{"id":515,"date":"2017-10-27T16:30:08","date_gmt":"2017-10-27T16:30:08","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/angular-acceleration\/"},"modified":"2017-11-08T03:24:47","modified_gmt":"2017-11-08T03:24:47","slug":"angular-acceleration","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/angular-acceleration\/","title":{"raw":"Angular Acceleration","rendered":"Angular Acceleration"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\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<p id=\"import-auto-id2400941\"><a href=\"\/contents\/3ef5dfb6-0a8d-433e-9c8f-b8c860a3903b@2\">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 [latex]\\omega [\/latex] was defined as the time rate of change of angle [latex]\\theta [\/latex]:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-257\">[latex]\\omega =\\frac{\\Delta \\theta }{\\Delta t}\\text{,}[\/latex]<\/div>\n<p id=\"import-auto-id2970170\">where [latex]\\theta [\/latex] is the angle of rotation as seen in <a href=\"#import-auto-id1941476\" class=\"autogenerated-content\">(Figure)<\/a>. The relationship between angular velocity [latex]\\omega [\/latex] and linear velocity [latex]v[\/latex] was also defined in <a href=\"\/contents\/44216d83-fe7e-461d-a5f4-9f3e2ffa284b@7\">Rotation Angle and Angular Velocity<\/a> as<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]v=\\mathrm{r\\omega }[\/latex]<\/div>\n<p id=\"eip-179\">or<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-363\">[latex]\\omega =\\frac{v}{r},[\/latex]<\/div>\n<p id=\"import-auto-id2655439\">where [latex]r[\/latex] is the radius of curvature, also seen in <a href=\"#import-auto-id1941476\" class=\"autogenerated-content\">(Figure)<\/a>. According to the sign convention, the counter clockwise direction is considered as positive direction and clockwise direction as negative <\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1941476\">\n<div class=\"bc-figcaption figcaption\">This figure shows uniform circular motion and some of its defined quantities.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id3233391\" data-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.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_11_01_01a.jpg\" data-media-type=\"image\/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\"><\/span><\/p><\/div>\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 <span data-type=\"term\" id=\"import-auto-id2438108\">angular acceleration<\/span>, in which [latex]\\omega [\/latex] changes. The faster the change occurs, the greater the angular acceleration. Angular acceleration [latex]\\alpha [\/latex] is defined as the rate of change of angular velocity. In equation form, angular acceleration is expressed as follows:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-974\">[latex]\\alpha =\\frac{\\Delta \\omega }{\\Delta t}\\text{,}[\/latex]<\/div>\n<p id=\"import-auto-id2403123\">where [latex]\\Delta \\omega [\/latex] is the <span data-type=\"term\" id=\"import-auto-id3406201\"> change in angular velocity<\/span> and [latex]\\Delta t[\/latex] is the change in time. The units of angular acceleration are [latex]\\left(\\text{rad\/s}\\right)\\text{\/s}[\/latex], or [latex]{\\text{rad\/s}}^{2}[\/latex]. If [latex]\\omega [\/latex] increases, then [latex]\\alpha [\/latex] is positive. If [latex]\\omega [\/latex] decreases, then [latex]\\alpha [\/latex] is negative.<\/p>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id3159590\">\n<div data-type=\"title\" class=\"title\">Calculating the Angular Acceleration and Deceleration of a Bike Wheel<\/div>\n<p id=\"import-auto-id3385431\">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 [latex]{\\text{rad\/s}}^{2}[\/latex]. (b) If she now slams on the brakes, causing an angular acceleration of [latex]\u201387.3\\phantom{\\rule{0.25em}{0ex}}{\\text{rad\/s}}^{2}[\/latex], 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]\\alpha =\\frac{\\Delta \\omega }{\\Delta t}[\/latex] because the final angular velocity and time are given. We see that [latex]\\Delta \\omega [\/latex] is 250 rpm and [latex]\\Delta t[\/latex] 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 data-type=\"equation\" class=\"equation\" id=\"eip-272\">[latex]\\begin{array}{lll}\\alpha &amp; =&amp; \\frac{\\Delta \\omega }{\\Delta t}\\\\ &amp; =&amp; \\frac{\\text{250 rpm}}{\\text{5.00 s}}\\text{.}\\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id3418450\">Because [latex]\\Delta \\omega [\/latex] is in revolutions per minute (rpm) and we want the standard units of [latex]{\\text{rad\/s}}^{2}[\/latex] for angular acceleration, we need to convert [latex]\\Delta \\omega [\/latex] from rpm to rad\/s:<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\begin{array}{lll}\\Delta \\omega &amp; =&amp; \\text{250}\\frac{\\text{rev}}{\\text{min}}\\cdot \\frac{\\text{2\u03c0 rad}}{\\text{rev}}\\cdot \\frac{\\text{1 min}}{\\text{60 sec}}\\\\ &amp; =&amp; \\text{26.2}\\frac{\\text{rad}}{\\text{s}}.\\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id3358717\">Entering this quantity into the expression for [latex]\\alpha [\/latex], we get<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-899\">[latex]\\begin{array}{lll}\\alpha &amp; =&amp; \\frac{\\Delta \\omega }{\\Delta t}\\\\ &amp; =&amp; \\frac{\\text{26.2&nbsp;rad\/s}}{\\text{5.00 s}}\\\\ &amp; =&amp; \\text{5.24}{\\text{&nbsp;rad\/s}}^{2}\\text{.}\\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 [latex]\\Delta t[\/latex], yielding<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-273\">[latex]\\Delta t=\\frac{\\Delta \\omega }{\\alpha }\\text{.}[\/latex]<\/div>\n<p id=\"import-auto-id2591053\"><strong>Solution for (b)<\/strong><\/p>\n<p id=\"import-auto-id1616167\">Here the angular velocity decreases from [latex]\\text{26.2 rad\/s}[\/latex] (250 rpm) to zero, so that <\/p>\n<p>[latex]\\Delta \\omega [\/latex] is <\/p>\n<p>[latex]\u2013\\text{26.2 rad\/s}[\/latex], and <\/p>\n<p>[latex]\\alpha [\/latex] is given to be <\/p>\n<p>[latex]\u2013\\text{87.3}\\phantom{\\rule{0.25em}{0ex}}{\\text{rad\/s}}^{2}[\/latex]. Thus,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-455\">[latex]\\begin{array}{lll}\\Delta t&amp; =&amp; \\frac{\u2013\\text{26.2 rad\/s}}{\u2013\\text{87.3}\\phantom{\\rule{0.25em}{0ex}}{\\text{rad\/s}}^{2}}\\\\ &amp; =&amp; \\text{0.300 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<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 data-effect=\"italics\">tangent<\/em> to the circle at the point of interest, as seen in <a href=\"#import-auto-id1019355\" class=\"autogenerated-content\">(Figure)<\/a>. Thus, linear acceleration is called <span data-type=\"term\" id=\"import-auto-id1997895\">tangential acceleration<\/span> [latex]{a}_{\\text{t}}[\/latex].<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1019355\">\n<div class=\"bc-figcaption figcaption\">In circular motion, linear acceleration [latex]a[\/latex], occurs as the magnitude of the velocity changes: [latex]a[\/latex] is tangent to the motion. In the context of circular motion, linear acceleration is also called tangential acceleration [latex]{a}_{\\text{t}}[\/latex].<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1449854\" data-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.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_11_01_02a.jpg\" data-media-type=\"image\/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.\" height=\"200\"><\/span><\/p><\/div>\n<p id=\"import-auto-id3063133\">Linear or tangential acceleration refers to changes in the magnitude of velocity but not its direction. We know from <a href=\"\/contents\/3ef5dfb6-0a8d-433e-9c8f-b8c860a3903b@2\">Uniform Circular Motion and Gravitation<\/a> that in circular motion centripetal acceleration, <\/p>\n<p>[latex]{a}_{\\text{c}}[\/latex], 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 href=\"#import-auto-id1995872\" class=\"autogenerated-content\">(Figure)<\/a>. Thus, <\/p>\n<p>[latex]{a}_{\\text{t}}[\/latex] and <\/p>\n<p>[latex]{a}_{\\text{c}}[\/latex] are perpendicular and independent of one another. Tangential acceleration <\/p>\n<p>[latex]{a}_{\\text{t}}[\/latex] is directly related to the angular acceleration <\/p>\n<p>[latex]\\alpha [\/latex] and is linked to an increase or decrease in the velocity, but not its direction.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1995872\">\n<div class=\"bc-figcaption figcaption\">Centripetal acceleration [latex]{a}_{\\text{c}}[\/latex] occurs as the direction of velocity changes; it is perpendicular to the circular motion. Centripetal and tangential acceleration are thus perpendicular to each other.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2648073\" data-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.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_11_01_03a.jpg\" data-media-type=\"image\/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.\" height=\"200\"><\/span><\/p><\/div>\n<p id=\"import-auto-id1993750\">Now we can find the exact relationship between linear acceleration [latex]{a}_{\\text{t}}[\/latex] and angular acceleration [latex]\\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=\"\/contents\/e12329e4-8d6c-49cf-aa45-6a05b26ebcba@2\">One-Dimensional Kinematics<\/a>) to be<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-85\">[latex]{a}_{\\text{t}}=\\frac{\\Delta v}{\\Delta t}\\text{.}[\/latex]<\/div>\n<p id=\"import-auto-id2407460\">For circular motion, note that [latex]v=\\mathrm{r\\omega }[\/latex], so that<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]{a}_{\\text{t}}=\\frac{\\Delta \\left(\\mathrm{r\\omega }\\right)}{\\Delta t}\\text{.}[\/latex]<\/div>\n<p id=\"import-auto-id2660038\">The radius [latex]r[\/latex] is constant for circular motion, and so [latex]\\text{\u0394}\\left(\\mathrm{r\\omega }\\right)=r\\left(\\Delta \\omega \\right)[\/latex]. Thus,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-688\">[latex]{a}_{\\text{t}}=r\\frac{\\Delta \\omega }{\\Delta t}\\text{.}[\/latex]<\/div>\n<p id=\"import-auto-id1861378\">By definition, [latex]\\alpha =\\frac{\\Delta \\omega }{\\Delta t}[\/latex]. Thus,<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]{a}_{\\text{t}}=\\mathrm{r\\alpha },[\/latex]<\/div>\n<p>or<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\alpha =\\frac{{a}_{\\text{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]\\alpha [\/latex].<\/p>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id3217117\">\n<div data-type=\"title\" class=\"title\">Calculating the Angular Acceleration of a Motorcycle Wheel<\/div>\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 href=\"#import-auto-id2415283\" class=\"autogenerated-content\">(Figure)<\/a>.)<\/p>\n<div class=\"bc-figure figure\">\n<div class=\"bc-figcaption figcaption\">The linear acceleration of a motorcycle is accompanied by an angular acceleration of its wheels.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id3149360\" data-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.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_11_01_04a.jpg\" data-media-type=\"image\/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\"><\/span><\/p><\/div>\n<p id=\"import-auto-id3068565\"><strong data-effect=\"bold\">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 [latex]{a}_{\\text{t}}[\/latex]. Then, the expression [latex]\\alpha =\\frac{{a}_{\\text{t}}}{r}[\/latex] can be used to find the angular acceleration.<\/p>\n<p id=\"import-auto-id2451305\"><strong data-effect=\"bold\">Solution<\/strong><\/p>\n<p id=\"import-auto-id2577576\">The linear acceleration is<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-20\">[latex]\\begin{array}{lll}{a}_{\\text{t}}&amp; =&amp; \\frac{\\Delta v}{\\Delta t}\\\\ &amp; =&amp; \\frac{\\text{30.0 m\/s}}{\\text{4.20 s}}\\\\ &amp; =&amp; \\text{7.14}\\phantom{\\rule{0.25em}{0ex}}{\\text{m\/s}}^{2}.\\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id3209774\">We also know the radius of the wheels. Entering the values for [latex]{a}_{\\text{t}}[\/latex] and [latex]r[\/latex] into <\/p>\n<p>[latex]\\alpha =\\frac{{a}_{\\text{t}}}{r}[\/latex], we get<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-155\">[latex]\\begin{array}{lll}\\alpha &amp; =&amp; \\frac{{a}_{\\text{t}}}{r}\\\\ &amp; =&amp; \\frac{\\text{7.14}\\phantom{\\rule{0.25em}{0ex}}{\\text{m\/s}}^{2}}{\\text{0.320 m}}\\\\ &amp; =&amp; \\text{22.3}\\phantom{\\rule{0.25em}{0ex}}{\\text{rad\/s}}^{2}.\\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id3062550\"><strong data-effect=\"bold\">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<p id=\"import-auto-id1117830\">So far, we have defined three rotational quantities\u2014 [latex]\\theta \\mathrm{,&nbsp;}\\omega [\/latex], and [latex]\\alpha [\/latex]. These quantities are analogous to the translational quantities [latex]x\\mathrm{,&nbsp;}v[\/latex], and [latex]a[\/latex]. <a href=\"#import-auto-id1572984\" class=\"autogenerated-content\">(Figure)<\/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<caption><span data-type=\"title\">Rotational and Translational Quantities<\/span><\/caption>\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]\\theta [\/latex]<\/td>\n<td>[latex]x[\/latex]<\/td>\n<td>[latex]\\theta =\\frac{x}{r}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\omega [\/latex]<\/td>\n<td>[latex]v[\/latex]<\/td>\n<td>[latex]\\omega =\\frac{v}{r}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\alpha [\/latex]<\/td>\n<td>[latex]a[\/latex]<\/td>\n<td>[latex]\\alpha =\\frac{{a}_{t}}{r}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id3292730\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Making Connections: Take-Home Experiment<\/div>\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 data-type=\"exercise\" class=\"exercise\" id=\"fs-id1870686\" data-element-type=\"check-understanding\" data-label=\"\">\n<div data-type=\"title\">Check Your Understanding<\/div>\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3093647\">\n<p id=\"import-auto-id1945320\">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 data-type=\"solution\" class=\"solution\" id=\"fs-id3103812\" data-print-placement=\"here\">\n<p id=\"import-auto-id2403563\">The magnitude of angular acceleration is [latex]\\alpha [\/latex] and its most common units are [latex]{\\text{rad\/s}}^{2}[\/latex]. 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<\/div>\n<\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"eip-669\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">PhET Explorations: Ladybug Revolution<\/div>\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<div class=\"bc-figure figure\" id=\"eip-id2624753\">\n<div class=\"bc-figcaption figcaption\"><a href=\"\/resources\/7c52f36f755df0a4d1bf4e75814c8135735d7055\/rotation_en.jar\">Ladybug Revolution<\/a><\/div>\n<p><span data-type=\"media\" id=\"Phet_module_11.1\" data-alt=\"\"><a href=\"\/resources\/7c52f36f755df0a4d1bf4e75814c8135735d7055\/rotation_en.jar\" data-type=\"image\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/PhET_Icon.png\" data-media-type=\"image\/png\" alt=\"\" data-print=\"false\" width=\"450\"><\/a><span data-media-type=\"image\/png\" data-print=\"true\" data-src=\"\/resources\/075500ad9f71890a85fe3f7a4137ac08e2b7907c\/PhET_Icon.png\" data-type=\"image\"><\/span><\/span><\/p><\/div>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id3034915\">\n<h1 data-type=\"title\">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]\\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]\\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]{a}_{\\text{t}}=\\frac{\\Delta v}{\\Delta t}[\/latex].<\/li>\n<li id=\"import-auto-id3035395\">For circular motion, note that [latex]v=\\mathrm{r\\omega }[\/latex], so that\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1588138\">[latex]{a}_{\\mathrm{\\text{t}}}=\\frac{\\text{\u0394}\\left(\\mathrm{r\\omega }\\right)}{\\Delta t}.[\/latex]<\/div>\n<\/li>\n<li id=\"import-auto-id1549399\">The radius r is constant for circular motion, and so [latex]\\mathrm{\\text{\u0394}}\\left(\\mathrm{r\\omega }\\right)=r\\Delta \\omega [\/latex]. Thus,\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id3232862\">[latex]{a}_{\\text{t}}=r\\frac{\\Delta \\omega }{\\Delta t}.[\/latex]<\/div>\n<\/li>\n<li id=\"import-auto-id3154788\">By definition, [latex]\\Delta \\omega \/\\Delta t=\\alpha [\/latex]. Thus,\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id3077640\">[latex]{a}_{\\text{t}}=\\mathrm{r\\alpha }[\/latex]<\/div>\n<p id=\"import-auto-id3064052\">or<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id3025466\">[latex]\\alpha =\\frac{{a}_{\\text{t}}}{r}.[\/latex]<\/div>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" id=\"fs-id1577741\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3006045\">\n<p id=\"import-auto-id2442420\">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 data-type=\"exercise\" class=\"exercise\" id=\"fs-id1867019\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2970163\">\n<p id=\"import-auto-id2052574\">Explain why centripetal acceleration changes the direction of velocity in circular motion but not its magnitude.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3046867\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1610119\">\n<p id=\"import-auto-id2442809\">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 data-type=\"exercise\" class=\"exercise\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3037278\">\n<p id=\"import-auto-id2639905\">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<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id1596665\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1571972\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2392409\">\n<p id=\"import-auto-id3355286\">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 data-type=\"solution\" class=\"solution\" id=\"fs-id1215980\" data-element-type=\"problems-exercises\">\n<p id=\"import-auto-id909629\">[latex]\\omega =0\\text{.}\\text{737 rev\/s}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2980135\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3177097\">\n<p id=\"import-auto-id1863817\"><strong>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 [latex]{\\text{rad\/s}}^{2}[\/latex]? (b) What is the tangential acceleration of a point 9.50 cm from the axis of rotation? (c) What is the radial acceleration in [latex]{\\text{m\/s}}^{2}[\/latex] and multiples of [latex]g[\/latex] of this point at full rpm?\n    <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1019355\">\n<p id=\"import-auto-id1548295\"><strong>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?\n    <\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id3081538\" data-element-type=\"problems-exercises\">\n<p id=\"import-auto-id2010945\">(a) [latex]-0\\text{.}{\\text{26 rad\/s}}^{2}[\/latex]<\/p>\n<p id=\"import-auto-id2206615\">(b) [latex]\\text{27}\\phantom{\\rule{0.25em}{0ex}}\\text{rev}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3063092\">\n<p id=\"import-auto-id955951\"><strong>Unreasonable Results<\/strong><\/p>\n<p id=\"eip-id1555320\">You are told that a basketball player spins the ball with an angular acceleration of [latex]\\text{100}{\\text{&nbsp;rad\/s}}^{2}[\/latex]. (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?\n    <\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\n<dl class=\"definition\" id=\"import-auto-id2410236\">\n<dt>angular acceleration<\/dt>\n<dd id=\"fs-id3257384\">the rate of change of angular velocity with time<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id3224451\">\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 class=\"definition\" id=\"import-auto-id3258416\">\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\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\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<p id=\"import-auto-id2400941\"><a href=\"\/contents\/3ef5dfb6-0a8d-433e-9c8f-b8c860a3903b@2\">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 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8ffb415af81ab9c23c1d2e7ec67d29c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#111;&#109;&#101;&#103;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> was defined as the time rate of change of angle <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-761998727948942ceb1b5763e45f01e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/>:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-257\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d01aba2cd8b97cb87d1980339640f77c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#116;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#44;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"61\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id2970170\">where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-761998727948942ceb1b5763e45f01e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> is the angle of rotation as seen in <a href=\"#import-auto-id1941476\" class=\"autogenerated-content\">(Figure)<\/a>. The relationship between angular velocity <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8ffb415af81ab9c23c1d2e7ec67d29c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#111;&#109;&#101;&#103;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> and linear velocity <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ef71511c70f0e4b25cc6bd69f3bc20c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> was also defined in <a href=\"\/contents\/44216d83-fe7e-461d-a5f4-9f3e2ffa284b@7\">Rotation Angle and Angular Velocity<\/a> as<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-82a38e9bb2b9b4db97145fa2d14dd7b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#61;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#114;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"9\" width=\"51\" style=\"vertical-align: -1px;\" \/><\/div>\n<p id=\"eip-179\">or<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-363\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-aad8d9834b0c73adc102744ba787098b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#118;&#125;&#123;&#114;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"50\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id2655439\">where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c409433a9e2dfcdb83360a974d243f18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/> is the radius of curvature, also seen in <a href=\"#import-auto-id1941476\" class=\"autogenerated-content\">(Figure)<\/a>. According to the sign convention, the counter clockwise direction is considered as positive direction and clockwise direction as negative <\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1941476\">\n<div class=\"bc-figcaption figcaption\">This figure shows uniform circular motion and some of its defined quantities.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id3233391\" data-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.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_11_01_01a.jpg\" data-media-type=\"image\/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\" \/><\/span><\/p>\n<\/div>\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 <span data-type=\"term\" id=\"import-auto-id2438108\">angular acceleration<\/span>, in which <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8ffb415af81ab9c23c1d2e7ec67d29c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#111;&#109;&#101;&#103;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> changes. The faster the change occurs, the greater the angular acceleration. Angular acceleration <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-946f8144d4e3d460c8621773145884d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> is defined as the rate of change of angular velocity. In equation form, angular acceleration is expressed as follows:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-974\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6693b414133f60945650caa87fc7a261_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#116;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#44;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"63\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id2403123\">where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f6f2f7f638b0a5b95214cc8bc6703182_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#92;&#111;&#109;&#101;&#103;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"26\" style=\"vertical-align: 0px;\" \/> is the <span data-type=\"term\" id=\"import-auto-id3406201\"> change in angular velocity<\/span> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0714636704a254c71bede042781bc57a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/> is the change in time. The units of angular acceleration are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3835f1cad98e318dc94b615103f5f8d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#97;&#100;&#47;&#115;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#116;&#101;&#120;&#116;&#123;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"73\" style=\"vertical-align: -4px;\" \/>, or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-edb731b5b9d3876ab828a1844ce41827_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#97;&#100;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"49\" style=\"vertical-align: -4px;\" \/>. If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8ffb415af81ab9c23c1d2e7ec67d29c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#111;&#109;&#101;&#103;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> increases, then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-946f8144d4e3d460c8621773145884d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> is positive. If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8ffb415af81ab9c23c1d2e7ec67d29c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#111;&#109;&#101;&#103;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> decreases, then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-946f8144d4e3d460c8621773145884d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> is negative.<\/p>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id3159590\">\n<div data-type=\"title\" class=\"title\">Calculating the Angular Acceleration and Deceleration of a Bike Wheel<\/div>\n<p id=\"import-auto-id3385431\">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 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-edb731b5b9d3876ab828a1844ce41827_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#97;&#100;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"49\" style=\"vertical-align: -4px;\" \/>. (b) If she now slams on the brakes, causing an angular acceleration of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-72c999921983a2cb5455ac2b9da2fd4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#56;&#55;&#46;&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#97;&#100;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"97\" style=\"vertical-align: -4px;\" \/>, 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 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c14a0b270a83c8a2c7d1d60d1161b79a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#116;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"58\" style=\"vertical-align: -6px;\" \/> because the final angular velocity and time are given. We see that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f6f2f7f638b0a5b95214cc8bc6703182_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#92;&#111;&#109;&#101;&#103;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"26\" style=\"vertical-align: 0px;\" \/> is 250 rpm and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0714636704a254c71bede042781bc57a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/> 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 data-type=\"equation\" class=\"equation\" id=\"eip-272\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2fcb15d0288837610a85b4e03655253c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#108;&#108;&#125;&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#116;&#125;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#53;&#48;&#32;&#114;&#112;&#109;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#46;&#48;&#48;&#32;&#115;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"115\" style=\"vertical-align: -18px;\" \/><\/div>\n<p id=\"import-auto-id3418450\">Because <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f6f2f7f638b0a5b95214cc8bc6703182_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#92;&#111;&#109;&#101;&#103;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"26\" style=\"vertical-align: 0px;\" \/> is in revolutions per minute (rpm) and we want the standard units of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-edb731b5b9d3876ab828a1844ce41827_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#97;&#100;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"49\" style=\"vertical-align: -4px;\" \/> for angular acceleration, we need to convert <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f6f2f7f638b0a5b95214cc8bc6703182_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#92;&#111;&#109;&#101;&#103;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"26\" style=\"vertical-align: 0px;\" \/> from rpm to rad\/s:<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a788256c5ea784acbc08bac17616a26f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#108;&#108;&#125;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#53;&#48;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#101;&#118;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#105;&#110;&#125;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&pi;&#32;&#114;&#97;&#100;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#101;&#118;&#125;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#32;&#109;&#105;&#110;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#48;&#32;&#115;&#101;&#99;&#125;&#125;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#54;&#46;&#50;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#97;&#100;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#125;&#125;&#46;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"226\" style=\"vertical-align: -18px;\" \/><\/div>\n<p id=\"import-auto-id3358717\">Entering this quantity into the expression for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-946f8144d4e3d460c8621773145884d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/>, we get<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-899\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e9ad83542ed08d6143864a208f3bddc0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#108;&#108;&#125;&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#116;&#125;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#54;&#46;&#50;&#32;&#114;&#97;&#100;&#47;&#115;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#46;&#48;&#48;&#32;&#115;&#125;&#125;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#46;&#50;&#52;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#114;&#97;&#100;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"68\" width=\"148\" style=\"vertical-align: -28px;\" \/><\/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 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0714636704a254c71bede042781bc57a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/>, yielding<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-273\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c34f6b79d8a0dece20bd26f7a4b748a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#116;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;&#123;&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"73\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id2591053\"><strong>Solution for (b)<\/strong><\/p>\n<p id=\"import-auto-id1616167\">Here the angular velocity decreases from <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7cf604affae685fc743d72e284713a2e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#54;&#46;&#50;&#32;&#114;&#97;&#100;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"79\" style=\"vertical-align: -4px;\" \/> (250 rpm) to zero, so that <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f6f2f7f638b0a5b95214cc8bc6703182_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#92;&#111;&#109;&#101;&#103;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"26\" style=\"vertical-align: 0px;\" \/> is <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f4fc412d8cd6a153abc1500a0ff2c79e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#54;&#46;&#50;&#32;&#114;&#97;&#100;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"92\" style=\"vertical-align: -4px;\" \/>, and <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-946f8144d4e3d460c8621773145884d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> is given to be <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7673c68b049848d650e4b63b4a84fbe9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#55;&#46;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#97;&#100;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"97\" style=\"vertical-align: -4px;\" \/>. Thus,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-455\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-18788b2bd17c3c2add4a1e300b041fcb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#108;&#108;&#125;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#116;&#38;&#32;&#61;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#54;&#46;&#50;&#32;&#114;&#97;&#100;&#47;&#115;&#125;&#125;&#123;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#55;&#46;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#97;&#100;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#46;&#51;&#48;&#48;&#32;&#115;&#46;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"47\" width=\"148\" style=\"vertical-align: -15px;\" \/><\/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<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 data-effect=\"italics\">tangent<\/em> to the circle at the point of interest, as seen in <a href=\"#import-auto-id1019355\" class=\"autogenerated-content\">(Figure)<\/a>. Thus, linear acceleration is called <span data-type=\"term\" id=\"import-auto-id1997895\">tangential acceleration<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-607369a67dd3a711377b38daba5ec458_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"14\" style=\"vertical-align: -3px;\" \/>.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1019355\">\n<div class=\"bc-figcaption figcaption\">In circular motion, linear acceleration <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/>, occurs as the magnitude of the velocity changes: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> is tangent to the motion. In the context of circular motion, linear acceleration is also called tangential acceleration <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-607369a67dd3a711377b38daba5ec458_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"14\" style=\"vertical-align: -3px;\" \/>.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1449854\" data-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.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_11_01_02a.jpg\" data-media-type=\"image\/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.\" height=\"200\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id3063133\">Linear or tangential acceleration refers to changes in the magnitude of velocity but not its direction. We know from <a href=\"\/contents\/3ef5dfb6-0a8d-433e-9c8f-b8c860a3903b@2\">Uniform Circular Motion and Gravitation<\/a> that in circular motion centripetal acceleration, <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ccd326c97d9f06d92f343270136cc684_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"15\" style=\"vertical-align: -3px;\" \/>, 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 href=\"#import-auto-id1995872\" class=\"autogenerated-content\">(Figure)<\/a>. Thus, <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-607369a67dd3a711377b38daba5ec458_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"14\" style=\"vertical-align: -3px;\" \/> and <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ccd326c97d9f06d92f343270136cc684_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"15\" style=\"vertical-align: -3px;\" \/> are perpendicular and independent of one another. Tangential acceleration <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-607369a67dd3a711377b38daba5ec458_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"14\" style=\"vertical-align: -3px;\" \/> is directly related to the angular acceleration <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-946f8144d4e3d460c8621773145884d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> and is linked to an increase or decrease in the velocity, but not its direction.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1995872\">\n<div class=\"bc-figcaption figcaption\">Centripetal acceleration <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ccd326c97d9f06d92f343270136cc684_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"15\" style=\"vertical-align: -3px;\" \/> occurs as the direction of velocity changes; it is perpendicular to the circular motion. Centripetal and tangential acceleration are thus perpendicular to each other.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2648073\" data-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.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_11_01_03a.jpg\" data-media-type=\"image\/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.\" height=\"200\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id1993750\">Now we can find the exact relationship between linear acceleration <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-607369a67dd3a711377b38daba5ec458_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"14\" style=\"vertical-align: -3px;\" \/> and angular acceleration <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-946f8144d4e3d460c8621773145884d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/>. Because linear acceleration is proportional to a change in the magnitude of the velocity, it is defined (as it was in <a href=\"\/contents\/e12329e4-8d6c-49cf-aa45-6a05b26ebcba@2\">One-Dimensional Kinematics<\/a>) to be<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-85\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-15b3eabeddecf2db7df5609c4b9ca3b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#118;&#125;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#116;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"66\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id2407460\">For circular motion, note that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-82a38e9bb2b9b4db97145fa2d14dd7b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#61;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#114;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"9\" width=\"51\" style=\"vertical-align: -1px;\" \/>, so that<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f0db1ad969fe34600eda11fb9e789285_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#114;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#116;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"84\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id2660038\">The radius <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c409433a9e2dfcdb83360a974d243f18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/> is constant for circular motion, and so <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c47c4191fac81ebbee63a95af180eca3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#114;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#114;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"106\" style=\"vertical-align: -4px;\" \/>. Thus,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-688\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0998842b398aa0d07da9d9efa647384f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#125;&#125;&#61;&#114;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#116;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"76\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id1861378\">By definition, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c14a0b270a83c8a2c7d1d60d1161b79a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#116;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"58\" style=\"vertical-align: -6px;\" \/>. Thus,<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1953969d3b2057926debf71c09ae17b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#125;&#125;&#61;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#114;&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"62\" style=\"vertical-align: -4px;\" \/><\/div>\n<p>or<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-963fb5c561390638951e7d4d5dcda693_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#97;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#125;&#125;&#125;&#123;&#114;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"56\" style=\"vertical-align: -6px;\" \/><\/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 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-946f8144d4e3d460c8621773145884d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/>.<\/p>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id3217117\">\n<div data-type=\"title\" class=\"title\">Calculating the Angular Acceleration of a Motorcycle Wheel<\/div>\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 href=\"#import-auto-id2415283\" class=\"autogenerated-content\">(Figure)<\/a>.)<\/p>\n<div class=\"bc-figure figure\">\n<div class=\"bc-figcaption figcaption\">The linear acceleration of a motorcycle is accompanied by an angular acceleration of its wheels.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id3149360\" data-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.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_11_01_04a.jpg\" data-media-type=\"image\/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\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id3068565\"><strong data-effect=\"bold\">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 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-607369a67dd3a711377b38daba5ec458_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"14\" style=\"vertical-align: -3px;\" \/>. Then, the expression <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c0ff50ff7d64a93cc77763f6d9536231_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#97;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#125;&#125;&#125;&#123;&#114;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"50\" style=\"vertical-align: -6px;\" \/> can be used to find the angular acceleration.<\/p>\n<p id=\"import-auto-id2451305\"><strong data-effect=\"bold\">Solution<\/strong><\/p>\n<p id=\"import-auto-id2577576\">The linear acceleration is<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-20\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3268b0f670103494f4741f19f3f43099_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#108;&#108;&#125;&#123;&#97;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#125;&#125;&#38;&#32;&#61;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#118;&#125;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#116;&#125;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#48;&#46;&#48;&#32;&#109;&#47;&#115;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#46;&#50;&#48;&#32;&#115;&#125;&#125;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#55;&#46;&#49;&#52;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;&#46;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"68\" width=\"140\" style=\"vertical-align: -28px;\" \/><\/div>\n<p id=\"import-auto-id3209774\">We also know the radius of the wheels. Entering the values for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-607369a67dd3a711377b38daba5ec458_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"14\" style=\"vertical-align: -3px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c409433a9e2dfcdb83360a974d243f18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/> into <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c0ff50ff7d64a93cc77763f6d9536231_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#97;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#125;&#125;&#125;&#123;&#114;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"50\" style=\"vertical-align: -6px;\" \/>, we get<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-155\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d5c74405ad2824e0c80a1b61d0ca87f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#108;&#108;&#125;&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#97;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#125;&#125;&#125;&#123;&#114;&#125;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#55;&#46;&#49;&#52;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#46;&#51;&#50;&#48;&#32;&#109;&#125;&#125;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#50;&#46;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#97;&#100;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;&#46;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"67\" width=\"147\" style=\"vertical-align: -29px;\" \/><\/div>\n<p id=\"import-auto-id3062550\"><strong data-effect=\"bold\">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<p id=\"import-auto-id1117830\">So far, we have defined three rotational quantities\u2014 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9eabf713625ce418bb4ebfe6101a685d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#44;&#32;&#125;&#92;&#111;&#109;&#101;&#103;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"25\" style=\"vertical-align: -4px;\" \/>, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-946f8144d4e3d460c8621773145884d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/>. These quantities are analogous to the translational quantities <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9f6ec05afaadb20714a82258565e28e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#44;&#32;&#125;&#118;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"24\" style=\"vertical-align: -4px;\" \/>, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/>. <a href=\"#import-auto-id1572984\" class=\"autogenerated-content\">(Figure)<\/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<caption><span data-type=\"title\">Rotational and Translational Quantities<\/span><\/caption>\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><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-761998727948942ceb1b5763e45f01e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0072250d0f2f9c95162f29f5e57d326d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#114;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"42\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8ffb415af81ab9c23c1d2e7ec67d29c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#111;&#109;&#101;&#103;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ef71511c70f0e4b25cc6bd69f3bc20c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c11aef5737922d9f1519e4324aa2cd4d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#118;&#125;&#123;&#114;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"45\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-946f8144d4e3d460c8621773145884d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3bfafeda010be7b6eda97251265b0f3a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#97;&#125;&#95;&#123;&#116;&#125;&#125;&#123;&#114;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"50\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id3292730\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Making Connections: Take-Home Experiment<\/div>\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 data-type=\"exercise\" class=\"exercise\" id=\"fs-id1870686\" data-element-type=\"check-understanding\" data-label=\"\">\n<div data-type=\"title\">Check Your Understanding<\/div>\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3093647\">\n<p id=\"import-auto-id1945320\">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 data-type=\"solution\" class=\"solution\" id=\"fs-id3103812\" data-print-placement=\"here\">\n<p id=\"import-auto-id2403563\">The magnitude of angular acceleration is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-946f8144d4e3d460c8621773145884d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> and its most common units are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-edb731b5b9d3876ab828a1844ce41827_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#97;&#100;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"49\" style=\"vertical-align: -4px;\" \/>. 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<\/div>\n<\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"eip-669\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">PhET Explorations: Ladybug Revolution<\/div>\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<div class=\"bc-figure figure\" id=\"eip-id2624753\">\n<div class=\"bc-figcaption figcaption\"><a href=\"\/resources\/7c52f36f755df0a4d1bf4e75814c8135735d7055\/rotation_en.jar\">Ladybug Revolution<\/a><\/div>\n<p><span data-type=\"media\" id=\"Phet_module_11.1\" data-alt=\"\"><a href=\"\/resources\/7c52f36f755df0a4d1bf4e75814c8135735d7055\/rotation_en.jar\" data-type=\"image\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/PhET_Icon.png\" data-media-type=\"image\/png\" alt=\"\" data-print=\"false\" width=\"450\" \/><\/a><span data-media-type=\"image\/png\" data-print=\"true\" data-src=\"\/resources\/075500ad9f71890a85fe3f7a4137ac08e2b7907c\/PhET_Icon.png\" data-type=\"image\"><\/span><\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id3034915\">\n<h1 data-type=\"title\">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 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-18f370eb61e3f538640a1b6b881f60b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#116;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"56\" style=\"vertical-align: -6px;\" \/>.<\/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 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c14a0b270a83c8a2c7d1d60d1161b79a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#116;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"58\" style=\"vertical-align: -6px;\" \/>.<\/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 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a85ea6dd9f18574d270c4503024a0f26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#118;&#125;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#116;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"60\" style=\"vertical-align: -6px;\" \/>.<\/li>\n<li id=\"import-auto-id3035395\">For circular motion, note that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-82a38e9bb2b9b4db97145fa2d14dd7b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#61;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#114;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"9\" width=\"51\" style=\"vertical-align: -1px;\" \/>, so that\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1588138\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c32bae40863614a37edf63c7eec25c21_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#125;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#114;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#116;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"72\" style=\"vertical-align: -6px;\" \/><\/div>\n<\/li>\n<li id=\"import-auto-id1549399\">The radius r is constant for circular motion, and so <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-cf23f20c707c613b25f25919916e9ccd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#114;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#114;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#92;&#111;&#109;&#101;&#103;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"89\" style=\"vertical-align: -4px;\" \/>. Thus,\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id3232862\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5bd72dcc1897bdba899b665119572e78_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#125;&#125;&#61;&#114;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;&#123;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#116;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"76\" style=\"vertical-align: -6px;\" \/><\/div>\n<\/li>\n<li id=\"import-auto-id3154788\">By definition, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-af03220587e8ed6347a56bac772f237a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#47;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#116;&#61;&#92;&#97;&#108;&#112;&#104;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"91\" style=\"vertical-align: -5px;\" \/>. Thus,\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id3077640\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f8c4e33f687f2014517165300efb8583_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#125;&#125;&#61;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#114;&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"57\" style=\"vertical-align: -3px;\" \/><\/div>\n<p id=\"import-auto-id3064052\">or<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id3025466\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-963fb5c561390638951e7d4d5dcda693_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#97;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#125;&#125;&#125;&#123;&#114;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"56\" style=\"vertical-align: -6px;\" \/><\/div>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" id=\"fs-id1577741\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3006045\">\n<p id=\"import-auto-id2442420\">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 data-type=\"exercise\" class=\"exercise\" id=\"fs-id1867019\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2970163\">\n<p id=\"import-auto-id2052574\">Explain why centripetal acceleration changes the direction of velocity in circular motion but not its magnitude.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3046867\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1610119\">\n<p id=\"import-auto-id2442809\">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 data-type=\"exercise\" class=\"exercise\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3037278\">\n<p id=\"import-auto-id2639905\">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<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id1596665\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1571972\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2392409\">\n<p id=\"import-auto-id3355286\">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 data-type=\"solution\" class=\"solution\" id=\"fs-id1215980\" data-element-type=\"problems-exercises\">\n<p id=\"import-auto-id909629\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-99383efc76b4eb9be8ec2ad8b45a9070_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#61;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#55;&#51;&#55;&#32;&#114;&#101;&#118;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"122\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2980135\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3177097\">\n<p id=\"import-auto-id1863817\"><strong>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 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-edb731b5b9d3876ab828a1844ce41827_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#97;&#100;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"49\" style=\"vertical-align: -4px;\" \/>? (b) What is the tangential acceleration of a point 9.50 cm from the axis of rotation? (c) What is the radial acceleration in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e37733fb743dcabc7c6a40d6150c0344_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"38\" style=\"vertical-align: -4px;\" \/> and multiples of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d208fd391fa57c168dc0f151de829fee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> of this point at full rpm?\n    <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1019355\">\n<p id=\"import-auto-id1548295\"><strong>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?\n    <\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id3081538\" data-element-type=\"problems-exercises\">\n<p id=\"import-auto-id2010945\">(a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-64d8cb05afbacb9f967ba0a1651aaa9d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#54;&#32;&#114;&#97;&#100;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"99\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"import-auto-id2206615\">(b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-10605b8ec0fb23154dd1bba7c20841e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#55;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#101;&#118;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"46\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3063092\">\n<p id=\"import-auto-id955951\"><strong>Unreasonable Results<\/strong><\/p>\n<p id=\"eip-id1555320\">You are told that a basketball player spins the ball with an angular acceleration of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-860858f8873e287e0cdf4a84432bbcd9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#48;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#114;&#97;&#100;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"80\" style=\"vertical-align: -4px;\" \/>. (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?\n    <\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\n<dl class=\"definition\" id=\"import-auto-id2410236\">\n<dt>angular acceleration<\/dt>\n<dd id=\"fs-id3257384\">the rate of change of angular velocity with time<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id3224451\">\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 class=\"definition\" id=\"import-auto-id3258416\">\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","protected":false},"author":211,"menu_order":1,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":"all-rights-reserved"},"chapter-type":[],"contributor":[],"license":[56],"class_list":["post-515","chapter","type-chapter","status-publish","hentry","license-all-rights-reserved"],"part":506,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/515","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/users\/211"}],"version-history":[{"count":1,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/515\/revisions"}],"predecessor-version":[{"id":516,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/515\/revisions\/516"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/parts\/506"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/515\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/media?parent=515"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapter-type?post=515"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/contributor?post=515"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/license?post=515"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}