncenter\"><a href=\"\/resources\/7c52f36f755df0a4d1bf4e75814c8135735d7055\/rotation_en.jar\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1107\/wp-content\/uploads\/sites\/1184\/2020\/11\/PhET_Icon-5-1.png\" alt=\"image\" width=\"450\" height=\"147\" \/><\/a><figcaption class=\"wp-caption-text\"><strong>Figure 6.<\/strong><a href=\"https:\/\/phet.colorado.edu\/en\/simulation\/rotation\"> Ladybug Revolution<\/a><\/figcaption><\/figure>\n<\/figure>\n<p id=\"eip-id1169738118030\">Join the ladybug in an exploration of rotational motion. Rotate the merry-go-round to change its angle, or choose a constant angular velocity or angular acceleration. Explore how circular motion relates to the bug&#8217;s x,y position, velocity, and acceleration using vectors or graphs.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<section id=\"fs-id3399151\" class=\"section-summary\">\n<h1>Section Summary<\/h1>\n<ul id=\"fs-id1104163\">\n<li id=\"import-auto-id2603294\">Uniform circular motion is motion in a circle at constant speed. The rotation angle<strong> \u0394<em>\u03b8<\/em><\/strong> is defined as the ratio of the arc length to the radius of curvature:\n<div class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{\\Delta\\theta\\:=}[\/latex][latex]\\boldsymbol{\\frac{\\Delta{s}}{r}},[\/latex]<\/div>\n<p id=\"import-auto-id3062885\">where arc length <strong>\u0394<em>s <\/em><\/strong>is distance traveled along a circular path and <em><strong>r<\/strong><\/em> is the radius of curvature of the circular path. The quantity <strong>\u0394<em>\u03b8<\/em><\/strong> is measured in units of radians (rad), for which<\/p>\n<div id=\"eip-567\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{2\\pi\\textbf{ rad}=360^0=1\\textbf{ revolution}}.[\/latex]<\/div>\n<\/li>\n<li id=\"import-auto-id2384053\">The conversion between radians and degrees is <strong>1 rad = 57.3\u00b0<\/strong>.<\/li>\n<li id=\"import-auto-id2442213\">Angular velocity <strong>\u03c9<\/strong> is the rate of change of an angle,\n<div id=\"eip-969\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{\\omega\\:=}[\/latex][latex]\\boldsymbol{\\frac{\\Delta\\theta}{\\Delta{t}}},[\/latex]<\/div>\n<p id=\"import-auto-id1439148\">where a rotation <strong>\u0394<em>\u03b8<\/em><\/strong> takes place in a time <strong>\u0394<em>t<\/em><\/strong>. The units of angular velocity are radians per second (rad\/s). Linear velocity <em><strong>v<\/strong><\/em> and angular velocity <strong>\u03c9<\/strong> are related by<\/p>\n<div id=\"eip-513\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{v=r\\omega\\textbf{ or }\\omega\\:=}[\/latex][latex]\\boldsymbol{\\frac{v}{r}}.[\/latex]<\/div>\n<\/li>\n<\/ul>\n<\/section>\n<section id=\"fs-id3103314\" class=\"conceptual-questions\">\n<div id=\"fs-id3119404\" class=\"exercise\">\n<div id=\"fs-id2969010\" class=\"problem\">\n<div class=\"bcc-box bcc-info\">\n<h3>Conceptual Questions<\/h3>\n<p><strong>1: <\/strong>There is an analogy between rotational and linear physical quantities. What rotational quantities are analogous to distance and velocity?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id2381548\" class=\"problems-exercises\">\n<div class=\"bcc-box bcc-info\">\n<h3>Problems &amp; Exercises<\/h3>\n<div id=\"fs-id3004274\" class=\"exercise\">\n<div id=\"fs-id3253787\" class=\"problem\">\n<p id=\"import-auto-id3415350\"><strong>1: <\/strong>Semi-trailer trucks have an odometer on one hub of a trailer wheel. The hub is weighted so that it does not rotate, but it contains gears to count the number of wheel revolutions\u2014it then calculates the distance traveled. If the wheel has a 1.15 m diameter and goes through 200,000 rotations, how many kilometers should the odometer read?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1004074\" class=\"exercise\">\n<div id=\"fs-id2448666\" class=\"problem\">\n<p id=\"fs-id2052623\"><strong>2: <\/strong>Microwave ovens rotate at a rate of about 6 rev\/min. What is this in revolutions per second? What is the angular velocity in radians per second?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1921627\" class=\"exercise\">\n<div id=\"fs-id2953385\" class=\"problem\">\n<p id=\"fs-id1574958\"><strong>3: <\/strong>An automobile with 0.260 m radius tires travels 80,000 km before wearing them out. How many revolutions do the tires make, neglecting any backing up and any change in radius due to wear?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1524972\" class=\"exercise\">\n<div id=\"fs-id3254586\" class=\"problem\">\n<p id=\"fs-id3035846\"><strong>4: <\/strong>(a) What is the period of rotation of Earth in seconds? (b) What is the angular velocity of Earth? (c) Given that Earth has a radius of 6.4 \u00d7 10<sup>6<\/sup> m at its equator, what is the linear velocity at Earth\u2019s surface?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2979194\" class=\"exercise\">\n<div id=\"fs-id1361145\" class=\"problem\">\n<p id=\"fs-id3093713\"><strong>5: <\/strong>A baseball pitcher brings his arm forward during a pitch, rotating the forearm about the elbow. If the velocity of the ball in the pitcher\u2019s hand is 35.0 m\/s and the ball is 0.300 m from the elbow joint, what is the angular velocity of the forearm?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id954942\" class=\"exercise\">\n<div id=\"fs-id3181723\" class=\"problem\">\n<p id=\"fs-id2595491\"><strong>6: <\/strong>In lacrosse, a ball is thrown from a net on the end of a stick by rotating the stick and forearm about the elbow. If the angular velocity of the ball about the elbow joint is 30.0 rad\/s and the ball is 1.30 m from the elbow joint, what is the velocity of the ball?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2678694\" class=\"exercise\">\n<div id=\"fs-id1576562\" class=\"problem\">\n<p id=\"fs-id2979343\"><strong>7: <\/strong>A truck with 0.420-m-radius tires travels at 32.0 m\/s. What is the angular velocity of the rotating tires in radians per second? What is this in rev\/min?<\/p>\n<p><strong>8: <\/strong>A rotating space station is said to create \u201cartificial gravity\u201d\u2014a loosely-defined term used for an acceleration that would be crudely similar to gravity. The outer wall of the rotating space station would become a floor for the astronauts, and centripetal acceleration supplied by the floor would allow astronauts to exercise and maintain muscle and bone strength more naturally than in non-rotating space environments. If the space station is 200 m in diameter, what angular velocity would produce an \u201cartificial gravity\u201d of 9.80 m\/s<sup>2<\/sup> at the rim?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2017206\" class=\"exercise\">\n<div id=\"fs-id2400941\" class=\"problem\">\n<p id=\"fs-id2953765\"><strong>9: <\/strong>An ordinary workshop grindstone has a radius of 7.50 cm and rotates at 6500 rev\/min.<\/p>\n<p id=\"fs-id1990575\">(a) Calculate the magnitude of the centripetal acceleration at its edge in meters per second squared and convert it to multiples of[latex]\\boldsymbol{g}.[\/latex]<\/p>\n<p id=\"fs-id3122407\">(b) What is the linear speed of a point on its edge?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1474903\" class=\"exercise\">\n<div id=\"fs-id1272517\" class=\"problem\">\n<p id=\"fs-id3424715\"><strong>10: <\/strong>Helicopter blades withstand tremendous stresses. In addition to supporting the weight of a helicopter, they are spun at rapid rates and experience large centripetal accelerations, especially at the tip.<\/p>\n<p id=\"fs-id3054343\">(a) Calculate the magnitude of the centripetal acceleration at the tip of a 4.00 m long helicopter blade that rotates at 300 rev\/min.<\/p>\n<p id=\"fs-id3033654\">(b) Compare the linear speed of the tip with the speed of sound (taken to be 340 m\/s).<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id3224044\" class=\"exercise\">\n<div id=\"fs-id1514610\" class=\"problem\">\n<p id=\"fs-id1004097\"><strong>11: <\/strong>Olympic ice skaters are able to spin at about 5 rev\/s.<\/p>\n<p id=\"fs-id3358931\">(a) What is their angular velocity in radians per second?<\/p>\n<p id=\"fs-id1355573\">(b) What is the centripetal acceleration of the skater\u2019s nose if it is 0.120 m from the axis of rotation?<\/p>\n<p id=\"fs-id1947068\">(c) An exceptional skater named Dick Button was able to spin much faster in the 1950s than anyone since\u2014at about 9 rev\/s. What was the centripetal acceleration of the tip of his nose, assuming it is at 0.120 m radius?<\/p>\n<p>(d) Comment on the magnitudes of the accelerations found. It is reputed that Button ruptured small blood vessels during his spins.<\/p>\n<p><strong>12: <\/strong>Verify that the linear speed of an ultracentrifuge is about 0.50 km\/s, and Earth in its orbit is about 30 km\/s by calculating:<\/p>\n<p id=\"fs-id3138086\">(a) The linear speed of a point on an ultracentrifuge 0.100 m from its center, rotating at 50,000 rev\/min.<\/p>\n<p>(b) The linear speed of Earth in its orbit about the Sun (use data from the text on the radius of Earth\u2019s orbit and approximate it as being circular).<br \/>\n<strong><br \/>\n13: <\/strong>At takeoff, a commercial jet has a 60.0 m\/s speed. Its tires have a diameter of 0.850 m.<\/p>\n<p id=\"fs-id1869365\">(a) At how many rev\/min are the tires rotating?<\/p>\n<p id=\"fs-id1436140\">(b) What is the centripetal acceleration at the edge of the tire?<\/p>\n<p id=\"fs-id1486366\">(c) With what force must a determined 1.00 \u00d7 10<sup>-15<\/sup> kg bacterium cling to the rim?<\/p>\n<p id=\"fs-id2400686\">(d) Take the ratio of this force to the bacterium\u2019s weight.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1429548\" class=\"exercise\">\n<div id=\"fs-id1926043\" class=\"problem\">\n<p id=\"fs-id2001055\"><strong>14: Integrated Concepts<\/strong><\/p>\n<p>When kicking a football, the kicker rotates his leg about the hip joint.<\/p>\n<p id=\"fs-id1997458\">(a) If the velocity of the tip of the kicker\u2019s shoe is 35.0 m\/s and the hip joint is 1.05 m from the tip of the shoe, what is the shoe tip\u2019s angular velocity?<\/p>\n<p id=\"fs-id1934232\">(b) The shoe is in contact with the initially stationary 0.500 kg football for 20.0 ms. What average force is exerted on the football to give it a velocity of 20.0 m\/s?<\/p>\n<p id=\"fs-id2057369\">(c) Find the maximum range of the football, neglecting air resistance.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2578682\" class=\"exercise\">\n<div id=\"fs-id1545891\" class=\"problem\">\n<p id=\"fs-id2654282\"><strong>15: Construct Your Own Problem<\/strong><\/p>\n<p id=\"eip-id2601357\">Consider an amusement park ride in which participants are rotated about a vertical axis in a cylinder with vertical walls. Once the angular velocity reaches its full value, the floor drops away and friction between the walls and the riders prevents them from sliding down. Construct a problem in which you calculate the necessary angular velocity that assures the riders will not slide down the wall. Include a free body diagram of a single rider. Among the variables to consider are the radius of the cylinder and the coefficients of friction between the riders\u2019 clothing and the wall.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<div>\n<h2>Glossary<\/h2>\n<dl id=\"import-auto-id3161313\" class=\"definition\">\n<dt>arc length<\/dt>\n<dd id=\"fs-id1350787\"><strong>\u0394<em>s<\/em><\/strong>, the distance traveled by an object along a circular path<\/dd>\n<\/dl>\n<dl id=\"import-auto-id1818172\" class=\"definition\">\n<dt>pit<\/dt>\n<dd id=\"fs-id3191636\">a tiny indentation on the spiral track moulded into the top of the polycarbonate layer of CD<\/dd>\n<\/dl>\n<dl id=\"import-auto-id2010216\" class=\"definition\">\n<dt>rotation angle<\/dt>\n<dd id=\"fs-id2836750\">the ratio of the arc length to the radius of curvature on a circular path:<\/p>\n<p id=\"import-auto-id2448776\">[latex]\\boldsymbol{\\Delta\\theta\\:=}[\/latex][latex]\\boldsymbol{\\frac{\\Delta{s}}{r}}[\/latex]<\/p>\n<\/dd>\n<\/dl>\n<dl id=\"import-auto-id3046840\" class=\"definition\">\n<dt>radius of curvature<\/dt>\n<dd id=\"fs-id1927592\">radius of a circular path<\/dd>\n<\/dl>\n<dl id=\"import-auto-id2031924\" class=\"definition\">\n<dt>radians<\/dt>\n<dd id=\"fs-id3400298\">a unit of angle measurement<\/dd>\n<\/dl>\n<dl id=\"import-auto-id1019372\" class=\"definition\">\n<dt>angular velocity<\/dt>\n<dd id=\"fs-id3010164\"><strong>\u03c9<\/strong>, the rate of change of the angle with which an object moves on a circular path<\/dd>\n<\/dl>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Solutions<\/h3>\n<p><strong>Problems &amp; Exercises<\/strong><\/p>\n<p><strong>1:<\/strong> [latex]\\boldsymbol{723\\textbf{ km}}[\/latex]<\/p>\n<p><strong>3:<\/strong> [latex]\\boldsymbol{5\\times10^7\\textbf{ rotations}}[\/latex]<\/p>\n<p><strong>5:<\/strong> [latex]\\boldsymbol{117\\textbf{ rad\/s}}[\/latex]<\/p>\n<p id=\"eip-id2409910\"><strong>7:<\/strong> [latex]\\boldsymbol{76.2\\textbf{ rad\/s }728\\textbf{ rpm}}[\/latex]<\/p>\n<p id=\"eip-id2587151\"><strong>9: <\/strong>(a) [latex]\\boldsymbol{3.47\\times10^4\\textbf{ m\/s}^2},\\boldsymbol{\\:3.55\\times10^3\\textbf{ g}}[\/latex] (b) [latex]\\boldsymbol{51.1\\textbf{ m\/s}}[\/latex]<\/p>\n<p id=\"fs-id3121982\"><strong>11: <\/strong>(a) [latex]\\boldsymbol{31.4\\textbf{ rad\/s}}[\/latex] (b) [latex]\\boldsymbol{118\\textbf{ m\/s}}[\/latex] (c) [latex]\\boldsymbol{384\\textbf{ m\/s}}[\/latex] (d) The centripetal acceleration felt by Olympic skaters is 12 times larger than the acceleration due to gravity. That\u2019s quite a lot of acceleration in itself. The centripetal acceleration felt by Button\u2019s nose was 39.2 times larger than the acceleration due to gravity. It is no wonder that he ruptured small blood vessels in his spins.<\/p>\n<p><strong>13: <\/strong>(a) [latex]\\boldsymbol{1.35\\times10^3\\textbf{ rpm}}[\/latex] (b) [latex]\\boldsymbol{8.47\\times10^3\\textbf{ m\/s}^2}[\/latex] (c) [latex]\\boldsymbol{8.47\\times10^{-12}\\textbf{ N}}[\/latex] (d) [latex]\\boldsymbol{865}[\/latex]<\/p>\n<p id=\"eip-id3252463\"><strong>14:<\/strong> (a) [latex]\\boldsymbol{33.3\\textbf{ rad\/s}}[\/latex] (b) [latex]\\boldsymbol{500\\textbf{ N}}[\/latex] (c) [latex]\\boldsymbol{40.8\\textbf{ m}}[\/latex]<\/p>\n<\/div>\n","protected":false},"author":9,"menu_order":1,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-477","chapter","type-chapter","status-publish","hentry"],"part":470,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/douglasphys1108\/wp-json\/pressbooks\/v2\/chapters\/477","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/douglasphys1108\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/douglasphys1108\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/douglasphys1108\/wp-json\/wp\/v2\/users\/9"}],"version-history":[{"count":1,"href":"https:\/\/pressbooks.bccampus.ca\/douglasphys1108\/wp-json\/pressbooks\/v2\/chapters\/477\/revisions"}],"predecessor-version":[{"id":478,"href":"https:\/\/pressbooks.bccampus.ca\/douglasphys1108\/wp-json\/pressbooks\/v2\/chapters\/477\/revisions\/478"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/douglasphys1108\/wp-json\/pressbooks\/v2\/parts\/470"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/douglasphys1108\/wp-json\/pressbooks\/v2\/chapters\/477\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/douglasphys1108\/wp-json\/wp\/v2\/media?parent=477"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/douglasphys1108\/wp-json\/pressbooks\/v2\/chapter-type?post=477"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/douglasphys1108\/wp-json\/wp\/v2\/contributor?post=477"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/douglasphys1108\/wp-json\/wp\/v2\/license?post=477"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}