{"id":300,"date":"2017-09-18T18:09:00","date_gmt":"2017-09-18T22:09:00","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/chapter\/5-1-centripetal-acceleration\/"},"modified":"2021-05-09T16:19:44","modified_gmt":"2021-05-09T20:19:44","slug":"5-1-centripetal-acceleration","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/chapter\/5-1-centripetal-acceleration\/","title":{"raw":"5.1 Centripetal Acceleration","rendered":"5.1 Centripetal Acceleration"},"content":{"raw":"<div>\n<div class=\"bcc-box bcc-highlight\">\n<h3>Summary<\/h3>\n<div>\n<ul>\n \t<li>Establish the expression for centripetal acceleration.<\/li>\n \t<li>Explain the centrifuge.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"import-auto-id3008127\">We know from kinematics that acceleration is a change in velocity, either in its magnitude or in its direction, or both. In uniform circular motion, the direction of the velocity changes constantly, so there is always an associated acceleration, even though the magnitude of the velocity might be constant. You experience this acceleration yourself when you turn a corner in your car. (If you hold the wheel steady during a turn and move at constant speed, you are in uniform circular motion.) What you notice is a sideways acceleration because you and the car are changing direction. The sharper the curve and the greater your speed, the more noticeable this acceleration will become. In this section we examine the direction and magnitude of that acceleration.<\/p>\n<p id=\"import-auto-id2422195\"><a class=\"autogenerated-content\" href=\"#import-auto-id1561888\">Figure 1<\/a> shows an object moving in a circular path at constant speed. The direction of the instantaneous velocity is shown at two points along the path. Acceleration is in the direction of the change in velocity, which points directly toward the center of rotation (the center of the circular path). This pointing is shown with the vector diagram in the figure. 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.\u201d<\/p>\n\n<figure id=\"import-auto-id1561888\"><figcaption><\/figcaption>\n\n[caption id=\"\" align=\"aligncenter\" width=\"330\" class=\"center\"]<img class=\"\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2017\/06\/Figure_07_02_01a-1.jpg\" alt=\"The given figure shows a circle, with a triangle having vertices A B C made from the center to the boundry. A is at the center and B and C points are at the circle path. Lines A B and A C act as radii and B C is a chord. Delta theta is shown inside the triangle, and the arc length delta s and the chord length delta r are also given. At point B, velocity of object is shown as v one and at point C, velocity of object is shown as v two. Along the circle an equation is shown as delta v equals v sub 2 minus v sub 1.\" width=\"330\" height=\"395\"> <strong>Figure 1.<\/strong> The directions of the velocity of an object at two different points are shown, and the change in velocity <strong>\u0394v<\/strong> is seen to point directly toward the center of curvature. (See small inset.) Because <strong>ac = \u0394<em>v<\/em>\/\u0394<em>t<\/em><\/strong>, the acceleration is also toward the center; ac is called centripetal acceleration. (Because<strong> \u0394<em>\u03b8<\/em> <\/strong>is very small, the arc length <strong>\u0394<em>s<\/em><\/strong> is equal to the chord length <strong>\u0394<em>r<\/em><\/strong> for small time differences.)[\/caption]<\/figure>\n<p id=\"import-auto-id723408\">The direction of centripetal acceleration is toward the center of curvature, but what is its magnitude? Note that the triangle formed by the velocity vectors and the one formed by the radii <em><strong>r<\/strong><\/em> and <strong>\u0394<em>s<\/em><\/strong> are similar. Both the triangles ABC and PQR are isosceles triangles (two equal sides). The two equal sides of the velocity vector triangle are the speeds <strong><em>v<\/em><sub>1<\/sub> = <em>v<\/em><sub>2<\/sub> =<em> v<\/em><\/strong>. Using the properties of two similar triangles, we obtain<\/p>\n\n<div id=\"eip-829\" class=\"equation\" style=\"text-align: center\">[latex size=\"2\"]\\boldsymbol{\\frac{\\Delta{v}}{v}}[\/latex][latex]\\boldsymbol{=}[\/latex][latex size=\"2\"]\\boldsymbol{\\frac{\\Delta{s}}{r}}.[\/latex]<\/div>\n<p id=\"import-auto-id2588045\">Acceleration is <strong>\u0394<em>v<\/em>\/\u0394<em>t<\/em><\/strong>, and so we first solve this expression for <strong>\u0394<em>v<\/em><\/strong>:<\/p>\n\n<div id=\"eip-0\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{\\Delta{v}\\:=}[\/latex][latex size=\"2\"]\\boldsymbol{\\frac{v}{r}}[\/latex][latex]\\boldsymbol{\\Delta{s}}.[\/latex]<\/div>\n<p id=\"import-auto-id2398724\">Then we divide this by <strong>\u0394<em>t<\/em><\/strong>, yielding<\/p>\n\n<div id=\"eip-593\" class=\"equation\" style=\"text-align: center\">[latex size=\"2\"]\\boldsymbol{\\frac{\\Delta{v}}{\\Delta{t}}}[\/latex][latex]\\boldsymbol{=}[\/latex][latex size=\"2\"]\\boldsymbol{\\frac{v}{r}\\times\\frac{\\Delta{s}}{\\Delta{t}}}.[\/latex]<\/div>\n<p id=\"import-auto-id3489731\">Finally, noting that <strong>\u0394<em>v<\/em>\/\u0394<em>t<\/em> = <em>a<\/em><sub>c<\/sub><\/strong> and that <strong>\u0394<em>s<\/em>\/\u0394<em>t<\/em> = <em>v<\/em><\/strong>, the linear or tangential speed, we see that the magnitude of the centripetal acceleration is<\/p>\n\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>\n<p id=\"import-auto-id2429569\">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>. So, centripetal acceleration is greater at high speeds and in sharp curves (smaller radius), as you have noticed when driving a car. But it is a bit surprising that <strong><em>a<\/em><sub>c<\/sub><\/strong> is proportional to speed squared, implying, for example, that it is four times as hard to take a curve at 100 km\/h than at 50 km\/h. A sharp corner has a small radius, so that <strong><em>a<\/em><sub>c<\/sub><\/strong> is greater for tighter turns, as you have probably noticed.<\/p>\n<p id=\"import-auto-id2441429\">A <span id=\"import-auto-id3256561\">centrifuge<\/span> (see <a class=\"autogenerated-content\" href=\"#import-auto-id3033074\">Figure 2<\/a>b) is a rotating device used to separate specimens of different densities. High centripetal acceleration significantly decreases the time it takes for separation to occur, and makes separation possible with small samples. Centrifuges are used in a variety of applications in science and medicine, including the separation of single cell suspensions such as bacteria, viruses, and blood cells from a liquid medium and the separation of macromolecules, such as DNA and protein, from a solution. Centrifuges are often rated in terms of their centripetal acceleration relative to acceleration due to gravity (<em><strong>g<\/strong><\/em>); maximum centripetal acceleration of several hundred thousand <em><strong>g<\/strong><\/em> is possible in a vacuum. Human centrifuges, extremely large centrifuges, have been used to test the tolerance of astronauts to the effects of accelerations larger than that of Earth\u2019s gravity.<\/p>\n\n<div class=\"textbox shaded\">\n<div id=\"fs-id3386265\" class=\"example\">\n<h3 id=\"import-auto-id3086245\">Example 1: How Does the Centripetal Acceleration of a Car Around a Curve Compare with That Due to Gravity?<\/h3>\nWhat is the magnitude of the centripetal acceleration of a car following a curve of radius 500 m at a speed of 25.0 m\/s (about 90 km\/h)? Compare the acceleration with that due to gravity for this fairly gentle curve taken at highway speed. See <a class=\"autogenerated-content\" href=\"#import-auto-id3033074\">Figure 2<\/a>(a).\n<p id=\"import-auto-id3079956\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id3358891\">Because <em><strong>v<\/strong><\/em> and <em><strong>r<\/strong><\/em> are given, we use the expression [latex]\\boldsymbol{a_{\\textbf{c}}=\\dfrac{v^2}{r}}[\/latex].<\/p>\n<p id=\"import-auto-id1447129\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id2057486\">Entering the given values of <strong><em>v<\/em> = 25.0 m\/s<\/strong> and <strong><em>r<\/em> = 500 m<\/strong> into the first expression for <strong><em>a<\/em><sub>c<\/sub><\/strong> gives<\/p>\n<p style=\"text-align: center\">[latex]\\boldsymbol{a_\\textbf{c}} = \\boldsymbol{\\dfrac{v^2}{r}} = \\boldsymbol{\\dfrac{25.0^2}{500} = 1.25}~\\textbf{m\/s}^2[\/latex]<\/p>\n\n<div id=\"eip-122\" class=\"equation\" style=\"text-align: center\"><\/div>\n<p id=\"import-auto-id1816308\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id1425882\">To compare this with the acceleration due to gravity (<strong><em>g<\/em> = 9.80 m\/s<sup>2<\/sup><\/strong>), we take the ratio of <strong><em>a<\/em><sub>c<\/sub> \/<em>g<\/em> = (1.25 m\/s<sup>2<\/sup>)\/(9.80m\/s<sup>2<\/sup>) = 0.128<\/strong>. Thus, <strong><em>a<\/em><sub>c <\/sub>= 0.128 <em>g<\/em><\/strong> and is noticeable especially if you were not wearing a seat belt.<\/p>\n\n<\/div>\n<\/div>\n<figure id=\"import-auto-id3033074\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"250\"]<img src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/Figure_07_02_02a-1.jpg\" alt=\"In figure a, a car shown from top is running on a circular road around a circular path. The center of the park is termed as the center of this circle and the distance from this point to the car is taken as radius r. The linear velocity is shown in perpendicular direction toward the front of the car, shown as v the centripetal acceleration is shown with an arrow pointed towards the center of rotation. In figure b, a centrifuge is shown an object of mass m is rotating in it at a constant speed. The object is at the distance equal to the radius, r, of the centrifuge. The centripetal acceleration is shown towards the center of rotation, and the velocity, v is shown perpendicular to the object in the clockwise direction.\" width=\"250\" height=\"606\"> <strong>Figure 2.<\/strong> (a) The car following a circular path at constant speed is accelerated perpendicular to its velocity, as shown. The magnitude of this centripetal acceleration is found in <a href=\"#import-auto-id3086245\">Example 1<\/a>. (b) A particle of mass in a centrifuge is rotating at constant angular velocity . It must be accelerated perpendicular to its velocity or it would continue in a straight line. The magnitude of the necessary acceleration is found in <a href=\"#fs-id2598952\">Example 2<\/a>.[\/caption]<\/figure>\n<div class=\"textbox shaded\">\n<div id=\"fs-id2598952\" class=\"example\">\n<h3><a id=\"ca_ex2\" href=\"\"><\/a>Example 2: What is the speed of an Ultracentrifuge?<\/h3>\nCalculate the speed of a point 7.50 cm from the axis of a spinning <strong><span id=\"import-auto-id2449410\">ultracentrifuge<\/span> <\/strong>if the centripetal acceleration at that point is [latex]\\boldsymbol{4.63\\times10^6}~\\textbf{m\/s}^2[\/latex]\n<p id=\"import-auto-id3105981\"><strong>Strategy<\/strong><\/p>\nBoth <em><strong>r<\/strong><\/em> and <em><strong>a<sub>c<\/sub> <\/strong><\/em>are given, so we can use the equation [latex] \\boldsymbol{a_{\\textbf{c}} = \\dfrac{v^2}{r}}[\/latex] to determine the speed of the centrifuge.\n<p id=\"import-auto-id1140598\"><strong>Solution<\/strong><\/p>\nFirst we solve for the speed of the centrifugre\n<p style=\"text-align: center\">[latex] \\boldsymbol{a_{\\textbf{c}} = \\dfrac{v^2}{r} \\rightarrow v =\\sqrt{a_{\\textbf{c}}r}} [\/latex]<\/p>\n\n<div class=\"equation\" style=\"text-align: center\"><\/div>\n<p id=\"import-auto-id1926483\">Converting 7.50 cm to meters and substituting known values gives<\/p>\n\n<div class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{v =\\sqrt{a_{\\textbf{c}}r}=\\sqrt{(4.63\\times10^6)(0.0750)} = 590~\\textbf{m\/s}}.[\/latex]<\/div>\n<div class=\"equation\" style=\"text-align: center\"><\/div>\n<p id=\"import-auto-id871156\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id2604418\">The given centripetal acceleration is 472,000 times as strong as <em><strong>g<\/strong><\/em>. It is no wonder that such high velocity 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<p id=\"import-auto-id2584043\">Of course, a net external force is needed to cause any acceleration, just as Newton proposed in his second law of motion. So a net external force is needed to cause a centripetal acceleration. In the next section on <a href=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1107\/chapter\/5-3-centripetal-force\/\">centripetal force<\/a>, we will consider the forces involved in circular motion.<\/p>\n\n<div id=\"eip-401\" class=\"note\">\n<div class=\"textbox shaded\">\n<div class=\"note\">\n<h3 class=\"title\">PHET EXPLORATIONS: LADYBUG MOTION 2D<span style=\"text-decoration: underline\">\n<\/span><\/h3>\n<p id=\"eip-id1169738009397\">Learn about position, velocity and acceleration vectors. Move the ladybug by setting the position, velocity or acceleration, and see how the vectors change. Choose linear, circular or elliptical motion, and record and playback the motion to analyze the behavior.<\/p>\n\n<figure id=\"eip-id2886860\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"450\"]<a href=\"\/resources\/215d08d5a1c5b522d8139353a8976529f896768b\/ladybug-motion-2d_en.jar\"><img src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/PhET_Icon-6-1.png\" alt=\"image\" width=\"450\" height=\"147\"><\/a> <strong>Figure 3.<\/strong> <a href=\"https:\/\/phet.colorado.edu\/en\/simulation\/ladybug-motion-2d\">Ladybug Motion in 2D<\/a>[\/caption]<\/figure>\n<\/div>\n<\/div>\n<\/div>\n<section id=\"fs-id2588123\" class=\"section-summary\">\n<h1>Section Summary<\/h1>\n<ul id=\"fs-id2407037\">\n \t<li id=\"import-auto-id1963039\">Centripetal acceleration <strong><em>a<\/em><sub>c<\/sub><\/strong> is the acceleration experienced while in uniform circular motion. It always points toward the center of rotation. It is perpendicular to the linear velocity[latex]\\boldsymbol{v}[\/latex]and has the magnitude<\/li>\n<\/ul>\n<\/section>\n<div id=\"eip-440\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{a_{\\textbf{c}}\\:=}[\/latex][latex size=\"2\"]\\boldsymbol{\\frac{v^2}{r}}[\/latex]<\/div>\n<ul>\n \t<li id=\"import-auto-id2624755\">The unit of centripetal acceleration is <strong>m\/s<sup>2<\/sup><\/strong>.<\/li>\n<\/ul>\n<section id=\"fs-id2654169\" class=\"conceptual-questions\">\n<div id=\"fs-id1348631\" class=\"exercise\">\n<div id=\"fs-id1386165\" class=\"problem\">\n<div class=\"bcc-box bcc-info\">\n<h3>Conceptual Questions<\/h3>\n<strong>1: <\/strong>Can centripetal acceleration change the speed of circular motion? Explain.\n\n<\/div>\n<\/div>\n<\/div>\n<\/section><section id=\"fs-id2601360\" class=\"problems-exercises\">\n<div class=\"bcc-box bcc-info\">\n<h3>Problems &amp; Exercises<\/h3>\n<div id=\"fs-id2617114\" class=\"exercise\">\n<div id=\"fs-id3112760\" class=\"problem\">\n<p id=\"import-auto-id3055430\"><strong>1: <\/strong>A fairground ride spins its occupants inside a flying saucer-shaped container. If the horizontal circular path the riders follow has an 8.00 m radius, at how many revolutions per minute will the riders be subjected to a centripetal acceleration whose magnitude is 1.50 times that due to gravity?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2936845\" class=\"exercise\">\n<div id=\"fs-id3177731\" class=\"problem\">\n\n<strong>2: <\/strong>A runner taking part in the 200 m dash must run around the end of a track that has a circular arc with a radius of curvature of 30 m. If he completes the 200 m dash in 23.2 s and runs at constant speed throughout the race, what is the magnitude of his centripetal acceleration as he runs the curved portion of the track?\n\n<\/div>\n<\/div>\n<div id=\"fs-id3116567\" class=\"exercise\">\n<div id=\"fs-id1488579\" class=\"problem\">\n<p id=\"import-auto-id2209974\"><strong>3: <\/strong>Taking the age of Earth to be about 4 \u00d7 10<sup>9<\/sup> years and assuming its orbital radius of 1.5 \u00d7 10<sup>11<\/sup> has not changed and is circular, calculate the approximate total distance Earth has traveled since its birth (in a frame of reference stationary with respect to the Sun). <em>Recall that the circumference of a circle is [latex]2\\pi r[\/latex].<\/em><\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2688068\" class=\"exercise\">\n<div id=\"fs-id1909383\" class=\"problem\">\n<p id=\"fs-id3354583\"><strong>4: <\/strong>The propeller of a World War II fighter plane is 2.30 m in diameter and the tip of the blade spins at 145 m\/s. What is the centripetal acceleration of the propeller tip under these conditions? Calculate it in meters per second squared and convert to multiples of <em>g<\/em>.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1363119\" class=\"exercise\">\n<div class=\"problem\">\n<p id=\"fs-id1375950\"><strong>5: Integrated Concepts<\/strong><\/p>\n<p id=\"fs-id1911278\">Riders in an amusement park ride shaped like a Viking ship hung from a large pivot are rotated back and forth like a rigid pendulum. Sometime near the middle of the ride, the ship is momentarily motionless at the top of its circular arc. The ship then swings down under the influence of gravity.<\/p>\n<p id=\"fs-id1932647\">(a) Assuming negligible friction, find the speed of the riders at the bottom of its arc, given the system's center of mass travels in an arc having a radius of 14.0 m and the riders are near the center of mass.<\/p>\n<p id=\"fs-id1920371\">(b) What is the centripetal acceleration at the bottom of the arc?<\/p>\n<p id=\"fs-id3454986\">(c) Draw a free body diagram of the forces acting on a rider at the bottom of the arc.<\/p>\n<p id=\"fs-id3079092\">(d) Find the force exerted by the ride on a 60.0 kg rider and compare it to her weight.<\/p>\n<p id=\"eip-id1171510831484\">(e) Discuss whether the answer seems reasonable.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id3170075\" class=\"exercise\">\n<div id=\"fs-id1842848\" class=\"problem\">\n<p id=\"fs-id3034751\"><strong>6: Unreasonable Results<\/strong><\/p>\n<p id=\"fs-id1909775\">A mother pushes her child on a swing so that his speed is 9.00 m\/s at the lowest point of his path. The swing is suspended 2.00 m above the child\u2019s center of mass.<\/p>\n<p id=\"fs-id1471686\">(a) What is the magnitude of the centripetal acceleration of the child at the low point?<\/p>\n(b) What is the magnitude of the force the child exerts on the seat if his mass is 18.0 kg?\n<p id=\"fs-id3006542\">(c) What is unreasonable about these results?<\/p>\n<p id=\"fs-id3004371\">(d) Which premises are unreasonable or inconsistent?<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<div>\n<h2>Glossary<\/h2>\n<dl id=\"import-auto-id2577991\" class=\"definition\">\n \t<dt>centripetal acceleration<\/dt>\n \t<dd id=\"fs-id1549236\">the acceleration of an object moving in a circle, directed toward the center<\/dd>\n<\/dl>\n<dl id=\"import-auto-id2603595\" class=\"definition\">\n \t<dt>ultracentrifuge<\/dt>\n \t<dd id=\"fs-id1587970\">a centrifuge optimized for spinning a rotor at very high speeds<\/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{10.8~\\textbf{m\/s}}[\/latex]\n\n<strong>3: <\/strong>[latex]\\boldsymbol{4\\times10^{21}\\textbf{ m}}[\/latex]\n\n<strong>5: <\/strong>(a) [latex]\\boldsymbol{23.4\\textbf{ m\/s}}[\/latex] (b) [latex]\\boldsymbol{39.1\\textbf{ m\/s}^2}[\/latex] (c)\n<div id=\"fs-id3229252\" class=\"solution\">\n<figure id=\"eip-id1165317522148\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"150\"]<img src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/Figure_07_02_03a-1.jpg\" alt=\"A rectangle with a base longer than the height. A vertical line with arrowheads on both ends passes through the rectangle, bisecting the horizontal sides. The top of the arrow is labeled N, and the bottom is labeled w.\" width=\"150\" height=\"141\"> <strong>Figure 4.<\/strong>[\/caption]<\/figure>\n<p id=\"import-auto-id3137362\">(d) [latex]\\boldsymbol{1.76\\times10^3\\textbf{ N}\\text{ or }3.00\\:\\omega},[\/latex] that is, the normal force (upward) is three times her weight. (e) This answer seems reasonable, since she feels like she\u2019s being forced into the chair MUCH stronger than just by gravity.<\/p>\n\n<\/div>\n<p id=\"import-auto-id2668617\"><strong>6: <\/strong>(a) [latex]\\boldsymbol{40.5\\textbf{ m\/s}^2}[\/latex] (b) 905 N (c) The force in part (b) is very large. The acceleration in part (a) is too much, about 4 g. (d) The speed of the swing is too large. At the given velocity at the bottom of the swing, there is enough kinetic energy to send the child all the way over the top, ignoring friction.<\/p>\n\n<\/div>","rendered":"<div>\n<div class=\"bcc-box bcc-highlight\">\n<h3>Summary<\/h3>\n<div>\n<ul>\n<li>Establish the expression for centripetal acceleration.<\/li>\n<li>Explain the centrifuge.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"import-auto-id3008127\">We know from kinematics that acceleration is a change in velocity, either in its magnitude or in its direction, or both. In uniform circular motion, the direction of the velocity changes constantly, so there is always an associated acceleration, even though the magnitude of the velocity might be constant. You experience this acceleration yourself when you turn a corner in your car. (If you hold the wheel steady during a turn and move at constant speed, you are in uniform circular motion.) What you notice is a sideways acceleration because you and the car are changing direction. The sharper the curve and the greater your speed, the more noticeable this acceleration will become. In this section we examine the direction and magnitude of that acceleration.<\/p>\n<p id=\"import-auto-id2422195\"><a class=\"autogenerated-content\" href=\"#import-auto-id1561888\">Figure 1<\/a> shows an object moving in a circular path at constant speed. The direction of the instantaneous velocity is shown at two points along the path. Acceleration is in the direction of the change in velocity, which points directly toward the center of rotation (the center of the circular path). This pointing is shown with the vector diagram in the figure. 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.\u201d<\/p>\n<figure id=\"import-auto-id1561888\"><figcaption><\/figcaption><figure style=\"width: 330px\" class=\"wp-caption aligncenter center\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2017\/06\/Figure_07_02_01a-1.jpg\" alt=\"The given figure shows a circle, with a triangle having vertices A B C made from the center to the boundry. A is at the center and B and C points are at the circle path. Lines A B and A C act as radii and B C is a chord. Delta theta is shown inside the triangle, and the arc length delta s and the chord length delta r are also given. At point B, velocity of object is shown as v one and at point C, velocity of object is shown as v two. Along the circle an equation is shown as delta v equals v sub 2 minus v sub 1.\" width=\"330\" height=\"395\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 1.<\/strong> The directions of the velocity of an object at two different points are shown, and the change in velocity <strong>\u0394v<\/strong> is seen to point directly toward the center of curvature. (See small inset.) Because <strong>ac = \u0394<em>v<\/em>\/\u0394<em>t<\/em><\/strong>, the acceleration is also toward the center; ac is called centripetal acceleration. (Because<strong> \u0394<em>\u03b8<\/em> <\/strong>is very small, the arc length <strong>\u0394<em>s<\/em><\/strong> is equal to the chord length <strong>\u0394<em>r<\/em><\/strong> for small time differences.)<\/figcaption><\/figure>\n<\/figure>\n<p id=\"import-auto-id723408\">The direction of centripetal acceleration is toward the center of curvature, but what is its magnitude? Note that the triangle formed by the velocity vectors and the one formed by the radii <em><strong>r<\/strong><\/em> and <strong>\u0394<em>s<\/em><\/strong> are similar. Both the triangles ABC and PQR are isosceles triangles (two equal sides). The two equal sides of the velocity vector triangle are the speeds <strong><em>v<\/em><sub>1<\/sub> = <em>v<\/em><sub>2<\/sub> =<em> v<\/em><\/strong>. Using the properties of two similar triangles, we obtain<\/p>\n<div id=\"eip-829\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{\\frac{\\Delta{v}}{v}}[\/latex][latex]\\boldsymbol{=}[\/latex][latex]\\boldsymbol{\\frac{\\Delta{s}}{r}}.[\/latex]<\/div>\n<p id=\"import-auto-id2588045\">Acceleration is <strong>\u0394<em>v<\/em>\/\u0394<em>t<\/em><\/strong>, and so we first solve this expression for <strong>\u0394<em>v<\/em><\/strong>:<\/p>\n<div id=\"eip-0\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{\\Delta{v}\\:=}[\/latex][latex]\\boldsymbol{\\frac{v}{r}}[\/latex][latex]\\boldsymbol{\\Delta{s}}.[\/latex]<\/div>\n<p id=\"import-auto-id2398724\">Then we divide this by <strong>\u0394<em>t<\/em><\/strong>, yielding<\/p>\n<div id=\"eip-593\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{\\frac{\\Delta{v}}{\\Delta{t}}}[\/latex][latex]\\boldsymbol{=}[\/latex][latex]\\boldsymbol{\\frac{v}{r}\\times\\frac{\\Delta{s}}{\\Delta{t}}}.[\/latex]<\/div>\n<p id=\"import-auto-id3489731\">Finally, noting that <strong>\u0394<em>v<\/em>\/\u0394<em>t<\/em> = <em>a<\/em><sub>c<\/sub><\/strong> and that <strong>\u0394<em>s<\/em>\/\u0394<em>t<\/em> = <em>v<\/em><\/strong>, the linear or tangential speed, we see 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 id=\"import-auto-id2429569\">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>. So, centripetal acceleration is greater at high speeds and in sharp curves (smaller radius), as you have noticed when driving a car. But it is a bit surprising that <strong><em>a<\/em><sub>c<\/sub><\/strong> is proportional to speed squared, implying, for example, that it is four times as hard to take a curve at 100 km\/h than at 50 km\/h. A sharp corner has a small radius, so that <strong><em>a<\/em><sub>c<\/sub><\/strong> is greater for tighter turns, as you have probably noticed.<\/p>\n<p id=\"import-auto-id2441429\">A <span id=\"import-auto-id3256561\">centrifuge<\/span> (see <a class=\"autogenerated-content\" href=\"#import-auto-id3033074\">Figure 2<\/a>b) is a rotating device used to separate specimens of different densities. High centripetal acceleration significantly decreases the time it takes for separation to occur, and makes separation possible with small samples. Centrifuges are used in a variety of applications in science and medicine, including the separation of single cell suspensions such as bacteria, viruses, and blood cells from a liquid medium and the separation of macromolecules, such as DNA and protein, from a solution. Centrifuges are often rated in terms of their centripetal acceleration relative to acceleration due to gravity (<em><strong>g<\/strong><\/em>); maximum centripetal acceleration of several hundred thousand <em><strong>g<\/strong><\/em> is possible in a vacuum. Human centrifuges, extremely large centrifuges, have been used to test the tolerance of astronauts to the effects of accelerations larger than that of Earth\u2019s gravity.<\/p>\n<div class=\"textbox shaded\">\n<div id=\"fs-id3386265\" class=\"example\">\n<h3 id=\"import-auto-id3086245\">Example 1: How Does the Centripetal Acceleration of a Car Around a Curve Compare with That Due to Gravity?<\/h3>\n<p>What is the magnitude of the centripetal acceleration of a car following a curve of radius 500 m at a speed of 25.0 m\/s (about 90 km\/h)? Compare the acceleration with that due to gravity for this fairly gentle curve taken at highway speed. See <a class=\"autogenerated-content\" href=\"#import-auto-id3033074\">Figure 2<\/a>(a).<\/p>\n<p id=\"import-auto-id3079956\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id3358891\">Because <em><strong>v<\/strong><\/em> and <em><strong>r<\/strong><\/em> are given, we use the expression [latex]\\boldsymbol{a_{\\textbf{c}}=\\dfrac{v^2}{r}}[\/latex].<\/p>\n<p id=\"import-auto-id1447129\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id2057486\">Entering the given values of <strong><em>v<\/em> = 25.0 m\/s<\/strong> and <strong><em>r<\/em> = 500 m<\/strong> into the first expression for <strong><em>a<\/em><sub>c<\/sub><\/strong> gives<\/p>\n<p style=\"text-align: center\">[latex]\\boldsymbol{a_\\textbf{c}} = \\boldsymbol{\\dfrac{v^2}{r}} = \\boldsymbol{\\dfrac{25.0^2}{500} = 1.25}~\\textbf{m\/s}^2[\/latex]<\/p>\n<div id=\"eip-122\" class=\"equation\" style=\"text-align: center\"><\/div>\n<p id=\"import-auto-id1816308\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id1425882\">To compare this with the acceleration due to gravity (<strong><em>g<\/em> = 9.80 m\/s<sup>2<\/sup><\/strong>), we take the ratio of <strong><em>a<\/em><sub>c<\/sub> \/<em>g<\/em> = (1.25 m\/s<sup>2<\/sup>)\/(9.80m\/s<sup>2<\/sup>) = 0.128<\/strong>. Thus, <strong><em>a<\/em><sub>c <\/sub>= 0.128 <em>g<\/em><\/strong> and is noticeable especially if you were not wearing a seat belt.<\/p>\n<\/div>\n<\/div>\n<figure id=\"import-auto-id3033074\">\n<figure style=\"width: 250px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/Figure_07_02_02a-1.jpg\" alt=\"In figure a, a car shown from top is running on a circular road around a circular path. The center of the park is termed as the center of this circle and the distance from this point to the car is taken as radius r. The linear velocity is shown in perpendicular direction toward the front of the car, shown as v the centripetal acceleration is shown with an arrow pointed towards the center of rotation. In figure b, a centrifuge is shown an object of mass m is rotating in it at a constant speed. The object is at the distance equal to the radius, r, of the centrifuge. The centripetal acceleration is shown towards the center of rotation, and the velocity, v is shown perpendicular to the object in the clockwise direction.\" width=\"250\" height=\"606\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 2.<\/strong> (a) The car following a circular path at constant speed is accelerated perpendicular to its velocity, as shown. The magnitude of this centripetal acceleration is found in <a href=\"#import-auto-id3086245\">Example 1<\/a>. (b) A particle of mass in a centrifuge is rotating at constant angular velocity . It must be accelerated perpendicular to its velocity or it would continue in a straight line. The magnitude of the necessary acceleration is found in <a href=\"#fs-id2598952\">Example 2<\/a>.<\/figcaption><\/figure>\n<\/figure>\n<div class=\"textbox shaded\">\n<div id=\"fs-id2598952\" class=\"example\">\n<h3><a id=\"ca_ex2\" href=\"\"><\/a>Example 2: What is the speed of an Ultracentrifuge?<\/h3>\n<p>Calculate the speed of a point 7.50 cm from the axis of a spinning <strong><span id=\"import-auto-id2449410\">ultracentrifuge<\/span> <\/strong>if the centripetal acceleration at that point is [latex]\\boldsymbol{4.63\\times10^6}~\\textbf{m\/s}^2[\/latex]<\/p>\n<p id=\"import-auto-id3105981\"><strong>Strategy<\/strong><\/p>\n<p>Both <em><strong>r<\/strong><\/em> and <em><strong>a<sub>c<\/sub> <\/strong><\/em>are given, so we can use the equation [latex]\\boldsymbol{a_{\\textbf{c}} = \\dfrac{v^2}{r}}[\/latex] to determine the speed of the centrifuge.<\/p>\n<p id=\"import-auto-id1140598\"><strong>Solution<\/strong><\/p>\n<p>First we solve for the speed of the centrifugre<\/p>\n<p style=\"text-align: center\">[latex]\\boldsymbol{a_{\\textbf{c}} = \\dfrac{v^2}{r} \\rightarrow v =\\sqrt{a_{\\textbf{c}}r}}[\/latex]<\/p>\n<div class=\"equation\" style=\"text-align: center\"><\/div>\n<p id=\"import-auto-id1926483\">Converting 7.50 cm to meters and substituting known values gives<\/p>\n<div class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{v =\\sqrt{a_{\\textbf{c}}r}=\\sqrt{(4.63\\times10^6)(0.0750)} = 590~\\textbf{m\/s}}.[\/latex]<\/div>\n<div class=\"equation\" style=\"text-align: center\"><\/div>\n<p id=\"import-auto-id871156\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id2604418\">The given centripetal acceleration is 472,000 times as strong as <em><strong>g<\/strong><\/em>. It is no wonder that such high velocity 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<p id=\"import-auto-id2584043\">Of course, a net external force is needed to cause any acceleration, just as Newton proposed in his second law of motion. So a net external force is needed to cause a centripetal acceleration. In the next section on <a href=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1107\/chapter\/5-3-centripetal-force\/\">centripetal force<\/a>, we will consider the forces involved in circular motion.<\/p>\n<div id=\"eip-401\" class=\"note\">\n<div class=\"textbox shaded\">\n<div class=\"note\">\n<h3 class=\"title\">PHET EXPLORATIONS: LADYBUG MOTION 2D<span style=\"text-decoration: underline\"><br \/>\n<\/span><\/h3>\n<p id=\"eip-id1169738009397\">Learn about position, velocity and acceleration vectors. Move the ladybug by setting the position, velocity or acceleration, and see how the vectors change. Choose linear, circular or elliptical motion, and record and playback the motion to analyze the behavior.<\/p>\n<figure id=\"eip-id2886860\">\n<figure style=\"width: 450px\" class=\"wp-caption aligncenter\"><a href=\"\/resources\/215d08d5a1c5b522d8139353a8976529f896768b\/ladybug-motion-2d_en.jar\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/PhET_Icon-6-1.png\" alt=\"image\" width=\"450\" height=\"147\" \/><\/a><figcaption class=\"wp-caption-text\"><strong>Figure 3.<\/strong> <a href=\"https:\/\/phet.colorado.edu\/en\/simulation\/ladybug-motion-2d\">Ladybug Motion in 2D<\/a><\/figcaption><\/figure>\n<\/figure>\n<\/div>\n<\/div>\n<\/div>\n<section id=\"fs-id2588123\" class=\"section-summary\">\n<h1>Section Summary<\/h1>\n<ul id=\"fs-id2407037\">\n<li id=\"import-auto-id1963039\">Centripetal acceleration <strong><em>a<\/em><sub>c<\/sub><\/strong> is the acceleration experienced while in uniform circular motion. It always points toward the center of rotation. It is perpendicular to the linear velocity[latex]\\boldsymbol{v}[\/latex]and has the magnitude<\/li>\n<\/ul>\n<\/section>\n<div id=\"eip-440\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{a_{\\textbf{c}}\\:=}[\/latex][latex]\\boldsymbol{\\frac{v^2}{r}}[\/latex]<\/div>\n<ul>\n<li id=\"import-auto-id2624755\">The unit of centripetal acceleration is <strong>m\/s<sup>2<\/sup><\/strong>.<\/li>\n<\/ul>\n<section id=\"fs-id2654169\" class=\"conceptual-questions\">\n<div id=\"fs-id1348631\" class=\"exercise\">\n<div id=\"fs-id1386165\" class=\"problem\">\n<div class=\"bcc-box bcc-info\">\n<h3>Conceptual Questions<\/h3>\n<p><strong>1: <\/strong>Can centripetal acceleration change the speed of circular motion? Explain.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id2601360\" class=\"problems-exercises\">\n<div class=\"bcc-box bcc-info\">\n<h3>Problems &amp; Exercises<\/h3>\n<div id=\"fs-id2617114\" class=\"exercise\">\n<div id=\"fs-id3112760\" class=\"problem\">\n<p id=\"import-auto-id3055430\"><strong>1: <\/strong>A fairground ride spins its occupants inside a flying saucer-shaped container. If the horizontal circular path the riders follow has an 8.00 m radius, at how many revolutions per minute will the riders be subjected to a centripetal acceleration whose magnitude is 1.50 times that due to gravity?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2936845\" class=\"exercise\">\n<div id=\"fs-id3177731\" class=\"problem\">\n<p><strong>2: <\/strong>A runner taking part in the 200 m dash must run around the end of a track that has a circular arc with a radius of curvature of 30 m. If he completes the 200 m dash in 23.2 s and runs at constant speed throughout the race, what is the magnitude of his centripetal acceleration as he runs the curved portion of the track?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id3116567\" class=\"exercise\">\n<div id=\"fs-id1488579\" class=\"problem\">\n<p id=\"import-auto-id2209974\"><strong>3: <\/strong>Taking the age of Earth to be about 4 \u00d7 10<sup>9<\/sup> years and assuming its orbital radius of 1.5 \u00d7 10<sup>11<\/sup> has not changed and is circular, calculate the approximate total distance Earth has traveled since its birth (in a frame of reference stationary with respect to the Sun). <em>Recall that the circumference of a circle is [latex]2\\pi r[\/latex].<\/em><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2688068\" class=\"exercise\">\n<div id=\"fs-id1909383\" class=\"problem\">\n<p id=\"fs-id3354583\"><strong>4: <\/strong>The propeller of a World War II fighter plane is 2.30 m in diameter and the tip of the blade spins at 145 m\/s. What is the centripetal acceleration of the propeller tip under these conditions? Calculate it in meters per second squared and convert to multiples of <em>g<\/em>.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1363119\" class=\"exercise\">\n<div class=\"problem\">\n<p id=\"fs-id1375950\"><strong>5: Integrated Concepts<\/strong><\/p>\n<p id=\"fs-id1911278\">Riders in an amusement park ride shaped like a Viking ship hung from a large pivot are rotated back and forth like a rigid pendulum. Sometime near the middle of the ride, the ship is momentarily motionless at the top of its circular arc. The ship then swings down under the influence of gravity.<\/p>\n<p id=\"fs-id1932647\">(a) Assuming negligible friction, find the speed of the riders at the bottom of its arc, given the system&#8217;s center of mass travels in an arc having a radius of 14.0 m and the riders are near the center of mass.<\/p>\n<p id=\"fs-id1920371\">(b) What is the centripetal acceleration at the bottom of the arc?<\/p>\n<p id=\"fs-id3454986\">(c) Draw a free body diagram of the forces acting on a rider at the bottom of the arc.<\/p>\n<p id=\"fs-id3079092\">(d) Find the force exerted by the ride on a 60.0 kg rider and compare it to her weight.<\/p>\n<p id=\"eip-id1171510831484\">(e) Discuss whether the answer seems reasonable.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id3170075\" class=\"exercise\">\n<div id=\"fs-id1842848\" class=\"problem\">\n<p id=\"fs-id3034751\"><strong>6: Unreasonable Results<\/strong><\/p>\n<p id=\"fs-id1909775\">A mother pushes her child on a swing so that his speed is 9.00 m\/s at the lowest point of his path. The swing is suspended 2.00 m above the child\u2019s center of mass.<\/p>\n<p id=\"fs-id1471686\">(a) What is the magnitude of the centripetal acceleration of the child at the low point?<\/p>\n<p>(b) What is the magnitude of the force the child exerts on the seat if his mass is 18.0 kg?<\/p>\n<p id=\"fs-id3006542\">(c) What is unreasonable about these results?<\/p>\n<p id=\"fs-id3004371\">(d) Which premises are unreasonable or inconsistent?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<div>\n<h2>Glossary<\/h2>\n<dl id=\"import-auto-id2577991\" class=\"definition\">\n<dt>centripetal acceleration<\/dt>\n<dd id=\"fs-id1549236\">the acceleration of an object moving in a circle, directed toward the center<\/dd>\n<\/dl>\n<dl id=\"import-auto-id2603595\" class=\"definition\">\n<dt>ultracentrifuge<\/dt>\n<dd id=\"fs-id1587970\">a centrifuge optimized for spinning a rotor at very high speeds<\/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{10.8~\\textbf{m\/s}}[\/latex]<\/p>\n<p><strong>3: <\/strong>[latex]\\boldsymbol{4\\times10^{21}\\textbf{ m}}[\/latex]<\/p>\n<p><strong>5: <\/strong>(a) [latex]\\boldsymbol{23.4\\textbf{ m\/s}}[\/latex] (b) [latex]\\boldsymbol{39.1\\textbf{ m\/s}^2}[\/latex] (c)<\/p>\n<div id=\"fs-id3229252\" class=\"solution\">\n<figure id=\"eip-id1165317522148\">\n<figure style=\"width: 150px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/Figure_07_02_03a-1.jpg\" alt=\"A rectangle with a base longer than the height. A vertical line with arrowheads on both ends passes through the rectangle, bisecting the horizontal sides. The top of the arrow is labeled N, and the bottom is labeled w.\" width=\"150\" height=\"141\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 4.<\/strong><\/figcaption><\/figure>\n<\/figure>\n<p id=\"import-auto-id3137362\">(d) [latex]\\boldsymbol{1.76\\times10^3\\textbf{ N}\\text{ or }3.00\\:\\omega},[\/latex] that is, the normal force (upward) is three times her weight. (e) This answer seems reasonable, since she feels like she\u2019s being forced into the chair MUCH stronger than just by gravity.<\/p>\n<\/div>\n<p id=\"import-auto-id2668617\"><strong>6: <\/strong>(a) [latex]\\boldsymbol{40.5\\textbf{ m\/s}^2}[\/latex] (b) 905 N (c) The force in part (b) is very large. The acceleration in part (a) is too much, about 4 g. (d) The speed of the swing is too large. At the given velocity at the bottom of the swing, there is enough kinetic energy to send the child all the way over the top, ignoring friction.<\/p>\n<\/div>\n","protected":false},"author":9,"menu_order":1,"comment_status":"closed","ping_status":"closed","template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-300","chapter","type-chapter","status-publish","hentry"],"part":295,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-json\/pressbooks\/v2\/chapters\/300","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-json\/wp\/v2\/users\/9"}],"replies":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-json\/wp\/v2\/comments?post=300"}],"version-history":[{"count":1,"href":"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-json\/pressbooks\/v2\/chapters\/300\/revisions"}],"predecessor-version":[{"id":301,"href":"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-json\/pressbooks\/v2\/chapters\/300\/revisions\/301"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-json\/pressbooks\/v2\/parts\/295"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-json\/pressbooks\/v2\/chapters\/300\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-json\/wp\/v2\/media?parent=300"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-json\/pressbooks\/v2\/chapter-type?post=300"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-json\/wp\/v2\/contributor?post=300"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-json\/wp\/v2\/license?post=300"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}