{"id":484,"date":"2020-01-13T15:11:20","date_gmt":"2020-01-13T20:11:20","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/douglasphys1108\/chapter\/9-2-angular-acceleration\/"},"modified":"2020-11-07T09:32:26","modified_gmt":"2020-11-07T14:32:26","slug":"9-2-angular-acceleration","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/douglasphys1108\/chapter\/9-2-angular-acceleration\/","title":{"raw":"9.2 Angular Acceleration","rendered":"9.2 Angular Acceleration"},"content":{"raw":"<div>\n<div class=\"bcc-box bcc-highlight\">\n<h3>Summary<\/h3>\n<div>\n<ul>\n \t<li>Describe uniform circular motion.<\/li>\n \t<li>Explain non-uniform circular motion.<\/li>\n \t<li>Calculate angular acceleration of an object.<\/li>\n \t<li>Observe the link between linear and angular acceleration.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"import-auto-id2400941\"><a href=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1108\/part\/chapter-6-uniform-circular-motion-and-gravitation\/\">Chapter 5 Uniform Circular Motion and Gravitation<\/a> discussed only uniform circular motion, which is motion in a circle at constant speed and, hence, constant angular velocity. Recall that angular velocity\u00a0<strong>\u03c9<\/strong> was defined as the time rate of change of angle <em><strong>\u03b8<\/strong><\/em>:<\/p>\n\n<div id=\"eip-257\" 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-id2970170\">where <em><strong>\u03b8<\/strong><\/em> is the angle of rotation as seen in <a class=\"autogenerated-content\" href=\"#import-auto-id1941476\">Figure 1<\/a>. The relationship between angular velocity <strong>\u03c9<\/strong> and linear velocity <em><strong>v<\/strong><\/em> was also defined in <a href=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1108\/chapter\/9-1-rotation-angle-and-angular-velocity\/\">Chapter 9.1 Rotation Angle and Angular Velocity<\/a> as<\/p>\n\n<div class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{v=r\\omega}[\/latex]<\/div>\n<p id=\"eip-179\">or<\/p>\n\n<div id=\"eip-363\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{\\omega\\:=}[\/latex][latex size=\"2\"]\\boldsymbol{\\frac{v}{r}},[\/latex]<\/div>\n<p id=\"import-auto-id2655439\">where <em><strong>r<\/strong><\/em> is the radius of curvature, also seen in <a class=\"autogenerated-content\" href=\"#import-auto-id1941476\">Figure 1<\/a>. According to the sign convention, the counter clockwise direction is considered as positive direction and clockwise direction as negative<\/p>\n\n<figure id=\"import-auto-id1941476\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"200\"]<img src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1107\/wp-content\/uploads\/sites\/1184\/2017\/09\/Figure_11_01_01a-1.jpg\" alt=\"The given figure shows counterclockwise circular motion with a horizontal line, depicting radius r, drawn from the center of the circle to the right side on its circumference and another line is drawn in such a manner that it makes an acute angle delta theta with the horizontal line. Tangential velocity vectors are indicated at the end of the two lines. At the bottom right side of the figure, the formula for angular velocity is given as v upon r.\" width=\"200\" height=\"302\"> <strong>Figure 1.<\/strong> This figure shows uniform circular motion and some of its defined quantities.[\/caption]<\/figure>\n<p id=\"import-auto-id2670067\">Angular velocity is not constant when a skater pulls in her arms, when a child starts up a merry-go-round from rest, or when a computer\u2019s hard disk slows to a halt when switched off. In all these cases, there is an<strong> <span id=\"import-auto-id2438108\">angular acceleration<\/span><\/strong>, in which <strong>\u03c9<\/strong> changes. The faster the change occurs, the greater the angular acceleration. Angular acceleration [latex]\\boldsymbol{\\alpha}[\/latex] is defined as the rate of change of angular velocity. In equation form, angular acceleration is expressed as follows:<\/p>\n\n<div id=\"eip-974\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{\\alpha\\:=}[\/latex][latex size=\"2\"]\\boldsymbol{\\frac{\\Delta\\omega}{\\Delta{t}}},[\/latex]<\/div>\n<p id=\"import-auto-id2403123\">where\u00a0<strong>\u0394\u03c9<\/strong> is the <span id=\"import-auto-id3406201\"> change in angular velocity<\/span> and <strong>\u0394<em>t<\/em><\/strong> is the change in time. The units of angular acceleration are (<strong>rad\/s)\/s<\/strong>, or <strong>rad\/s<sup>2<\/sup><\/strong>. If <strong>\u03c9<\/strong> increases, then [latex]\\boldsymbol{\\alpha}[\/latex] is positive. If <strong>\u03c9<\/strong> decreases, then [latex]\\boldsymbol{\\alpha}[\/latex] is negative.<\/p>\n\n<div class=\"textbox shaded\">\n<div id=\"fs-id3159590\" class=\"example\">\n<h3 id=\"import-auto-id3385431\">Example 1: Calculating the Angular Acceleration and Deceleration of a Bike Wheel<\/h3>\nSuppose a teenager puts her bicycle on its back and starts the rear wheel spinning from rest to a final angular velocity of 250 rpm in 5.00 s. (a) Calculate the angular acceleration in <strong>rad\/s<sup>2<\/sup><\/strong>. (b) If she now slams on the brakes, causing an angular acceleration of <strong>-87.3 rad\/s<sup>2<\/sup><\/strong>, how long does it take the wheel to stop?\n<p id=\"import-auto-id1954754\"><strong>Strategy for (a)<\/strong><\/p>\n<p id=\"import-auto-id1917815\">The angular acceleration can be found directly from its definition in [latex]\\boldsymbol{\\alpha=\\frac{\\Delta\\omega}{\\Delta{t}}}[\/latex] because the final angular velocity and time are given. We see that <strong>\u0394\u03c9<\/strong> is 250 rpm and <strong>\u0394<em>t<\/em><\/strong> is 5.00 s.<\/p>\n<p id=\"import-auto-id1840465\"><strong>Solution for (a)<\/strong><\/p>\n<p id=\"import-auto-id2930128\">Entering known information into the definition of angular acceleration, we get<\/p>\n\n<div id=\"eip-272\" class=\"equation\" style=\"text-align: center\">[latex]\\begin{array}{lcl} \\boldsymbol{\\alpha} &amp; \\boldsymbol{=} &amp; \\boldsymbol{\\frac{\\Delta\\omega}{\\Delta{t}}} \\\\ {} &amp; \\boldsymbol{=} &amp; \\boldsymbol{\\frac{250\\textbf{ rpm}}{5.00\\textbf{ s.}}} \\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id3418450\">Because <strong>\u0394\u03c9<\/strong> is in revolutions per minute (rpm) and we want the standard units of <strong>rad\/s<sup>2<\/sup><\/strong> for angular acceleration, we need to convert <strong>\u0394\u03c9<\/strong> from rpm to rad\/s:<\/p>\n\n<div class=\"equation\" style=\"text-align: center\">[latex]\\begin{array}{lcl} \\boldsymbol{\\Delta\\omega} &amp; \\boldsymbol{=} &amp; \\boldsymbol{250\\frac{\\textbf{rev}}{\\textbf{min}}\\cdotp\\frac{2\\pi\\textbf{ rad}}{\\textbf{rev}}\\cdotp\\frac{1\\textbf{ min}}{60\\textbf{ sec}}} \\\\ {} &amp; \\boldsymbol{=} &amp; \\boldsymbol{26.2\\textbf{ rads.}} \\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id3358717\">Entering this quantity into the expression for [latex]\\boldsymbol{\\alpha}[\/latex], we get<\/p>\n\n<div id=\"eip-899\" class=\"equation\" style=\"text-align: center\">[latex]\\begin{array}{lcl} \\boldsymbol{\\alpha} &amp; \\boldsymbol{=} &amp; \\boldsymbol{\\frac{\\Delta\\omega}{\\Delta{t}}} \\\\ {} &amp; \\boldsymbol{=} &amp; \\boldsymbol{\\frac{26.2\\textbf{ rad\/s}}{5.00\\textbf{ s}}} \\\\ {} &amp; \\boldsymbol{=} &amp; \\boldsymbol{5.24\\textbf{ rad\/s.}^2} \\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id2625933\"><strong>Strategy for (b)<\/strong><\/p>\n<p id=\"import-auto-id2438018\">In this part, we know the angular acceleration and the initial angular velocity. We can find the stoppage time by using the definition of angular acceleration and solving for <strong>\u0394<em>t<\/em><\/strong>, yielding<\/p>\n\n<div id=\"eip-273\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{\\Delta{t}\\:=}[\/latex][latex size=\"2\"]\\boldsymbol{\\frac{\\Delta\\omega}{\\alpha}}.[\/latex]<\/div>\n<p id=\"import-auto-id2591053\"><strong>Solution for (b)<\/strong><\/p>\nHere the angular velocity decreases from <strong>26.2 rad\/s<\/strong> (250 rpm) to zero, so that <strong>\u0394\u03c9<\/strong> is <strong>-26.2 rad\/s<\/strong>, and [latex]\\boldsymbol{\\alpha}[\/latex] is given to be <strong>-87.3 rad\/s<sup>2<\/sup><\/strong>. Thus,\n<div id=\"eip-455\" class=\"equation\" style=\"text-align: center\">[latex]\\begin{array}{lcl} \\boldsymbol{\\Delta{t}} &amp; \\boldsymbol{=} &amp; \\boldsymbol{\\frac{-26.2\\textbf{ rad\/s}}{-87.3\\textbf{ rad\/s}^2}} \\\\ {} &amp; \\boldsymbol{=} &amp; \\boldsymbol{0.300\\textbf{ s.}} \\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id3025168\"><strong>Discussion<\/strong><\/p>\nNote that the angular acceleration as the girl spins the wheel is small and positive; it takes 5 s to produce an appreciable angular velocity. When she hits the brake, the angular acceleration is large and negative. The angular velocity quickly goes to zero. In both cases, the relationships are analogous to what happens with linear motion. For example, there is a large deceleration when you crash into a brick wall\u2014the velocity change is large in a short time interval.\n\n<\/div>\n<\/div>\n<p id=\"import-auto-id2595516\">If the bicycle in the preceding example had been on its wheels instead of upside-down, it would first have accelerated along the ground and then come to a stop. This connection between circular motion and linear motion needs to be explored. For example, it would be useful to know how linear and angular acceleration are related. In circular motion, linear acceleration is <em>tangent<\/em> to the circle at the point of interest, as seen in <a class=\"autogenerated-content\" href=\"#import-auto-id1019355\">Figure 2<\/a>. Thus, linear acceleration is called <span id=\"import-auto-id1997895\">tangential acceleration <strong><em>a<\/em><\/strong><\/span><strong><sub>t<\/sub><\/strong>.<\/p>\n\n<figure id=\"import-auto-id1019355\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"315\"]<img src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1107\/wp-content\/uploads\/sites\/1184\/2020\/11\/Figure_11_01_02a-1.jpg\" alt=\"In the figure, a semicircle is drawn, with its radius r, shown here as a line segment. The anti-clockwise motion of the circle is shown with an arrow on the path of the circle. Tangential velocity vector, v, of the point, which is on the meeting point of radius with the circle, is shown as a green arrow and the linear acceleration, a-t is shown as a yellow arrow in the same direction along v.\" width=\"315\" height=\"200\"> <strong>Figure 2.<\/strong> In circular motion, linear acceleration <em><strong>a<\/strong><\/em>, occurs as the magnitude of the velocity changes: <em><strong>a<\/strong><\/em> is tangent to the motion. In the context of circular motion, linear acceleration is also called tangential acceleration <strong><em>a<\/em><sub>t<\/sub><\/strong>.[\/caption]<\/figure>\nLinear or tangential acceleration refers to changes in the magnitude of velocity but not its direction. We know from uniform circular motion that in circular motion centripetal acceleration, <span id=\"import-auto-id1997895\"><strong><em>a<\/em><\/strong><\/span><strong><sub>c<\/sub><\/strong>, refers to changes in the direction of the velocity but not its magnitude. An object undergoing circular motion experiences centripetal acceleration, as seen in <a class=\"autogenerated-content\" href=\"#import-auto-id1995872\">Figure 3<\/a>. Thus, <span id=\"import-auto-id1997895\"><strong><em>a<\/em><\/strong><\/span><strong><sub>t<\/sub><\/strong> and <span id=\"import-auto-id1997895\"><strong><em>a<\/em><\/strong><\/span><strong><sub>c<\/sub><\/strong> are perpendicular and independent of one another. Tangential acceleration <span id=\"import-auto-id1997895\"><strong><em>a<\/em><\/strong><\/span><strong><sub>t<\/sub><\/strong> is directly related to the angular acceleration [latex]\\boldsymbol{\\alpha}[\/latex] and is linked to an increase or decrease in the velocity, but not its direction.\n<figure id=\"import-auto-id1995872\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"350\"]<img src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1107\/wp-content\/uploads\/sites\/1184\/2020\/11\/Figure_11_01_03a-1.jpg\" alt=\"In the figure, a semicircle is drawn, with its radius r, shown here as a line segment. The anti-clockwise motion of the circle is shown with an arrow on the path of the circle. Tangential velocity vector, v, of the point, which is on the meeting point of radius with the circle, is shown as a green arrow and the linear acceleration, a sub t is shown as a yellow arrow in the same direction along v. The centripetal acceleration, a sub c, is also shown as a yellow arrow drawn perpendicular to a sub t, toward the direction of the center of the circle. A label in the figures states a sub t affects magnitude and a sub c affects direction.\" width=\"350\" height=\"200\"> <strong>Figure 3.<\/strong> Centripetal acceleration <strong><em>a<\/em><sub>c<\/sub><\/strong> occurs as the direction of velocity changes; it is perpendicular to the circular motion. Centripetal and tangential acceleration are thus perpendicular to each other.[\/caption]<\/figure>\n<p id=\"import-auto-id1993750\">Now we can find the exact relationship between linear acceleration <span id=\"import-auto-id1997895\"><strong><em>a<\/em><\/strong><\/span><strong><sub>t<\/sub><\/strong> and angular acceleration [latex]\\boldsymbol{\\alpha}[\/latex]. Because linear acceleration is proportional to a change in the magnitude of the velocity, it is defined (as it was in <a href=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1108\/part\/chapter-2-one-dimensional-kinematics\/\">Chapter 2 One-Dimensional Kinematics<\/a>) to be<\/p>\n\n<div id=\"eip-85\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{a_{\\textbf{t}}\\:=}[\/latex][latex size=\"2\"]\\boldsymbol{\\frac{\\Delta{v}}{\\Delta{t}}.}[\/latex]<\/div>\n<p id=\"import-auto-id2407460\">For circular motion, note that <strong><em>v<\/em>=<em>r<\/em>\u03c9<\/strong>, so that<\/p>\n\n<div class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{a_{\\textbf{t}}\\:=}[\/latex][latex size=\"2\"]\\boldsymbol{\\frac{\\Delta(r\\omega)}{\\Delta{t}}.}[\/latex]<\/div>\n<p id=\"import-auto-id2660038\">The radius <strong><em>r<\/em><\/strong> is constant for circular motion, and so <strong>\u0394(<em>r<\/em>\u03c9)=<em>r<\/em>(\u0394\u03c9)<\/strong>. Thus,<\/p>\n\n<div id=\"eip-688\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{a_{\\textbf{t}}=r}[\/latex][latex size=\"2\"]\\boldsymbol{\\frac{\\Delta\\omega}{\\Delta{t}}.}[\/latex]<\/div>\n<p id=\"import-auto-id1861378\">By definition, [latex]\\boldsymbol{\\alpha=\\frac{\\Delta\\omega}{\\Delta{t}}}.[\/latex] Thus,<\/p>\n\n<div class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{a_{\\textbf{t}}=r\\alpha},[\/latex]<\/div>\nor\n<div class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{\\alpha\\:=}[\/latex][latex size=\"2\"]\\boldsymbol{\\frac{a_{\\textbf{t}}}{r}.}[\/latex]<\/div>\n<p id=\"import-auto-id1977486\">These equations mean that linear acceleration and angular acceleration are directly proportional. The greater the angular acceleration is, the larger the linear (tangential) acceleration is, and vice versa. For example, the greater the angular acceleration of a car\u2019s drive wheels, the greater the acceleration of the car. The radius also matters. For example, the smaller a wheel, the smaller its linear acceleration for a given angular acceleration [latex]\\boldsymbol{\\alpha}[\/latex].<\/p>\n\n<div class=\"textbox shaded\">\n<div id=\"fs-id3217117\" class=\"example\">\n<h3 class=\"title\">Example 2: Calculating the Angular Acceleration of a Motorcycle Wheel<\/h3>\n<p id=\"import-auto-id3008868\">A powerful motorcycle can accelerate from 0 to 30.0 m\/s (about 108 km\/h) in 4.20 s. What is the angular acceleration of its 0.320-m-radius wheels? (See <a class=\"autogenerated-content\" href=\"#import-auto-id2415283\">Figure 4<\/a>.)<\/p>\n\n<figure>\n\n[caption id=\"\" align=\"aligncenter\" width=\"225\"]<img src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1107\/wp-content\/uploads\/sites\/1184\/2020\/11\/Figure_11_01_04a-1.jpg\" alt=\"The figure shows the right side view of a man riding a motorcycle hence, depicting linear acceleration a of the motorcycle pointing toward the front of the bike as a horizontal arrow and the angular acceleration alpha of its wheels, shown here as curved arrows along the front of both the wheels pointing downward.\" width=\"225\" height=\"236\"> <strong>Figure 4<\/strong>. The linear acceleration of a motorcycle is accompanied by an angular acceleration of its wheels.[\/caption]<\/figure>\n<p id=\"import-auto-id3068565\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id3149756\">We are given information about the linear velocities of the motorcycle. Thus, we can find its linear acceleration <span id=\"import-auto-id1997895\"><strong><em>a<\/em><\/strong><\/span><strong><sub>t<\/sub><\/strong>. Then, the expression [latex]\\boldsymbol{\\alpha=\\frac{a_{\\textbf{t}}}{r}}[\/latex] can be used to find the angular acceleration.<\/p>\n<p id=\"import-auto-id2451305\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id2577576\">The linear acceleration is<\/p>\n\n<div id=\"eip-20\" class=\"equation\" style=\"text-align: center\">[latex]\\begin{array}{lcl} \\boldsymbol{a_{\\textbf{t}}} &amp; \\boldsymbol{=} &amp; \\boldsymbol{\\frac{\\Delta{v}}{\\Delta{t}}} \\\\ {} &amp; \\boldsymbol{=} &amp; \\boldsymbol{\\frac{30.0\\textbf{ m\/s}}{4.20\\textbf{ s}}} \\\\ {} &amp; \\boldsymbol{=} &amp; \\boldsymbol{7.14\\textbf{ m\/s}^2.} \\end{array}[\/latex]<\/div>\nWe also know the radius of the wheels. Entering the values for <span id=\"import-auto-id1997895\"><strong><em>a<\/em><\/strong><\/span><strong><sub>t<\/sub><\/strong> and <em><strong>r<\/strong><\/em> into [latex]\\boldsymbol{\\alpha=\\frac{a_{\\textbf{t}}}{r}},[\/latex] we get\n<div id=\"eip-155\" class=\"equation\" style=\"text-align: center\">[latex]\\begin{array}{lcl} \\boldsymbol{\\alpha} &amp; \\boldsymbol{=} &amp; \\boldsymbol{\\frac{a_{\\textbf{t}}}{r}} \\\\ {} &amp; \\boldsymbol{=} &amp; \\boldsymbol{\\frac{7.14\\textbf{ m\/s}}{20.320\\textbf{ m}}} \\\\ {} &amp; \\boldsymbol{=} &amp; \\boldsymbol{22.3\\textbf{ rad\/s}^2.} \\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id3062550\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id2017158\">Units of radians are dimensionless and appear in any relationship between angular and linear quantities.<\/p>\n\n<\/div>\n<\/div>\n<p id=\"import-auto-id1117830\">So far, we have defined three rotational quantities\u2014 <em><strong>\u03b8<\/strong><\/em>, <strong>\u03c9<\/strong>, and [latex]\\boldsymbol{\\alpha}[\/latex]. These quantities are analogous to the translational quantities <em><strong>x<\/strong><\/em>, <em><strong>v<\/strong><\/em>, and <em><strong>a<\/strong><\/em>. <a class=\"autogenerated-content\" href=\"#import-auto-id1572984\">Table 1<\/a> displays rotational quantities, the analogous translational quantities, and the relationships between them.<\/p>\n\n<table id=\"import-auto-id1572984\" summary=\"Rotational and Translational Quantities\">\n<thead>\n<tr>\n<th>Rotational<\/th>\n<th>Translational<\/th>\n<th>Relationship<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]\\boldsymbol{\\theta}[\/latex]<\/td>\n<td>[latex]\\boldsymbol{x}[\/latex]<\/td>\n<td>[latex]\\boldsymbol{\\theta=\\frac{x}{r}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\boldsymbol{\\omega}[\/latex]<\/td>\n<td>[latex]\\boldsymbol{v}[\/latex]<\/td>\n<td>[latex]\\boldsymbol{\\omega=\\frac{v}{r}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\boldsymbol{\\alpha}[\/latex]<\/td>\n<td>[latex]\\boldsymbol{a}[\/latex]<\/td>\n<td>[latex]\\boldsymbol{\\alpha=\\frac{a_{\\textbf{t}}}{r}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<tbody>\n<tr>\n<td colspan=\"3\"><strong>Table 1.<\/strong> Rotational and Translational Quantities.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id3292730\" class=\"note\">\n<div class=\"textbox shaded\">\n<div class=\"note\">\n<h3 class=\"title\">MAKING CONNECTIONS: TAKE-HOME EXPERIMENT<span style=\"text-decoration: underline\">\n<\/span><\/h3>\n<p id=\"import-auto-id1909409\">Sit down with your feet on the ground on a chair that rotates. Lift one of your legs such that it is unbent (straightened out). Using the other leg, begin to rotate yourself by pushing on the ground. Stop using your leg to push the ground but allow the chair to rotate. From the origin where you began, sketch the angle, angular velocity, and angular acceleration of your leg as a function of time in the form of three separate graphs. Estimate the magnitudes of these quantities.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1870686\" class=\"exercise\">\n<div class=\"bcc-box bcc-info\">\n<h3>Check Your Understanding<\/h3>\n<div class=\"exercise\">\n<div id=\"fs-id3093647\" class=\"problem\">\n<p id=\"import-auto-id1945320\"><strong>1:<\/strong> Angular acceleration is a vector, having both magnitude and direction. How do we denote its magnitude and direction? Illustrate with an example.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3 class=\"title\">PHET EXPLORATIONS: LADYBUG REVOLUTION<\/h3>\n<p id=\"eip-id1325598\">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<figure id=\"eip-id2624753\">\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-23-1.png\" alt=\"image\" width=\"450\" height=\"147\"><\/a> Figure 5. <a href=\"https:\/\/phet.colorado.edu\/en\/simulation\/rotation\">Ladybug Revolution<\/a>[\/caption]<\/figure>\n<\/div>\n<\/div>\n<section id=\"fs-id3034915\" class=\"section-summary\">\n<h1>Section Summary<\/h1>\n<ul id=\"fs-id1947000\">\n \t<li id=\"import-auto-id2443525\">Uniform circular motion is the motion with a constant angular velocity [latex]\\boldsymbol{\\omega=\\frac{\\Delta\\theta}{\\Delta{t}}}.[\/latex]<\/li>\n \t<li id=\"import-auto-id3353297\">In non-uniform circular motion, the velocity changes with time and the rate of change of angular velocity (i.e. angular acceleration) is [latex]\\boldsymbol{\\alpha=\\frac{\\Delta\\omega}{\\Delta{t}}}.[\/latex]<\/li>\n \t<li id=\"import-auto-id1867184\">Linear or tangential acceleration refers to changes in the magnitude of velocity but not its direction, given as [latex]\\boldsymbol{a_{\\textbf{t}}=\\frac{\\Delta{v}}{\\Delta{t}}}.[\/latex]<\/li>\n \t<li id=\"import-auto-id3035395\">For circular motion, note that <strong><em>v<\/em>=<em>r<\/em>\u03c9<\/strong>, so that\n<div id=\"import-auto-id1588138\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{a_{\\textbf{t}}\\:=}[\/latex][latex size=\"2\"]\\boldsymbol{\\frac{\\Delta(r\\omega)}{\\Delta{t}}}.[\/latex]<\/div><\/li>\n \t<li id=\"import-auto-id1549399\">The radius r is constant for circular motion, and so <strong>\u0394(<em>r<\/em>\u03c9)=<em>r<\/em>\u0394\u03c9<\/strong>. Thus,\n<div id=\"import-auto-id3232862\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{a_{\\textbf{t}}=r}[\/latex][latex size=\"2\"]\\boldsymbol{\\frac{\\Delta\\omega}{\\Delta{t}}}.[\/latex]<\/div><\/li>\n \t<li id=\"import-auto-id3154788\">By definition, <strong>\u0394\u03c9\/\u0394<em>t<\/em>=<\/strong>[latex]\\boldsymbol{\\alpha}[\/latex]. Thus,\n<div id=\"import-auto-id3077640\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{a_{\\textbf{t}}=r\\alpha}[\/latex]<\/div>\n<p id=\"import-auto-id3064052\">or<\/p>\n\n<div id=\"import-auto-id3025466\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{\\alpha=}[\/latex][latex size=\"2\"]\\boldsymbol{\\frac{a_{\\textbf{t}}}{r}}.[\/latex]<\/div><\/li>\n<\/ul>\n<\/section><section id=\"fs-id1577741\" class=\"conceptual-questions\">\n<div class=\"bcc-box bcc-info\">\n<h3>Conceptual Questions<\/h3>\n<div class=\"exercise\">\n<div id=\"fs-id3006045\" class=\"problem\">\n<p id=\"import-auto-id2442420\"><strong>1: <\/strong>Analogies exist between rotational and translational physical quantities. Identify the rotational term analogous to each of the following: acceleration, force, mass, work, translational kinetic energy, linear momentum, impulse.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1867019\" class=\"exercise\">\n<div id=\"fs-id2970163\" class=\"problem\">\n<p id=\"import-auto-id2052574\"><strong>2: <\/strong>Explain why centripetal acceleration changes the direction of velocity in circular motion but not its magnitude.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id3046867\" class=\"exercise\">\n<div id=\"fs-id1610119\" class=\"problem\">\n<p id=\"import-auto-id2442809\"><strong>3: <\/strong>In circular motion, a tangential acceleration can change the magnitude of the velocity but not its direction. Explain your answer.<\/p>\n\n<\/div>\n<\/div>\n<div class=\"exercise\">\n<div id=\"fs-id3037278\" class=\"problem\">\n<p id=\"import-auto-id2639905\"><strong>4: <\/strong>Suppose a piece of food is on the edge of a rotating microwave oven plate. Does it experience nonzero tangential acceleration, centripetal acceleration, or both when: (a) The plate starts to spin? (b) The plate rotates at constant angular velocity? (c) The plate slows to a halt?<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/section><section id=\"fs-id1596665\" class=\"problems-exercises\">\n<div class=\"bcc-box bcc-info\">\n<h3>Problems &amp; Exercises<\/h3>\n<div id=\"fs-id1571972\" class=\"exercise\">\n<div id=\"fs-id2392409\" class=\"problem\">\n<p id=\"import-auto-id3355286\"><strong>1: <\/strong>At its peak, a tornado is 60.0 m in diameter and carries 500 km\/h winds. What is its angular velocity in revolutions per second?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2980135\" class=\"exercise\">\n<div id=\"fs-id3177097\" class=\"problem\">\n<p id=\"import-auto-id1863817\"><strong>2: Integrated Concepts<\/strong><\/p>\n<p id=\"eip-id2490643\">An ultracentrifuge accelerates from rest to 100,000 rpm in 2.00 min. (a) What is its angular acceleration in rad\/s<sup>2<\/sup>? (b) What is the tangential acceleration of a point 9.50 cm from the axis of rotation? (c) What is the radial acceleration in m\/s<sup>2<\/sup> and multiples of <em>g<\/em> of this point at full rpm?<\/p>\n\n<\/div>\n<\/div>\n<div class=\"exercise\">\n<div id=\"fs-id1019355\" class=\"problem\">\n<p id=\"import-auto-id1548295\"><strong>3: Integrated Concepts<\/strong><\/p>\n<p id=\"eip-id1172655289687\">You have a grindstone (a disk) that is 90.0 kg, has a 0.340-m radius, and is turning at 90.0 rpm, and you press a steel axe against it with a radial force of 20.0 N. (a) Assuming the kinetic coefficient of friction between steel and stone is 0.20, calculate the angular acceleration of the grindstone. (b) How many turns will the stone make before coming to rest?<\/p>\n\n<\/div>\n<\/div>\n<div class=\"exercise\">\n<div id=\"fs-id3063092\" class=\"problem\">\n<p id=\"import-auto-id955951\"><strong>4: Unreasonable Results<\/strong><\/p>\n<p id=\"eip-id1555320\">You are told that a basketball player spins the ball with an angular acceleration of 100 rad\/s<sup>2<\/sup>. (a) What is the ball\u2019s final angular velocity if the ball starts from rest and the acceleration lasts 2.00 s? (b) What is unreasonable about the result? (c) 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-id2410236\" class=\"definition\">\n \t<dt>angular acceleration<\/dt>\n \t<dd id=\"fs-id3257384\">the rate of change of angular velocity with time<\/dd>\n<\/dl>\n<dl id=\"import-auto-id3224451\" class=\"definition\">\n \t<dt>change in angular velocity<\/dt>\n \t<dd id=\"fs-id1930201\">the difference between final and initial values of angular velocity<\/dd>\n<\/dl>\n<dl id=\"import-auto-id3258416\" class=\"definition\">\n \t<dt>tangential acceleration<\/dt>\n \t<dd id=\"fs-id1999225\">the acceleration in a direction tangent to the circle at the point of interest in circular motion<\/dd>\n<\/dl>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Solutions<\/h3>\n<strong>Check Your Understanding\n<\/strong>\n\n<strong>1:<\/strong> The magnitude of angular acceleration is\u00a0\u03b1 and its most common units are rad\/s<sup>2<\/sup>. The direction of angular acceleration along a fixed axis is denoted by a + or a \u2013 sign, just as the direction of linear acceleration in one dimension is denoted by a + or a \u2013 sign. For example, consider a gymnast doing a forward flip. Her angular momentum would be parallel to the mat and to her left. The magnitude of her angular acceleration would be proportional to her angular velocity (spin rate) and her moment of inertia about her spin axis.\n\n<strong>Problems &amp; Exercises<\/strong>\n\n<strong>1: <\/strong>[latex]\\boldsymbol{\\omega=0.737\\textbf{ rev\/s}}[\/latex]\n<p id=\"import-auto-id2206615\"><strong>3: <\/strong>(a) [latex]\\boldsymbol{-0.26\\textbf{ rad\/s}^2}[\/latex] (b) [latex]\\boldsymbol{27\\textbf{ rev}}[\/latex]<\/p>\n\n<\/div>","rendered":"<div>\n<div class=\"bcc-box bcc-highlight\">\n<h3>Summary<\/h3>\n<div>\n<ul>\n<li>Describe uniform circular motion.<\/li>\n<li>Explain non-uniform circular motion.<\/li>\n<li>Calculate angular acceleration of an object.<\/li>\n<li>Observe the link between linear and angular acceleration.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"import-auto-id2400941\"><a href=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1108\/part\/chapter-6-uniform-circular-motion-and-gravitation\/\">Chapter 5 Uniform Circular Motion and Gravitation<\/a> discussed only uniform circular motion, which is motion in a circle at constant speed and, hence, constant angular velocity. Recall that angular velocity\u00a0<strong>\u03c9<\/strong> was defined as the time rate of change of angle <em><strong>\u03b8<\/strong><\/em>:<\/p>\n<div id=\"eip-257\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{\\omega\\:=}[\/latex][latex]\\boldsymbol{\\frac{\\Delta\\theta}{\\Delta{t}}},[\/latex]<\/div>\n<p id=\"import-auto-id2970170\">where <em><strong>\u03b8<\/strong><\/em> is the angle of rotation as seen in <a class=\"autogenerated-content\" href=\"#import-auto-id1941476\">Figure 1<\/a>. The relationship between angular velocity <strong>\u03c9<\/strong> and linear velocity <em><strong>v<\/strong><\/em> was also defined in <a href=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1108\/chapter\/9-1-rotation-angle-and-angular-velocity\/\">Chapter 9.1 Rotation Angle and Angular Velocity<\/a> as<\/p>\n<div class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{v=r\\omega}[\/latex]<\/div>\n<p id=\"eip-179\">or<\/p>\n<div id=\"eip-363\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{\\omega\\:=}[\/latex][latex]\\boldsymbol{\\frac{v}{r}},[\/latex]<\/div>\n<p id=\"import-auto-id2655439\">where <em><strong>r<\/strong><\/em> is the radius of curvature, also seen in <a class=\"autogenerated-content\" href=\"#import-auto-id1941476\">Figure 1<\/a>. According to the sign convention, the counter clockwise direction is considered as positive direction and clockwise direction as negative<\/p>\n<figure id=\"import-auto-id1941476\">\n<figure style=\"width: 200px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1107\/wp-content\/uploads\/sites\/1184\/2017\/09\/Figure_11_01_01a-1.jpg\" alt=\"The given figure shows counterclockwise circular motion with a horizontal line, depicting radius r, drawn from the center of the circle to the right side on its circumference and another line is drawn in such a manner that it makes an acute angle delta theta with the horizontal line. Tangential velocity vectors are indicated at the end of the two lines. At the bottom right side of the figure, the formula for angular velocity is given as v upon r.\" width=\"200\" height=\"302\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 1.<\/strong> This figure shows uniform circular motion and some of its defined quantities.<\/figcaption><\/figure>\n<\/figure>\n<p id=\"import-auto-id2670067\">Angular velocity is not constant when a skater pulls in her arms, when a child starts up a merry-go-round from rest, or when a computer\u2019s hard disk slows to a halt when switched off. In all these cases, there is an<strong> <span id=\"import-auto-id2438108\">angular acceleration<\/span><\/strong>, in which <strong>\u03c9<\/strong> changes. The faster the change occurs, the greater the angular acceleration. Angular acceleration [latex]\\boldsymbol{\\alpha}[\/latex] is defined as the rate of change of angular velocity. In equation form, angular acceleration is expressed as follows:<\/p>\n<div id=\"eip-974\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{\\alpha\\:=}[\/latex][latex]\\boldsymbol{\\frac{\\Delta\\omega}{\\Delta{t}}},[\/latex]<\/div>\n<p id=\"import-auto-id2403123\">where\u00a0<strong>\u0394\u03c9<\/strong> is the <span id=\"import-auto-id3406201\"> change in angular velocity<\/span> and <strong>\u0394<em>t<\/em><\/strong> is the change in time. The units of angular acceleration are (<strong>rad\/s)\/s<\/strong>, or <strong>rad\/s<sup>2<\/sup><\/strong>. If <strong>\u03c9<\/strong> increases, then [latex]\\boldsymbol{\\alpha}[\/latex] is positive. If <strong>\u03c9<\/strong> decreases, then [latex]\\boldsymbol{\\alpha}[\/latex] is negative.<\/p>\n<div class=\"textbox shaded\">\n<div id=\"fs-id3159590\" class=\"example\">\n<h3 id=\"import-auto-id3385431\">Example 1: Calculating the Angular Acceleration and Deceleration of a Bike Wheel<\/h3>\n<p>Suppose a teenager puts her bicycle on its back and starts the rear wheel spinning from rest to a final angular velocity of 250 rpm in 5.00 s. (a) Calculate the angular acceleration in <strong>rad\/s<sup>2<\/sup><\/strong>. (b) If she now slams on the brakes, causing an angular acceleration of <strong>-87.3 rad\/s<sup>2<\/sup><\/strong>, how long does it take the wheel to stop?<\/p>\n<p id=\"import-auto-id1954754\"><strong>Strategy for (a)<\/strong><\/p>\n<p id=\"import-auto-id1917815\">The angular acceleration can be found directly from its definition in [latex]\\boldsymbol{\\alpha=\\frac{\\Delta\\omega}{\\Delta{t}}}[\/latex] because the final angular velocity and time are given. We see that <strong>\u0394\u03c9<\/strong> is 250 rpm and <strong>\u0394<em>t<\/em><\/strong> is 5.00 s.<\/p>\n<p id=\"import-auto-id1840465\"><strong>Solution for (a)<\/strong><\/p>\n<p id=\"import-auto-id2930128\">Entering known information into the definition of angular acceleration, we get<\/p>\n<div id=\"eip-272\" class=\"equation\" style=\"text-align: center\">[latex]\\begin{array}{lcl} \\boldsymbol{\\alpha} & \\boldsymbol{=} & \\boldsymbol{\\frac{\\Delta\\omega}{\\Delta{t}}} \\\\ {} & \\boldsymbol{=} & \\boldsymbol{\\frac{250\\textbf{ rpm}}{5.00\\textbf{ s.}}} \\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id3418450\">Because <strong>\u0394\u03c9<\/strong> is in revolutions per minute (rpm) and we want the standard units of <strong>rad\/s<sup>2<\/sup><\/strong> for angular acceleration, we need to convert <strong>\u0394\u03c9<\/strong> from rpm to rad\/s:<\/p>\n<div class=\"equation\" style=\"text-align: center\">[latex]\\begin{array}{lcl} \\boldsymbol{\\Delta\\omega} & \\boldsymbol{=} & \\boldsymbol{250\\frac{\\textbf{rev}}{\\textbf{min}}\\cdotp\\frac{2\\pi\\textbf{ rad}}{\\textbf{rev}}\\cdotp\\frac{1\\textbf{ min}}{60\\textbf{ sec}}} \\\\ {} & \\boldsymbol{=} & \\boldsymbol{26.2\\textbf{ rads.}} \\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id3358717\">Entering this quantity into the expression for [latex]\\boldsymbol{\\alpha}[\/latex], we get<\/p>\n<div id=\"eip-899\" class=\"equation\" style=\"text-align: center\">[latex]\\begin{array}{lcl} \\boldsymbol{\\alpha} & \\boldsymbol{=} & \\boldsymbol{\\frac{\\Delta\\omega}{\\Delta{t}}} \\\\ {} & \\boldsymbol{=} & \\boldsymbol{\\frac{26.2\\textbf{ rad\/s}}{5.00\\textbf{ s}}} \\\\ {} & \\boldsymbol{=} & \\boldsymbol{5.24\\textbf{ rad\/s.}^2} \\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id2625933\"><strong>Strategy for (b)<\/strong><\/p>\n<p id=\"import-auto-id2438018\">In this part, we know the angular acceleration and the initial angular velocity. We can find the stoppage time by using the definition of angular acceleration and solving for <strong>\u0394<em>t<\/em><\/strong>, yielding<\/p>\n<div id=\"eip-273\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{\\Delta{t}\\:=}[\/latex][latex]\\boldsymbol{\\frac{\\Delta\\omega}{\\alpha}}.[\/latex]<\/div>\n<p id=\"import-auto-id2591053\"><strong>Solution for (b)<\/strong><\/p>\n<p>Here the angular velocity decreases from <strong>26.2 rad\/s<\/strong> (250 rpm) to zero, so that <strong>\u0394\u03c9<\/strong> is <strong>-26.2 rad\/s<\/strong>, and [latex]\\boldsymbol{\\alpha}[\/latex] is given to be <strong>-87.3 rad\/s<sup>2<\/sup><\/strong>. Thus,<\/p>\n<div id=\"eip-455\" class=\"equation\" style=\"text-align: center\">[latex]\\begin{array}{lcl} \\boldsymbol{\\Delta{t}} & \\boldsymbol{=} & \\boldsymbol{\\frac{-26.2\\textbf{ rad\/s}}{-87.3\\textbf{ rad\/s}^2}} \\\\ {} & \\boldsymbol{=} & \\boldsymbol{0.300\\textbf{ s.}} \\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id3025168\"><strong>Discussion<\/strong><\/p>\n<p>Note that the angular acceleration as the girl spins the wheel is small and positive; it takes 5 s to produce an appreciable angular velocity. When she hits the brake, the angular acceleration is large and negative. The angular velocity quickly goes to zero. In both cases, the relationships are analogous to what happens with linear motion. For example, there is a large deceleration when you crash into a brick wall\u2014the velocity change is large in a short time interval.<\/p>\n<\/div>\n<\/div>\n<p id=\"import-auto-id2595516\">If the bicycle in the preceding example had been on its wheels instead of upside-down, it would first have accelerated along the ground and then come to a stop. This connection between circular motion and linear motion needs to be explored. For example, it would be useful to know how linear and angular acceleration are related. In circular motion, linear acceleration is <em>tangent<\/em> to the circle at the point of interest, as seen in <a class=\"autogenerated-content\" href=\"#import-auto-id1019355\">Figure 2<\/a>. Thus, linear acceleration is called <span id=\"import-auto-id1997895\">tangential acceleration <strong><em>a<\/em><\/strong><\/span><strong><sub>t<\/sub><\/strong>.<\/p>\n<figure id=\"import-auto-id1019355\">\n<figure style=\"width: 315px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1107\/wp-content\/uploads\/sites\/1184\/2020\/11\/Figure_11_01_02a-1.jpg\" alt=\"In the figure, a semicircle is drawn, with its radius r, shown here as a line segment. The anti-clockwise motion of the circle is shown with an arrow on the path of the circle. Tangential velocity vector, v, of the point, which is on the meeting point of radius with the circle, is shown as a green arrow and the linear acceleration, a-t is shown as a yellow arrow in the same direction along v.\" width=\"315\" height=\"200\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 2.<\/strong> In circular motion, linear acceleration <em><strong>a<\/strong><\/em>, occurs as the magnitude of the velocity changes: <em><strong>a<\/strong><\/em> is tangent to the motion. In the context of circular motion, linear acceleration is also called tangential acceleration <strong><em>a<\/em><sub>t<\/sub><\/strong>.<\/figcaption><\/figure>\n<\/figure>\n<p>Linear or tangential acceleration refers to changes in the magnitude of velocity but not its direction. We know from uniform circular motion that in circular motion centripetal acceleration, <span id=\"import-auto-id1997895\"><strong><em>a<\/em><\/strong><\/span><strong><sub>c<\/sub><\/strong>, refers to changes in the direction of the velocity but not its magnitude. An object undergoing circular motion experiences centripetal acceleration, as seen in <a class=\"autogenerated-content\" href=\"#import-auto-id1995872\">Figure 3<\/a>. Thus, <span id=\"import-auto-id1997895\"><strong><em>a<\/em><\/strong><\/span><strong><sub>t<\/sub><\/strong> and <span id=\"import-auto-id1997895\"><strong><em>a<\/em><\/strong><\/span><strong><sub>c<\/sub><\/strong> are perpendicular and independent of one another. Tangential acceleration <span id=\"import-auto-id1997895\"><strong><em>a<\/em><\/strong><\/span><strong><sub>t<\/sub><\/strong> is directly related to the angular acceleration [latex]\\boldsymbol{\\alpha}[\/latex] and is linked to an increase or decrease in the velocity, but not its direction.<\/p>\n<figure id=\"import-auto-id1995872\">\n<figure style=\"width: 350px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1107\/wp-content\/uploads\/sites\/1184\/2020\/11\/Figure_11_01_03a-1.jpg\" alt=\"In the figure, a semicircle is drawn, with its radius r, shown here as a line segment. The anti-clockwise motion of the circle is shown with an arrow on the path of the circle. Tangential velocity vector, v, of the point, which is on the meeting point of radius with the circle, is shown as a green arrow and the linear acceleration, a sub t is shown as a yellow arrow in the same direction along v. The centripetal acceleration, a sub c, is also shown as a yellow arrow drawn perpendicular to a sub t, toward the direction of the center of the circle. A label in the figures states a sub t affects magnitude and a sub c affects direction.\" width=\"350\" height=\"200\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 3.<\/strong> Centripetal acceleration <strong><em>a<\/em><sub>c<\/sub><\/strong> occurs as the direction of velocity changes; it is perpendicular to the circular motion. Centripetal and tangential acceleration are thus perpendicular to each other.<\/figcaption><\/figure>\n<\/figure>\n<p id=\"import-auto-id1993750\">Now we can find the exact relationship between linear acceleration <span id=\"import-auto-id1997895\"><strong><em>a<\/em><\/strong><\/span><strong><sub>t<\/sub><\/strong> and angular acceleration [latex]\\boldsymbol{\\alpha}[\/latex]. Because linear acceleration is proportional to a change in the magnitude of the velocity, it is defined (as it was in <a href=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1108\/part\/chapter-2-one-dimensional-kinematics\/\">Chapter 2 One-Dimensional Kinematics<\/a>) to be<\/p>\n<div id=\"eip-85\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{a_{\\textbf{t}}\\:=}[\/latex][latex]\\boldsymbol{\\frac{\\Delta{v}}{\\Delta{t}}.}[\/latex]<\/div>\n<p id=\"import-auto-id2407460\">For circular motion, note that <strong><em>v<\/em>=<em>r<\/em>\u03c9<\/strong>, so that<\/p>\n<div class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{a_{\\textbf{t}}\\:=}[\/latex][latex]\\boldsymbol{\\frac{\\Delta(r\\omega)}{\\Delta{t}}.}[\/latex]<\/div>\n<p id=\"import-auto-id2660038\">The radius <strong><em>r<\/em><\/strong> is constant for circular motion, and so <strong>\u0394(<em>r<\/em>\u03c9)=<em>r<\/em>(\u0394\u03c9)<\/strong>. Thus,<\/p>\n<div id=\"eip-688\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{a_{\\textbf{t}}=r}[\/latex][latex]\\boldsymbol{\\frac{\\Delta\\omega}{\\Delta{t}}.}[\/latex]<\/div>\n<p id=\"import-auto-id1861378\">By definition, [latex]\\boldsymbol{\\alpha=\\frac{\\Delta\\omega}{\\Delta{t}}}.[\/latex] Thus,<\/p>\n<div class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{a_{\\textbf{t}}=r\\alpha},[\/latex]<\/div>\n<p>or<\/p>\n<div class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{\\alpha\\:=}[\/latex][latex]\\boldsymbol{\\frac{a_{\\textbf{t}}}{r}.}[\/latex]<\/div>\n<p id=\"import-auto-id1977486\">These equations mean that linear acceleration and angular acceleration are directly proportional. The greater the angular acceleration is, the larger the linear (tangential) acceleration is, and vice versa. For example, the greater the angular acceleration of a car\u2019s drive wheels, the greater the acceleration of the car. The radius also matters. For example, the smaller a wheel, the smaller its linear acceleration for a given angular acceleration [latex]\\boldsymbol{\\alpha}[\/latex].<\/p>\n<div class=\"textbox shaded\">\n<div id=\"fs-id3217117\" class=\"example\">\n<h3 class=\"title\">Example 2: Calculating the Angular Acceleration of a Motorcycle Wheel<\/h3>\n<p id=\"import-auto-id3008868\">A powerful motorcycle can accelerate from 0 to 30.0 m\/s (about 108 km\/h) in 4.20 s. What is the angular acceleration of its 0.320-m-radius wheels? (See <a class=\"autogenerated-content\" href=\"#import-auto-id2415283\">Figure 4<\/a>.)<\/p>\n<figure>\n<figure style=\"width: 225px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1107\/wp-content\/uploads\/sites\/1184\/2020\/11\/Figure_11_01_04a-1.jpg\" alt=\"The figure shows the right side view of a man riding a motorcycle hence, depicting linear acceleration a of the motorcycle pointing toward the front of the bike as a horizontal arrow and the angular acceleration alpha of its wheels, shown here as curved arrows along the front of both the wheels pointing downward.\" width=\"225\" height=\"236\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 4<\/strong>. The linear acceleration of a motorcycle is accompanied by an angular acceleration of its wheels.<\/figcaption><\/figure>\n<\/figure>\n<p id=\"import-auto-id3068565\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id3149756\">We are given information about the linear velocities of the motorcycle. Thus, we can find its linear acceleration <span id=\"import-auto-id1997895\"><strong><em>a<\/em><\/strong><\/span><strong><sub>t<\/sub><\/strong>. Then, the expression [latex]\\boldsymbol{\\alpha=\\frac{a_{\\textbf{t}}}{r}}[\/latex] can be used to find the angular acceleration.<\/p>\n<p id=\"import-auto-id2451305\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id2577576\">The linear acceleration is<\/p>\n<div id=\"eip-20\" class=\"equation\" style=\"text-align: center\">[latex]\\begin{array}{lcl} \\boldsymbol{a_{\\textbf{t}}} & \\boldsymbol{=} & \\boldsymbol{\\frac{\\Delta{v}}{\\Delta{t}}} \\\\ {} & \\boldsymbol{=} & \\boldsymbol{\\frac{30.0\\textbf{ m\/s}}{4.20\\textbf{ s}}} \\\\ {} & \\boldsymbol{=} & \\boldsymbol{7.14\\textbf{ m\/s}^2.} \\end{array}[\/latex]<\/div>\n<p>We also know the radius of the wheels. Entering the values for <span id=\"import-auto-id1997895\"><strong><em>a<\/em><\/strong><\/span><strong><sub>t<\/sub><\/strong> and <em><strong>r<\/strong><\/em> into [latex]\\boldsymbol{\\alpha=\\frac{a_{\\textbf{t}}}{r}},[\/latex] we get<\/p>\n<div id=\"eip-155\" class=\"equation\" style=\"text-align: center\">[latex]\\begin{array}{lcl} \\boldsymbol{\\alpha} & \\boldsymbol{=} & \\boldsymbol{\\frac{a_{\\textbf{t}}}{r}} \\\\ {} & \\boldsymbol{=} & \\boldsymbol{\\frac{7.14\\textbf{ m\/s}}{20.320\\textbf{ m}}} \\\\ {} & \\boldsymbol{=} & \\boldsymbol{22.3\\textbf{ rad\/s}^2.} \\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id3062550\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id2017158\">Units of radians are dimensionless and appear in any relationship between angular and linear quantities.<\/p>\n<\/div>\n<\/div>\n<p id=\"import-auto-id1117830\">So far, we have defined three rotational quantities\u2014 <em><strong>\u03b8<\/strong><\/em>, <strong>\u03c9<\/strong>, and [latex]\\boldsymbol{\\alpha}[\/latex]. These quantities are analogous to the translational quantities <em><strong>x<\/strong><\/em>, <em><strong>v<\/strong><\/em>, and <em><strong>a<\/strong><\/em>. <a class=\"autogenerated-content\" href=\"#import-auto-id1572984\">Table 1<\/a> displays rotational quantities, the analogous translational quantities, and the relationships between them.<\/p>\n<table id=\"import-auto-id1572984\" summary=\"Rotational and Translational Quantities\">\n<thead>\n<tr>\n<th>Rotational<\/th>\n<th>Translational<\/th>\n<th>Relationship<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]\\boldsymbol{\\theta}[\/latex]<\/td>\n<td>[latex]\\boldsymbol{x}[\/latex]<\/td>\n<td>[latex]\\boldsymbol{\\theta=\\frac{x}{r}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\boldsymbol{\\omega}[\/latex]<\/td>\n<td>[latex]\\boldsymbol{v}[\/latex]<\/td>\n<td>[latex]\\boldsymbol{\\omega=\\frac{v}{r}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\boldsymbol{\\alpha}[\/latex]<\/td>\n<td>[latex]\\boldsymbol{a}[\/latex]<\/td>\n<td>[latex]\\boldsymbol{\\alpha=\\frac{a_{\\textbf{t}}}{r}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<tbody>\n<tr>\n<td colspan=\"3\"><strong>Table 1.<\/strong> Rotational and Translational Quantities.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id3292730\" class=\"note\">\n<div class=\"textbox shaded\">\n<div class=\"note\">\n<h3 class=\"title\">MAKING CONNECTIONS: TAKE-HOME EXPERIMENT<span style=\"text-decoration: underline\"><br \/>\n<\/span><\/h3>\n<p id=\"import-auto-id1909409\">Sit down with your feet on the ground on a chair that rotates. Lift one of your legs such that it is unbent (straightened out). Using the other leg, begin to rotate yourself by pushing on the ground. Stop using your leg to push the ground but allow the chair to rotate. From the origin where you began, sketch the angle, angular velocity, and angular acceleration of your leg as a function of time in the form of three separate graphs. Estimate the magnitudes of these quantities.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1870686\" class=\"exercise\">\n<div class=\"bcc-box bcc-info\">\n<h3>Check Your Understanding<\/h3>\n<div class=\"exercise\">\n<div id=\"fs-id3093647\" class=\"problem\">\n<p id=\"import-auto-id1945320\"><strong>1:<\/strong> Angular acceleration is a vector, having both magnitude and direction. How do we denote its magnitude and direction? Illustrate with an example.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3 class=\"title\">PHET EXPLORATIONS: LADYBUG REVOLUTION<\/h3>\n<p id=\"eip-id1325598\">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<figure id=\"eip-id2624753\">\n<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-23-1.png\" alt=\"image\" width=\"450\" height=\"147\" \/><\/a><figcaption class=\"wp-caption-text\">Figure 5. <a href=\"https:\/\/phet.colorado.edu\/en\/simulation\/rotation\">Ladybug Revolution<\/a><\/figcaption><\/figure>\n<\/figure>\n<\/div>\n<\/div>\n<section id=\"fs-id3034915\" class=\"section-summary\">\n<h1>Section Summary<\/h1>\n<ul id=\"fs-id1947000\">\n<li id=\"import-auto-id2443525\">Uniform circular motion is the motion with a constant angular velocity [latex]\\boldsymbol{\\omega=\\frac{\\Delta\\theta}{\\Delta{t}}}.[\/latex]<\/li>\n<li id=\"import-auto-id3353297\">In non-uniform circular motion, the velocity changes with time and the rate of change of angular velocity (i.e. angular acceleration) is [latex]\\boldsymbol{\\alpha=\\frac{\\Delta\\omega}{\\Delta{t}}}.[\/latex]<\/li>\n<li id=\"import-auto-id1867184\">Linear or tangential acceleration refers to changes in the magnitude of velocity but not its direction, given as [latex]\\boldsymbol{a_{\\textbf{t}}=\\frac{\\Delta{v}}{\\Delta{t}}}.[\/latex]<\/li>\n<li id=\"import-auto-id3035395\">For circular motion, note that <strong><em>v<\/em>=<em>r<\/em>\u03c9<\/strong>, so that\n<div id=\"import-auto-id1588138\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{a_{\\textbf{t}}\\:=}[\/latex][latex]\\boldsymbol{\\frac{\\Delta(r\\omega)}{\\Delta{t}}}.[\/latex]<\/div>\n<\/li>\n<li id=\"import-auto-id1549399\">The radius r is constant for circular motion, and so <strong>\u0394(<em>r<\/em>\u03c9)=<em>r<\/em>\u0394\u03c9<\/strong>. Thus,\n<div id=\"import-auto-id3232862\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{a_{\\textbf{t}}=r}[\/latex][latex]\\boldsymbol{\\frac{\\Delta\\omega}{\\Delta{t}}}.[\/latex]<\/div>\n<\/li>\n<li id=\"import-auto-id3154788\">By definition, <strong>\u0394\u03c9\/\u0394<em>t<\/em>=<\/strong>[latex]\\boldsymbol{\\alpha}[\/latex]. Thus,\n<div id=\"import-auto-id3077640\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{a_{\\textbf{t}}=r\\alpha}[\/latex]<\/div>\n<p id=\"import-auto-id3064052\">or<\/p>\n<div id=\"import-auto-id3025466\" class=\"equation\" style=\"text-align: center\">[latex]\\boldsymbol{\\alpha=}[\/latex][latex]\\boldsymbol{\\frac{a_{\\textbf{t}}}{r}}.[\/latex]<\/div>\n<\/li>\n<\/ul>\n<\/section>\n<section id=\"fs-id1577741\" class=\"conceptual-questions\">\n<div class=\"bcc-box bcc-info\">\n<h3>Conceptual Questions<\/h3>\n<div class=\"exercise\">\n<div id=\"fs-id3006045\" class=\"problem\">\n<p id=\"import-auto-id2442420\"><strong>1: <\/strong>Analogies exist between rotational and translational physical quantities. Identify the rotational term analogous to each of the following: acceleration, force, mass, work, translational kinetic energy, linear momentum, impulse.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1867019\" class=\"exercise\">\n<div id=\"fs-id2970163\" class=\"problem\">\n<p id=\"import-auto-id2052574\"><strong>2: <\/strong>Explain why centripetal acceleration changes the direction of velocity in circular motion but not its magnitude.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id3046867\" class=\"exercise\">\n<div id=\"fs-id1610119\" class=\"problem\">\n<p id=\"import-auto-id2442809\"><strong>3: <\/strong>In circular motion, a tangential acceleration can change the magnitude of the velocity but not its direction. Explain your answer.<\/p>\n<\/div>\n<\/div>\n<div class=\"exercise\">\n<div id=\"fs-id3037278\" class=\"problem\">\n<p id=\"import-auto-id2639905\"><strong>4: <\/strong>Suppose a piece of food is on the edge of a rotating microwave oven plate. Does it experience nonzero tangential acceleration, centripetal acceleration, or both when: (a) The plate starts to spin? (b) The plate rotates at constant angular velocity? (c) The plate slows to a halt?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1596665\" class=\"problems-exercises\">\n<div class=\"bcc-box bcc-info\">\n<h3>Problems &amp; Exercises<\/h3>\n<div id=\"fs-id1571972\" class=\"exercise\">\n<div id=\"fs-id2392409\" class=\"problem\">\n<p id=\"import-auto-id3355286\"><strong>1: <\/strong>At its peak, a tornado is 60.0 m in diameter and carries 500 km\/h winds. What is its angular velocity in revolutions per second?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2980135\" class=\"exercise\">\n<div id=\"fs-id3177097\" class=\"problem\">\n<p id=\"import-auto-id1863817\"><strong>2: Integrated Concepts<\/strong><\/p>\n<p id=\"eip-id2490643\">An ultracentrifuge accelerates from rest to 100,000 rpm in 2.00 min. (a) What is its angular acceleration in rad\/s<sup>2<\/sup>? (b) What is the tangential acceleration of a point 9.50 cm from the axis of rotation? (c) What is the radial acceleration in m\/s<sup>2<\/sup> and multiples of <em>g<\/em> of this point at full rpm?<\/p>\n<\/div>\n<\/div>\n<div class=\"exercise\">\n<div id=\"fs-id1019355\" class=\"problem\">\n<p id=\"import-auto-id1548295\"><strong>3: Integrated Concepts<\/strong><\/p>\n<p id=\"eip-id1172655289687\">You have a grindstone (a disk) that is 90.0 kg, has a 0.340-m radius, and is turning at 90.0 rpm, and you press a steel axe against it with a radial force of 20.0 N. (a) Assuming the kinetic coefficient of friction between steel and stone is 0.20, calculate the angular acceleration of the grindstone. (b) How many turns will the stone make before coming to rest?<\/p>\n<\/div>\n<\/div>\n<div class=\"exercise\">\n<div id=\"fs-id3063092\" class=\"problem\">\n<p id=\"import-auto-id955951\"><strong>4: Unreasonable Results<\/strong><\/p>\n<p id=\"eip-id1555320\">You are told that a basketball player spins the ball with an angular acceleration of 100 rad\/s<sup>2<\/sup>. (a) What is the ball\u2019s final angular velocity if the ball starts from rest and the acceleration lasts 2.00 s? (b) What is unreasonable about the result? (c) 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-id2410236\" class=\"definition\">\n<dt>angular acceleration<\/dt>\n<dd id=\"fs-id3257384\">the rate of change of angular velocity with time<\/dd>\n<\/dl>\n<dl id=\"import-auto-id3224451\" class=\"definition\">\n<dt>change in angular velocity<\/dt>\n<dd id=\"fs-id1930201\">the difference between final and initial values of angular velocity<\/dd>\n<\/dl>\n<dl id=\"import-auto-id3258416\" class=\"definition\">\n<dt>tangential acceleration<\/dt>\n<dd id=\"fs-id1999225\">the acceleration in a direction tangent to the circle at the point of interest in circular motion<\/dd>\n<\/dl>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Solutions<\/h3>\n<p><strong>Check Your Understanding<br \/>\n<\/strong><\/p>\n<p><strong>1:<\/strong> The magnitude of angular acceleration is\u00a0\u03b1 and its most common units are rad\/s<sup>2<\/sup>. The direction of angular acceleration along a fixed axis is denoted by a + or a \u2013 sign, just as the direction of linear acceleration in one dimension is denoted by a + or a \u2013 sign. For example, consider a gymnast doing a forward flip. Her angular momentum would be parallel to the mat and to her left. The magnitude of her angular acceleration would be proportional to her angular velocity (spin rate) and her moment of inertia about her spin axis.<\/p>\n<p><strong>Problems &amp; Exercises<\/strong><\/p>\n<p><strong>1: <\/strong>[latex]\\boldsymbol{\\omega=0.737\\textbf{ rev\/s}}[\/latex]<\/p>\n<p id=\"import-auto-id2206615\"><strong>3: <\/strong>(a) [latex]\\boldsymbol{-0.26\\textbf{ rad\/s}^2}[\/latex] (b) [latex]\\boldsymbol{27\\textbf{ rev}}[\/latex]<\/p>\n<\/div>\n","protected":false},"author":9,"menu_order":2,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-484","chapter","type-chapter","status-publish","hentry"],"part":470,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/douglasphys1108\/wp-json\/pressbooks\/v2\/chapters\/484","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\/484\/revisions"}],"predecessor-version":[{"id":485,"href":"https:\/\/pressbooks.bccampus.ca\/douglasphys1108\/wp-json\/pressbooks\/v2\/chapters\/484\/revisions\/485"}],"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\/484\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/douglasphys1108\/wp-json\/wp\/v2\/media?parent=484"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/douglasphys1108\/wp-json\/pressbooks\/v2\/chapter-type?post=484"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/douglasphys1108\/wp-json\/wp\/v2\/contributor?post=484"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/douglasphys1108\/wp-json\/wp\/v2\/license?post=484"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}