{"id":311,"date":"2017-10-27T16:29:27","date_gmt":"2017-10-27T16:29:27","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/rotation-angle-and-angular-velocity\/"},"modified":"2017-11-08T03:24:15","modified_gmt":"2017-11-08T03:24:15","slug":"rotation-angle-and-angular-velocity","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/rotation-angle-and-angular-velocity\/","title":{"raw":"Rotation Angle and Angular Velocity","rendered":"Rotation Angle and Angular Velocity"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Define arc length, rotation angle, radius of curvature and angular velocity.<\/li>\n<li>Calculate the angular velocity of a car wheel spin.<\/li>\n<\/ul>\n<\/div>\n<p id=\"import-auto-id1571938\">In <a href=\"\/contents\/e12329e4-8d6c-49cf-aa45-6a05b26ebcba@2\">Kinematics<\/a>, we studied motion along a straight line and introduced such concepts as displacement, velocity, and acceleration. <a href=\"\/contents\/f309a0f9-63fb-46ca-9585-d1e1dc96a142@3\">Two-Dimensional Kinematics<\/a> dealt with motion in two dimensions. Projectile motion is a special case of two-dimensional kinematics in which the object is projected into the air, while being subject to the gravitational force, and lands a distance away. In this chapter, we consider situations where the object does not land but moves in a curve. We begin the study of uniform circular motion by defining two angular quantities needed to describe rotational motion.<\/p>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1488693\">\n<h1 data-type=\"title\">Rotation Angle<\/h1>\n<p id=\"import-auto-id1917384\">When objects rotate about some axis\u2014for example, when the CD (compact disc) in <a href=\"#import-auto-id3402904\" class=\"autogenerated-content\">(Figure)<\/a> rotates about its center\u2014each point in the object follows a circular arc. Consider a line from the center of the CD to its edge. Each <span data-type=\"term\" id=\"import-auto-id2654027\">pit<\/span> used to record sound along this line moves through the same angle in the same amount of time. The rotation angle is the amount of rotation and is analogous to linear distance. We define the <span data-type=\"term\" id=\"import-auto-id3255842\">rotation angle<\/span> [latex]\\text{\u0394}\\theta [\/latex] to be the ratio of the arc length to the radius of curvature:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-211\">[latex]\\text{\u0394}\\theta =\\frac{\\text{\u0394}s}{r}\\text{.}[\/latex]<\/div>\n<p id=\"import-auto-id2968424\">\n<\/p><div class=\"bc-figure figure\" id=\"import-auto-id3402904\">\n<div class=\"bc-figcaption figcaption\">All points on a CD travel in circular arcs. The pits along a line from the center to the edge all move through the same angle [latex]\\text{\u0394}\\theta [\/latex] in a time [latex]\\text{\u0394}t[\/latex].<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id3246430\" data-alt=\"The figure shows the back side of a compact disc. There is a scratched part on the upper right side of the C D, about one-fifth size of the whole area, with inner circular dots clearly visible. Two line segments are drawn enclosing the scratched area from the border of the C D to the middle plastic portion. A curved arrow is drawn between the two line segments near this middle portion and angle delta theta written alongside it.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_07_01_01aa.jpg\" data-media-type=\"image\/jpg\" alt=\"The figure shows the back side of a compact disc. There is a scratched part on the upper right side of the C D, about one-fifth size of the whole area, with inner circular dots clearly visible. Two line segments are drawn enclosing the scratched area from the border of the C D to the middle plastic portion. A curved arrow is drawn between the two line segments near this middle portion and angle delta theta written alongside it.\" width=\"225\"><\/span><\/p><\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id3418263\">\n<div class=\"bc-figcaption figcaption\">The radius of a circle is rotated through an angle [latex]\\text{\u0394}\\theta [\/latex]. The arc length [latex]\\text{\u0394s}[\/latex] is described on the circumference. <\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2601460\" data-alt=\"A circle of radius r and center O is shown. A radius O-A of the circle is rotated through angle delta theta about the center O to terminate as radius O-B. The arc length A-B is marked as delta s.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_07_01_01ab.jpg\" data-media-type=\"image\/png\" alt=\"A circle of radius r and center O is shown. A radius O-A of the circle is rotated through angle delta theta about the center O to terminate as radius O-B. The arc length A-B is marked as delta s.\" width=\"300\"><\/span><\/p><\/div>\n<p id=\"import-auto-id3025908\">The <span data-type=\"term\">arc length<\/span>[latex]\\phantom{\\rule{0.25em}{0ex}}\\text{\u0394}s[\/latex] is the distance traveled along a circular path as shown in <a href=\"#import-auto-id3418263\" class=\"autogenerated-content\">(Figure)<\/a> Note that [latex]r[\/latex] is the <span data-type=\"term\" id=\"import-auto-id2920709\">radius of curvature<\/span> of the circular path.<\/p>\n<p id=\"import-auto-id1471752\">We know that for one complete revolution, the arc length is the circumference of a circle of radius [latex]r[\/latex]. The circumference of a circle is [latex]2\\pi r[\/latex]. Thus for one complete revolution the rotation angle is<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\text{\u0394}\\theta =\\frac{2\\pi r}{r}=2\\pi \\text{.}[\/latex]<\/div>\n<p id=\"import-auto-id1842986\">This result is the basis for defining the units used to measure rotation angles, [latex]\\text{\u0394}\\theta [\/latex] to be <span data-type=\"term\" id=\"import-auto-id2625941\">radians<\/span> (rad), defined so that<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-135\">[latex]2\\pi \\phantom{\\rule{0.25em}{0ex}}\\text{rad}=\\text{1 revolution.}[\/latex]<\/div>\n<p id=\"eip-425\">A comparison of some useful angles expressed in both degrees and radians is shown in <a href=\"#import-auto-id2588905\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<table id=\"import-auto-id2588905\" summary=\"The table compares various angle measures in degrees (first column) and radians (second colum).\">\n<caption><span data-type=\"title\">Comparison of Angular Units<\/span><\/caption>\n<thead>\n<tr>\n<th>Degree Measures<\/th>\n<th>Radian Measure<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]\\text{30\u00ba}[\/latex]<\/td>\n<td>[latex]\\frac{\\pi }{6}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\text{60\u00ba}[\/latex]<\/td>\n<td>[latex]\\frac{\\pi }{3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\text{90\u00ba}[\/latex]<\/td>\n<td>[latex]\\frac{\\pi }{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\text{120\u00ba}[\/latex]<\/td>\n<td>[latex]\\frac{2\\pi }{3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\text{135\u00ba}[\/latex]<\/td>\n<td>[latex]\\frac{3\\pi }{4}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\text{180\u00ba}[\/latex]<\/td>\n<td>[latex]\\pi [\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"bc-figure figure\" id=\"import-auto-id2442865\">\n<div class=\"bc-figcaption figcaption\">Points 1 and 2 rotate through the same angle ([latex]\\text{\u0394}\\theta [\/latex]), but point 2 moves through a greater arc length [latex]\\left(\\text{\u0394}s\\right)[\/latex] because it is at a greater distance from the center of rotation [latex]\\left(r\\right)[\/latex]. <\/div>\n<p><span data-type=\"media\" id=\"import-auto-id3356541\" data-alt=\"A circle is shown. Two radii of the circle, inclined at an acute angle delta theta, are shown. On one of the radii, two points, one and two are marked. The point one is inside the circle through which an arc between the two radii is shown. The point two is on the cirumfenrence of the circle. The two arc lengths are delta s one and delta s two respectively for the two points.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_07_01_02a.jpg\" data-media-type=\"image\/jpg\" alt=\"A circle is shown. Two radii of the circle, inclined at an acute angle delta theta, are shown. On one of the radii, two points, one and two are marked. The point one is inside the circle through which an arc between the two radii is shown. The point two is on the cirumfenrence of the circle. The two arc lengths are delta s one and delta s two respectively for the two points.\" width=\"230\"><\/span><\/p><\/div>\n<p id=\"import-auto-id1930108\">If [latex]\\text{\u0394}\\theta =2\\pi [\/latex] rad, then the CD has made one complete revolution, and every point on the CD is back at its original position. Because there are [latex]\\text{360\u00ba}[\/latex] in a circle or one revolution, the relationship between radians and degrees is thus<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-808\">[latex]2\\pi \\phantom{\\rule{0.25em}{0ex}}\\text{rad}=\\text{360\u00ba}[\/latex]<\/div>\n<p id=\"import-auto-id2052087\">so that<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]1\\phantom{\\rule{0.25em}{0ex}}\\text{rad}=\\frac{\\text{360\u00ba}}{2\\pi }\\approx \\text{57.}3\u00ba\\text{.}[\/latex]<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id3104613\">\n<h1 data-type=\"title\">Angular Velocity<\/h1>\n<p id=\"import-auto-id2681279\">How fast is an object rotating? We define <span data-type=\"term\" id=\"import-auto-id2962847\">angular velocity<\/span> [latex]\\omega [\/latex] as the rate of change of an angle. In symbols, this is<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-759\">[latex]\\omega =\\frac{\\text{\u0394}\\theta }{\\text{\u0394}t}\\text{,}[\/latex]<\/div>\n<p>where an angular rotation [latex]\\text{\u0394}\\theta [\/latex] takes place in a time [latex]\\text{\u0394}t[\/latex]. The greater the rotation angle in a given amount of time, the greater the angular velocity. The units for angular velocity are radians per second (rad\/s).<\/p>\n<p id=\"import-auto-id2621168\">Angular velocity [latex]\\omega [\/latex] is analogous to linear velocity [latex]v[\/latex]. To get the precise relationship between angular and linear velocity, we again consider a pit on the rotating CD. This pit moves an arc length [latex]\\text{\u0394}s[\/latex] in a time [latex]\\text{\u0394}t[\/latex], and so it has a linear velocity<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]v=\\frac{\\text{\u0394}s}{\\text{\u0394}t}\\text{.}[\/latex]<\/div>\n<p id=\"import-auto-id1840944\">From [latex]\\text{\u0394}\\theta =\\frac{\\text{\u0394}s}{r}[\/latex] we see that [latex]\\text{\u0394}s=r\\text{\u0394}\\theta [\/latex]. Substituting this into the expression for [latex]v[\/latex] gives<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]v=\\frac{r\\text{\u0394}\\theta }{\\text{\u0394}t}=\\mathrm{r\\omega }\\text{.}[\/latex]<\/div>\n<p id=\"import-auto-id1549179\">We write this relationship in two different ways and gain two different insights:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-639\">[latex]v=\\mathrm{r\\omega }\\text{&nbsp;or&nbsp;}\\omega =\\frac{v}{r}\\text{.}[\/latex]<\/div>\n<p id=\"import-auto-id2680923\">The first relationship in [latex]v=\\mathrm{r\\omega }\\text{&nbsp;or&nbsp;}\\omega =\\frac{v}{r}[\/latex] states that the linear velocity [latex]v[\/latex] is proportional to the distance from the center of rotation, thus, it is largest for a point on the rim (largest [latex]r[\/latex]), as you might expect. We can also call this linear speed [latex]v[\/latex] of a point on the rim the <em data-effect=\"italics\">tangential speed<\/em>. The second relationship in [latex]v=\\mathrm{r\\omega }\\text{&nbsp;or&nbsp;}\\omega =\\frac{v}{r}[\/latex] can be illustrated by considering the tire of a moving car. Note that the speed of a point on the rim of the tire is the same as the speed [latex]v[\/latex] of the car. See <a href=\"#import-auto-id2931190\" class=\"autogenerated-content\">(Figure)<\/a>. So the faster the car moves, the faster the tire spins\u2014large [latex]v[\/latex] means a large [latex]\\omega [\/latex], because [latex]v=\\mathrm{r\\omega }[\/latex]. Similarly, a larger-radius tire rotating at the same angular velocity ([latex]\\omega [\/latex]) will produce a greater linear speed ([latex]v[\/latex]) for the car.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id2931190\">\n<div class=\"bc-figcaption figcaption\">A car moving at a velocity [latex]v[\/latex] to the right has a tire rotating with an angular velocity [latex]\\omega [\/latex].The speed of the tread of the tire relative to the axle is [latex]v[\/latex], the same as if the car were jacked up. Thus the car moves forward at linear velocity [latex]v=\\mathrm{r\\omega }[\/latex], where [latex]r[\/latex] is the tire radius. A larger angular velocity for the tire means a greater velocity for the car.\n      <\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1449564\" data-alt=\"The given figure shows the front wheel of a car. The radius of the car wheel, r, is shown as an arrow and the linear velocity, v, is shown with a green horizontal arrow pointing rightward. The angular velocity, omega, is shown with a clockwise-curved arrow over the wheel.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_07_01_03a.jpg\" data-media-type=\"image\/jpg\" alt=\"The given figure shows the front wheel of a car. The radius of the car wheel, r, is shown as an arrow and the linear velocity, v, is shown with a green horizontal arrow pointing rightward. The angular velocity, omega, is shown with a clockwise-curved arrow over the wheel.\" width=\"300\"><\/span><\/p><\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id2589253\">\n<div data-type=\"title\" class=\"title\">How Fast Does a Car Tire Spin?<\/div>\n<p id=\"import-auto-id3402720\">Calculate the angular velocity of a 0.300 m radius car tire when the car travels at [latex]\\text{15}\\text{.}0\\phantom{\\rule{0.25em}{0ex}}\\text{m\/s}[\/latex] (about [latex]\\text{54}\\phantom{\\rule{0.25em}{0ex}}\\text{km\/h}[\/latex]). See <a href=\"#import-auto-id2931190\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<p id=\"import-auto-id2968606\"><strong>Strategy<\/strong><\/p>\n<p id=\"fs-id1374481\">Because the linear speed of the tire rim is the same as the speed of the car, we have<br>\n[latex]v=\\text{15.0 m\/s}.[\/latex]  <\/p>\n<p>The radius of the tire is given to be<br>\n[latex]r=\\text{0.300 m}.[\/latex] Knowing <\/p>\n<p>[latex]v[\/latex] and [latex]r[\/latex], we can use the second relationship in [latex]v=\\mathrm{r\\omega }\\mathrm{,&nbsp;}\\omega =\\frac{v}{r}[\/latex] to calculate the angular velocity.<\/p>\n<p id=\"import-auto-id2949936\"><strong>Solution<\/strong><\/p>\n<p id=\"eip-105\">To calculate the angular velocity, we will use the following relationship:<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\omega =\\frac{v}{r}\\text{.}[\/latex]<\/div>\n<p id=\"import-auto-id3199878\">Substituting the knowns,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-451\">[latex]\\omega =\\frac{\\text{15}\\text{.}0\\phantom{\\rule{0.25em}{0ex}}\\text{m\/s}}{0\\text{.}\\text{300}\\phantom{\\rule{0.25em}{0ex}}\\text{m}}=\\text{50}\\text{.}0\\phantom{\\rule{0.25em}{0ex}}\\text{rad\/s.}[\/latex]<\/div>\n<p id=\"import-auto-id1889900\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id956895\">When we cancel units in the above calculation, we get 50.0\/s. But the angular velocity must have units of rad\/s. Because radians are actually unitless (radians are defined as a ratio of distance), we can simply insert them into the answer for the angular velocity. Also note that if an earth mover with much larger tires, say 1.20 m in radius, were moving at the same speed of 15.0 m\/s, its tires would rotate more slowly. They would have an angular velocity <\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-971\">[latex]\\omega =\\left(\\text{15}\\text{.}0\\phantom{\\rule{0.25em}{0ex}}\\text{m\/s}\\right)\/\\left(1\\text{.}\\text{20}\\phantom{\\rule{0.25em}{0ex}}\\text{m}\\right)=\\text{12}\\text{.}5\\phantom{\\rule{0.25em}{0ex}}\\text{rad\/s.}[\/latex]<\/div>\n<\/div>\n<p id=\"import-auto-id2415283\">Both [latex]\\omega [\/latex] and [latex]v[\/latex] have directions (hence they are angular and linear <em data-effect=\"italics\">velocities<\/em>, respectively). Angular velocity has only two directions with respect to the axis of rotation\u2014it is either clockwise or counterclockwise. Linear velocity is tangent to the path, as illustrated in <a href=\"#import-auto-id1452850\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id2584087\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Take-Home Experiment<\/div>\n<p id=\"import-auto-id1986367\">Tie an object to the end of a string and swing it around in a horizontal circle above your head (swing at your wrist). Maintain uniform speed as the object swings and measure the angular velocity of the motion. What is the approximate speed of the object? Identify a point close to your hand and take appropriate measurements to calculate the linear speed at this point. Identify other circular motions and measure their angular velocities.<\/p>\n<\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id1452850\">\n<div class=\"bc-figcaption figcaption\">As an object moves in a circle, here a fly on the edge of an old-fashioned vinyl record, its instantaneous velocity is always tangent to the circle. The direction of the angular velocity is clockwise in this case. <\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2604789\" data-alt=\"The given figure shows the top view of an old fashioned vinyl record. Two perpendicular line segments are drawn through the center of the circular record, one vertically upward and one horizontal to the right side. Two flies are shown at the end points of the vertical lines near the borders of the record. Two arrows are also drawn perpendicularly rightward through the end points of these vertical lines depicting linear velocities. A curved arrow is also drawn at the center circular part of the record which shows the angular velocity.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_07_01_04a.jpg\" data-media-type=\"image\/jpg\" alt=\"The given figure shows the top view of an old fashioned vinyl record. Two perpendicular line segments are drawn through the center of the circular record, one vertically upward and one horizontal to the right side. Two flies are shown at the end points of the vertical lines near the borders of the record. Two arrows are also drawn perpendicularly rightward through the end points of these vertical lines depicting linear velocities. A curved arrow is also drawn at the center circular part of the record which shows the angular velocity.\" width=\"250\"><\/span><\/p><\/div>\n<\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"eip-270\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">PhET Explorations: Ladybug Revolution<\/div>\n<div class=\"bc-figure figure\" id=\"eip-id1171550\">\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_7.1\" 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<p id=\"eip-id1169738118030\">Join the ladybug in an exploration of rotational motion. Rotate the merry-go-round to change its angle, or choose a constant angular velocity or angular acceleration. Explore how circular motion relates to the bug's x,y position, velocity, and acceleration using vectors or graphs.<\/p>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id3399151\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul id=\"fs-id1104163\">\n<li id=\"import-auto-id2603294\">Uniform circular motion is motion in a circle at constant speed. The rotation angle [latex]\\text{\u0394}\\theta [\/latex] is defined as the ratio of the arc length to the radius of curvature:\n<div data-type=\"equation\" class=\"equation\">[latex]\\text{\u0394}\\theta =\\frac{\\text{\u0394}s}{r}\\text{,}[\/latex]<\/div>\n<p id=\"import-auto-id3062885\">where arc length [latex]\\text{\u0394}s[\/latex] is distance traveled along a circular path and [latex]r[\/latex] is the radius of curvature of the circular path. The quantity [latex]\\text{\u0394}\\theta [\/latex] is measured in units of radians (rad), for which<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-567\">[latex]2\\pi \\phantom{\\rule{0.25em}{0ex}}\\text{rad}=\\text{360\u00ba}\\text{=&nbsp;}1\\text{&nbsp;revolution.}[\/latex]<\/div>\n<\/li>\n<li id=\"import-auto-id2384053\">The conversion between radians and degrees is [latex]1\\phantom{\\rule{0.25em}{0ex}}\\text{rad}=\\text{57}\\text{.}3\\text{\u00ba}[\/latex].<\/li>\n<li id=\"import-auto-id2442213\">Angular velocity [latex]\\omega [\/latex] is the rate of change of an angle,\n<div data-type=\"equation\" class=\"equation\" id=\"eip-969\">[latex]\\omega =\\frac{\\text{\u0394}\\theta }{\\text{\u0394}t}\\text{,}[\/latex]<\/div>\n<p id=\"import-auto-id1439148\">where a rotation [latex]\\text{\u0394}\\theta [\/latex] takes place in a time [latex]\\text{\u0394}t[\/latex]. The units of angular velocity are radians per second (rad\/s). Linear velocity [latex]v[\/latex] and angular velocity [latex]\\omega [\/latex] are related by<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-513\">[latex]v=\\mathrm{r\\omega }\\text{&nbsp;or&nbsp;}\\omega =\\frac{v}{r}\\text{.}[\/latex]<\/div>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" id=\"fs-id3103314\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3119404\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2969010\">\n<p id=\"import-auto-id2931754\">There is an analogy between rotational and linear physical quantities. What rotational quantities are analogous to distance and velocity?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id2381548\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problem Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3004274\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3253787\">\n<p id=\"import-auto-id3415350\">Semi-trailer trucks have an odometer on one hub of a trailer wheel. The hub is weighted so that it does not rotate, but it contains gears to count the number of wheel revolutions\u2014it then calculates the distance traveled. If the wheel has a 1.15 m diameter and goes through 200,000 rotations, how many kilometers should the odometer read?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1245988\">\n<p id=\"fs-id1930826\">723 km<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1004074\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2448666\">\n<p id=\"fs-id2052623\">Microwave ovens rotate at a rate of about 6 rev\/min. What is this in revolutions per second? What is the angular velocity in radians per second?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1921627\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2953385\">\n<p id=\"fs-id1574958\">An automobile with 0.260 m radius tires travels 80,000 km before wearing them out. How many revolutions do the tires make, neglecting any backing up and any change in radius due to wear?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id3257890\">\n<p id=\"fs-id2031098\">[latex]5\u00d7{\\text{10}}^{7}\\phantom{\\rule{0.25em}{0ex}}\\text{rotations}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1524972\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3254586\">\n<p id=\"fs-id3035846\">(a) What is the period of rotation of Earth in seconds? (b) What is the angular velocity of Earth? (c) Given that Earth has a radius of [latex]6\\text{.}4\u00d7{\\text{10}}^{6}\\phantom{\\rule{0.25em}{0ex}}\\text{m}[\/latex] at its equator, what is the linear velocity at Earth\u2019s surface?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2979194\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1361145\">\n<p id=\"fs-id3093713\">A baseball pitcher brings his arm forward during a pitch, rotating the forearm about the elbow. If the velocity of the ball in the pitcher\u2019s hand is 35.0 m\/s and the ball is 0.300 m from the elbow joint, what is the angular velocity of the forearm?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"eip-id1922676\">\n<p id=\"eip-id2474001\">117 rad\/s<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id954942\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3181723\">\n<p id=\"fs-id2595491\">In lacrosse, a ball is thrown from a net on the end of a stick by rotating the stick and forearm about the elbow. If the angular velocity of the ball about the elbow joint is 30.0 rad\/s and the ball is 1.30 m from the elbow joint, what is the velocity of the ball?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2678694\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1576562\">\n<p id=\"fs-id2979343\">A truck with 0.420-m-radius tires travels at 32.0 m\/s. What is the angular velocity of the rotating tires in radians per second? What is this in rev\/min?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1095106\" data-element-type=\"problmes-exercises\">\n<p id=\"fs-id1816633\">76.2 rad\/s<\/p>\n<p id=\"eip-id2409910\">728 rpm<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1429548\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1926043\">\n<p id=\"fs-id2001055\"><strong>Integrated Concepts<\/strong> When kicking a football, the kicker rotates his leg about the hip joint.<\/p>\n<p id=\"fs-id1997458\">(a) If the velocity of the tip of the kicker\u2019s shoe is 35.0 m\/s and the hip joint is 1.05 m from the tip of the shoe, what is the shoe tip\u2019s angular velocity?<\/p>\n<p id=\"fs-id1934232\">(b) The shoe is in contact with the initially stationary 0.500 kg football for 20.0 ms. What average force is exerted on the football to give it a velocity of 20.0 m\/s?<\/p>\n<p id=\"fs-id2057369\"> (c) Find the maximum range of the football, neglecting air resistance.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2401011\">\n<p id=\"eip-id3377877\">(a) 33.3 rad\/s<\/p>\n<p id=\"eip-id1533441\">(b) 500 N<\/p>\n<p id=\"eip-id3252463\">(c) 40.8 m<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2578682\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1545891\">\n<p><strong>Construct Your Own Problem<\/strong><\/p>\n<p id=\"eip-id2601357\">Consider an amusement park ride in which participants are rotated about a vertical axis in a cylinder with vertical walls. Once the angular velocity reaches its full value, the floor drops away and friction between the walls and the riders prevents them from sliding down. Construct a problem in which you calculate the necessary angular velocity that assures the riders will not slide down the wall. Include a free body diagram of a single rider. Among the variables to consider are the radius of the cylinder and the coefficients of friction between the riders\u2019 clothing and the wall.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\n<dl class=\"definition\" id=\"import-auto-id3161313\">\n<dt>arc length<\/dt>\n<dd id=\"fs-id1350787\">[latex]\\text{\u0394}s[\/latex], the distance traveled by an object along a circular path<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1818172\">\n<dt>pit<\/dt>\n<dd id=\"fs-id3191636\">a tiny indentation on the spiral track moulded into the top of the polycarbonate layer of CD<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id2010216\">\n<dt>rotation angle<\/dt>\n<dd id=\"fs-id2836750\">the ratio of the arc length to the radius of curvature on a circular path:\n<p id=\"import-auto-id2448776\">[latex]\\text{\u0394}\\theta =\\frac{\\text{\u0394}s}{r}[\/latex]<\/p>\n<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id3046840\">\n<dt>radius of curvature<\/dt>\n<dd id=\"fs-id1927592\">radius of a circular path<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id2031924\">\n<dt>radians<\/dt>\n<dd id=\"fs-id3400298\">a unit of angle measurement<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1019372\">\n<dt>angular velocity<\/dt>\n<dd id=\"fs-id3010164\">[latex]\\omega [\/latex],  the rate of change of the angle with which an object moves on a circular path<\/dd>\n<\/dl>\n<\/div>\n\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Define arc length, rotation angle, radius of curvature and angular velocity.<\/li>\n<li>Calculate the angular velocity of a car wheel spin.<\/li>\n<\/ul>\n<\/div>\n<p id=\"import-auto-id1571938\">In <a href=\"\/contents\/e12329e4-8d6c-49cf-aa45-6a05b26ebcba@2\">Kinematics<\/a>, we studied motion along a straight line and introduced such concepts as displacement, velocity, and acceleration. <a href=\"\/contents\/f309a0f9-63fb-46ca-9585-d1e1dc96a142@3\">Two-Dimensional Kinematics<\/a> dealt with motion in two dimensions. Projectile motion is a special case of two-dimensional kinematics in which the object is projected into the air, while being subject to the gravitational force, and lands a distance away. In this chapter, we consider situations where the object does not land but moves in a curve. We begin the study of uniform circular motion by defining two angular quantities needed to describe rotational motion.<\/p>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1488693\">\n<h1 data-type=\"title\">Rotation Angle<\/h1>\n<p id=\"import-auto-id1917384\">When objects rotate about some axis\u2014for example, when the CD (compact disc) in <a href=\"#import-auto-id3402904\" class=\"autogenerated-content\">(Figure)<\/a> rotates about its center\u2014each point in the object follows a circular arc. Consider a line from the center of the CD to its edge. Each <span data-type=\"term\" id=\"import-auto-id2654027\">pit<\/span> used to record sound along this line moves through the same angle in the same amount of time. The rotation angle is the amount of rotation and is analogous to linear distance. We define the <span data-type=\"term\" id=\"import-auto-id3255842\">rotation angle<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0ddc7b345bb78feb72965b853c582487_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> to be the ratio of the arc length to the radius of curvature:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-211\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b22bbd440399232949179eca6bde30f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#115;&#125;&#123;&#114;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"47\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id2968424\">\n<div class=\"bc-figure figure\" id=\"import-auto-id3402904\">\n<div class=\"bc-figcaption figcaption\">All points on a CD travel in circular arcs. The pits along a line from the center to the edge all move through the same angle <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0ddc7b345bb78feb72965b853c582487_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> in a time <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6e22acc0b91514a4aaf1954c300f3438_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\" \/>.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id3246430\" data-alt=\"The figure shows the back side of a compact disc. There is a scratched part on the upper right side of the C D, about one-fifth size of the whole area, with inner circular dots clearly visible. Two line segments are drawn enclosing the scratched area from the border of the C D to the middle plastic portion. A curved arrow is drawn between the two line segments near this middle portion and angle delta theta written alongside it.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_07_01_01aa.jpg\" data-media-type=\"image\/jpg\" alt=\"The figure shows the back side of a compact disc. There is a scratched part on the upper right side of the C D, about one-fifth size of the whole area, with inner circular dots clearly visible. Two line segments are drawn enclosing the scratched area from the border of the C D to the middle plastic portion. A curved arrow is drawn between the two line segments near this middle portion and angle delta theta written alongside it.\" width=\"225\" \/><\/span><\/p>\n<\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id3418263\">\n<div class=\"bc-figcaption figcaption\">The radius of a circle is rotated through an angle <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0ddc7b345bb78feb72965b853c582487_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/>. The arc length <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1e4bbd6a23cb25d84c191e5e80e40917_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"7\" style=\"vertical-align: 0px;\" \/> is described on the circumference. <\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2601460\" data-alt=\"A circle of radius r and center O is shown. A radius O-A of the circle is rotated through angle delta theta about the center O to terminate as radius O-B. The arc length A-B is marked as delta s.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_07_01_01ab.jpg\" data-media-type=\"image\/png\" alt=\"A circle of radius r and center O is shown. A radius O-A of the circle is rotated through angle delta theta about the center O to terminate as radius O-B. The arc length A-B is marked as delta s.\" width=\"300\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id3025908\">The <span data-type=\"term\">arc length<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ca329ff4fec1c133c02fc700f2b34275_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#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;&Delta;&#125;&#115;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/> is the distance traveled along a circular path as shown in <a href=\"#import-auto-id3418263\" class=\"autogenerated-content\">(Figure)<\/a> Note that <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 <span data-type=\"term\" id=\"import-auto-id2920709\">radius of curvature<\/span> of the circular path.<\/p>\n<p id=\"import-auto-id1471752\">We know that for one complete revolution, the arc length is the circumference of a circle of radius <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;\" \/>. The circumference of a circle is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9b97ad6a9137d9b457ba677c26d08ae3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#112;&#105;&#32;&#114;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"28\" style=\"vertical-align: 0px;\" \/>. Thus for one complete revolution the rotation angle is<\/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-446a3a3ebc8295f15d1c6cbc51d850cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#92;&#112;&#105;&#32;&#114;&#125;&#123;&#114;&#125;&#61;&#50;&#92;&#112;&#105;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"105\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id1842986\">This result is the basis for defining the units used to measure rotation angles, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0ddc7b345bb78feb72965b853c582487_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> to be <span data-type=\"term\" id=\"import-auto-id2625941\">radians<\/span> (rad), defined so that<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-135\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8f8c85d5f77f8e71da4a451236eeaa76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#112;&#105;&#32;&#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;&#97;&#100;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#32;&#114;&#101;&#118;&#111;&#108;&#117;&#116;&#105;&#111;&#110;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"171\" style=\"vertical-align: -1px;\" \/><\/div>\n<p id=\"eip-425\">A comparison of some useful angles expressed in both degrees and radians is shown in <a href=\"#import-auto-id2588905\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<table id=\"import-auto-id2588905\" summary=\"The table compares various angle measures in degrees (first column) and radians (second colum).\">\n<caption><span data-type=\"title\">Comparison of Angular Units<\/span><\/caption>\n<thead>\n<tr>\n<th>Degree Measures<\/th>\n<th>Radian Measure<\/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-7e404b535ec15d8300b0b1148704da97_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-bd29b35427ca42022ce4d17867d52f31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#112;&#105;&#32;&#125;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"9\" 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-9f58822796ed6c341868803a248de619_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c4c088e2fd672f57708f7b3a786a39f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#112;&#105;&#32;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"9\" 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-0ef094c705f55f76b4993ff72af9e73f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dd7f0884e8a2605196f0ecd35a8751a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#112;&#105;&#32;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"9\" 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-0bcf39e8728bf5c43d326b4e6534c715_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#50;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: -1px;\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9aa4d431d3f8bea910165c150a550748_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#92;&#112;&#105;&#32;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"16\" 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-abd108f7a58919f77d636d444036fadb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#51;&#53;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"25\" style=\"vertical-align: -1px;\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d15e80c55e2228d6fee1ccab768c8b3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#92;&#112;&#105;&#32;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"16\" 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-d99e5df9b7f723e173ac5bd5364c01e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#56;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: -1px;\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3868841096278f98ab5b439f54df304a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#105;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"bc-figure figure\" id=\"import-auto-id2442865\">\n<div class=\"bc-figcaption figcaption\">Points 1 and 2 rotate through the same angle (<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0ddc7b345bb78feb72965b853c582487_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/>), but point 2 moves through a greater arc length <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-61a7b57a5f7cba024aa5d1c90ed3ffaf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#115;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"20\" style=\"vertical-align: -4px;\" \/> because it is at a greater distance from the center of rotation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c3d342011bd7cd1ff0698c1506b07b48_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#114;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"20\" style=\"vertical-align: -4px;\" \/>. <\/div>\n<p><span data-type=\"media\" id=\"import-auto-id3356541\" data-alt=\"A circle is shown. Two radii of the circle, inclined at an acute angle delta theta, are shown. On one of the radii, two points, one and two are marked. The point one is inside the circle through which an arc between the two radii is shown. The point two is on the cirumfenrence of the circle. The two arc lengths are delta s one and delta s two respectively for the two points.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_07_01_02a.jpg\" data-media-type=\"image\/jpg\" alt=\"A circle is shown. Two radii of the circle, inclined at an acute angle delta theta, are shown. On one of the radii, two points, one and two are marked. The point one is inside the circle through which an arc between the two radii is shown. The point two is on the cirumfenrence of the circle. The two arc lengths are delta s one and delta s two respectively for the two points.\" width=\"230\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id1930108\">If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-68d9265d8de853a63caf2ca3ff53f17a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#61;&#50;&#92;&#112;&#105;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"52\" style=\"vertical-align: 0px;\" \/> rad, then the CD has made one complete revolution, and every point on the CD is back at its original position. Because there are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7a8b791c88f6d8f79b261520a8277402_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#54;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/> in a circle or one revolution, the relationship between radians and degrees is thus<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-808\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d6114430db3dbbecdf6a565833cee0fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#112;&#105;&#32;&#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;&#97;&#100;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#54;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"100\" style=\"vertical-align: -1px;\" \/><\/div>\n<p id=\"import-auto-id2052087\">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-08fbad78c5b03b19273ce1bb2e3e5bf7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#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;&#97;&#100;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#54;&#48;&ordm;&#125;&#125;&#123;&#50;&#92;&#112;&#105;&#32;&#125;&#92;&#97;&#112;&#112;&#114;&#111;&#120;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#55;&#46;&#125;&#51;&ordm;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"145\" style=\"vertical-align: -6px;\" \/><\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id3104613\">\n<h1 data-type=\"title\">Angular Velocity<\/h1>\n<p id=\"import-auto-id2681279\">How fast is an object rotating? We define <span data-type=\"term\" id=\"import-auto-id2962847\">angular velocity<\/span> <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;\" \/> as the rate of change of an angle. In symbols, this is<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-759\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d46c822995d75844a308d00ced2c273c_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;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#116;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#44;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"50\" style=\"vertical-align: -6px;\" \/><\/div>\n<p>where an angular rotation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0ddc7b345bb78feb72965b853c582487_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> takes place in a time <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6e22acc0b91514a4aaf1954c300f3438_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\" \/>. The greater the rotation angle in a given amount of time, the greater the angular velocity. The units for angular velocity are radians per second (rad\/s).<\/p>\n<p id=\"import-auto-id2621168\">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;\" \/> is analogous to 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;\" \/>. To get the precise relationship between angular and linear velocity, we again consider a pit on the rotating CD. This pit moves an arc length <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d2ea351b54d5015287c3f403c692b563_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#115;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/> in a time <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6e22acc0b91514a4aaf1954c300f3438_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\" \/>, and so it has a linear velocity<\/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-27d51f476b513022d68080e82937626d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#115;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#116;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"47\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id1840944\">From <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-865a6ed87d1522311f44146994262a18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#115;&#125;&#123;&#114;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"41\" style=\"vertical-align: -6px;\" \/> we see that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0ab31a14d0472a460858d4f0f5e1a843_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#115;&#61;&#114;&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"49\" style=\"vertical-align: 0px;\" \/>. Substituting this into the expression for <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;\" \/> gives<\/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-6c4b3eec87d9dd3a5a5557707a8f0958_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#114;&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#116;&#125;&#61;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#114;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"96\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id1549179\">We write this relationship in two different ways and gain two different insights:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-639\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3d22e9a9c7b142dae4c1e1982abc25b4_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;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#111;&#114;&#32;&#125;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#118;&#125;&#123;&#114;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"129\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id2680923\">The first relationship in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-11ae98f4fb15a01b3b76b2822523606a_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;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#111;&#114;&#32;&#125;&#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=\"124\" style=\"vertical-align: -6px;\" \/> states that the 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;\" \/> is proportional to the distance from the center of rotation, thus, it is largest for a point on the rim (largest <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;\" \/>), as you might expect. We can also call this linear speed <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;\" \/> of a point on the rim the <em data-effect=\"italics\">tangential speed<\/em>. The second relationship in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-11ae98f4fb15a01b3b76b2822523606a_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;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#111;&#114;&#32;&#125;&#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=\"124\" style=\"vertical-align: -6px;\" \/> can be illustrated by considering the tire of a moving car. Note that the speed of a point on the rim of the tire is the same as the speed <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;\" \/> of the car. See <a href=\"#import-auto-id2931190\" class=\"autogenerated-content\">(Figure)<\/a>. So the faster the car moves, the faster the tire spins\u2014large <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;\" \/> means a large <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;\" \/>, because <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;\" \/>. Similarly, a larger-radius tire rotating at the same 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;\" \/>) will produce a greater linear speed (<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;\" \/>) for the car.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id2931190\">\n<div class=\"bc-figcaption figcaption\">A car moving at a 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;\" \/> to the right has a tire rotating with an 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;\" \/>.The speed of the tread of the tire relative to the axle is <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;\" \/>, the same as if the car were jacked up. Thus the car moves forward at linear velocity <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;\" \/>, 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 tire radius. A larger angular velocity for the tire means a greater velocity for the car.\n      <\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1449564\" data-alt=\"The given figure shows the front wheel of a car. The radius of the car wheel, r, is shown as an arrow and the linear velocity, v, is shown with a green horizontal arrow pointing rightward. The angular velocity, omega, is shown with a clockwise-curved arrow over the wheel.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_07_01_03a.jpg\" data-media-type=\"image\/jpg\" alt=\"The given figure shows the front wheel of a car. The radius of the car wheel, r, is shown as an arrow and the linear velocity, v, is shown with a green horizontal arrow pointing rightward. The angular velocity, omega, is shown with a clockwise-curved arrow over the wheel.\" width=\"300\" \/><\/span><\/p>\n<\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id2589253\">\n<div data-type=\"title\" class=\"title\">How Fast Does a Car Tire Spin?<\/div>\n<p id=\"import-auto-id3402720\">Calculate the angular velocity of a 0.300 m radius car tire when the car travels at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1720664b41e258ef4d4a6623d3ab32a6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#53;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"66\" style=\"vertical-align: -4px;\" \/> (about <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-efae7f9fb40768962e4bceee639b3e94_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#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;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#109;&#47;&#104;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/>). See <a href=\"#import-auto-id2931190\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<p id=\"import-auto-id2968606\"><strong>Strategy<\/strong><\/p>\n<p id=\"fs-id1374481\">Because the linear speed of the tire rim is the same as the speed of the car, we have<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b64e5ee171c3a3223dbead3008fbc60f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#53;&#46;&#48;&#32;&#109;&#47;&#115;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"105\" style=\"vertical-align: -4px;\" \/>  <\/p>\n<p>The radius of the tire is given to be<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e56b3863f6a1fa18b1a08a45989c36b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#46;&#51;&#48;&#48;&#32;&#109;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"97\" style=\"vertical-align: 0px;\" \/> Knowing <\/p>\n<p><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;\" \/> 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;\" \/>, we can use the second relationship in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1d35829e8ed7898a02ccbcd6b62fa03e_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;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#44;&#32;&#125;&#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=\"102\" style=\"vertical-align: -6px;\" \/> to calculate the angular velocity.<\/p>\n<p id=\"import-auto-id2949936\"><strong>Solution<\/strong><\/p>\n<p id=\"eip-105\">To calculate the angular velocity, we will use the following relationship:<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0ffa3e1ad63863cbea1f4e5d8b32b56b_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;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"50\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id3199878\">Substituting the knowns,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-451\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f14c7b86c64508bdba05132efac343a4_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;&#116;&#101;&#120;&#116;&#123;&#49;&#53;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;&#125;&#123;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#48;&#48;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#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;&#97;&#100;&#47;&#115;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"197\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id1889900\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id956895\">When we cancel units in the above calculation, we get 50.0\/s. But the angular velocity must have units of rad\/s. Because radians are actually unitless (radians are defined as a ratio of distance), we can simply insert them into the answer for the angular velocity. Also note that if an earth mover with much larger tires, say 1.20 m in radius, were moving at the same speed of 15.0 m\/s, its tires would rotate more slowly. They would have an angular velocity <\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-971\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b5c68bb7219588b3f5c988e124b2f77d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#53;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#47;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#53;&#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;&#97;&#100;&#47;&#115;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"301\" style=\"vertical-align: -5px;\" \/><\/div>\n<\/div>\n<p id=\"import-auto-id2415283\">Both <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 <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;\" \/> have directions (hence they are angular and linear <em data-effect=\"italics\">velocities<\/em>, respectively). Angular velocity has only two directions with respect to the axis of rotation\u2014it is either clockwise or counterclockwise. Linear velocity is tangent to the path, as illustrated in <a href=\"#import-auto-id1452850\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id2584087\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Take-Home Experiment<\/div>\n<p id=\"import-auto-id1986367\">Tie an object to the end of a string and swing it around in a horizontal circle above your head (swing at your wrist). Maintain uniform speed as the object swings and measure the angular velocity of the motion. What is the approximate speed of the object? Identify a point close to your hand and take appropriate measurements to calculate the linear speed at this point. Identify other circular motions and measure their angular velocities.<\/p>\n<\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id1452850\">\n<div class=\"bc-figcaption figcaption\">As an object moves in a circle, here a fly on the edge of an old-fashioned vinyl record, its instantaneous velocity is always tangent to the circle. The direction of the angular velocity is clockwise in this case. <\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2604789\" data-alt=\"The given figure shows the top view of an old fashioned vinyl record. Two perpendicular line segments are drawn through the center of the circular record, one vertically upward and one horizontal to the right side. Two flies are shown at the end points of the vertical lines near the borders of the record. Two arrows are also drawn perpendicularly rightward through the end points of these vertical lines depicting linear velocities. A curved arrow is also drawn at the center circular part of the record which shows the angular velocity.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_07_01_04a.jpg\" data-media-type=\"image\/jpg\" alt=\"The given figure shows the top view of an old fashioned vinyl record. Two perpendicular line segments are drawn through the center of the circular record, one vertically upward and one horizontal to the right side. Two flies are shown at the end points of the vertical lines near the borders of the record. Two arrows are also drawn perpendicularly rightward through the end points of these vertical lines depicting linear velocities. A curved arrow is also drawn at the center circular part of the record which shows the angular velocity.\" width=\"250\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"eip-270\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">PhET Explorations: Ladybug Revolution<\/div>\n<div class=\"bc-figure figure\" id=\"eip-id1171550\">\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_7.1\" 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<p id=\"eip-id1169738118030\">Join the ladybug in an exploration of rotational motion. Rotate the merry-go-round to change its angle, or choose a constant angular velocity or angular acceleration. Explore how circular motion relates to the bug&#8217;s x,y position, velocity, and acceleration using vectors or graphs.<\/p>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id3399151\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul id=\"fs-id1104163\">\n<li id=\"import-auto-id2603294\">Uniform circular motion is motion in a circle at constant speed. The rotation angle <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0ddc7b345bb78feb72965b853c582487_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> is defined as the ratio of the arc length to the radius of curvature:\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8c8b6b4264b0f01e50ef628ae0249481_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#115;&#125;&#123;&#114;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#44;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"47\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id3062885\">where arc length <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d2ea351b54d5015287c3f403c692b563_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#115;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/> is distance traveled along a circular path 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;\" \/> is the radius of curvature of the circular path. The quantity <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0ddc7b345bb78feb72965b853c582487_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> is measured in units of radians (rad), for which<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-567\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6ff743a86b660323c11d3790687b8a57_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#112;&#105;&#32;&#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;&#97;&#100;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#54;&#48;&ordm;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#61;&#32;&#125;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#114;&#101;&#118;&#111;&#108;&#117;&#116;&#105;&#111;&#110;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"217\" style=\"vertical-align: -1px;\" \/><\/div>\n<\/li>\n<li id=\"import-auto-id2384053\">The conversion between radians and degrees is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-867db6eeb9992235ba691f184469533e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#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;&#97;&#100;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#55;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#51;&#92;&#116;&#101;&#120;&#116;&#123;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"93\" style=\"vertical-align: -1px;\" \/>.<\/li>\n<li id=\"import-auto-id2442213\">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;\" \/> is the rate of change of an angle,\n<div data-type=\"equation\" class=\"equation\" id=\"eip-969\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d46c822995d75844a308d00ced2c273c_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;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#116;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#44;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"50\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id1439148\">where a rotation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0ddc7b345bb78feb72965b853c582487_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> takes place in a time <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6e22acc0b91514a4aaf1954c300f3438_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\" \/>. The units of angular velocity are radians per second (rad\/s). 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;\" \/> and 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;\" \/> are related by<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-513\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3d22e9a9c7b142dae4c1e1982abc25b4_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;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#111;&#114;&#32;&#125;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#118;&#125;&#123;&#114;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"129\" style=\"vertical-align: -6px;\" \/><\/div>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" id=\"fs-id3103314\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3119404\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2969010\">\n<p id=\"import-auto-id2931754\">There is an analogy between rotational and linear physical quantities. What rotational quantities are analogous to distance and velocity?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id2381548\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problem Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3004274\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3253787\">\n<p id=\"import-auto-id3415350\">Semi-trailer trucks have an odometer on one hub of a trailer wheel. The hub is weighted so that it does not rotate, but it contains gears to count the number of wheel revolutions\u2014it then calculates the distance traveled. If the wheel has a 1.15 m diameter and goes through 200,000 rotations, how many kilometers should the odometer read?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1245988\">\n<p id=\"fs-id1930826\">723 km<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1004074\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2448666\">\n<p id=\"fs-id2052623\">Microwave ovens rotate at a rate of about 6 rev\/min. What is this in revolutions per second? What is the angular velocity in radians per second?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1921627\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2953385\">\n<p id=\"fs-id1574958\">An automobile with 0.260 m radius tires travels 80,000 km before wearing them out. How many revolutions do the tires make, neglecting any backing up and any change in radius due to wear?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id3257890\">\n<p id=\"fs-id2031098\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-117e258c791a456606306db4360aff2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#55;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#111;&#116;&#97;&#116;&#105;&#111;&#110;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"109\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1524972\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3254586\">\n<p id=\"fs-id3035846\">(a) What is the period of rotation of Earth in seconds? (b) What is the angular velocity of Earth? (c) Given that Earth has a radius of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3ea74dea1738e8fba7c9e74b68939fec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#52;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#54;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"68\" style=\"vertical-align: -1px;\" \/> at its equator, what is the linear velocity at Earth\u2019s surface?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2979194\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1361145\">\n<p id=\"fs-id3093713\">A baseball pitcher brings his arm forward during a pitch, rotating the forearm about the elbow. If the velocity of the ball in the pitcher\u2019s hand is 35.0 m\/s and the ball is 0.300 m from the elbow joint, what is the angular velocity of the forearm?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"eip-id1922676\">\n<p id=\"eip-id2474001\">117 rad\/s<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id954942\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3181723\">\n<p id=\"fs-id2595491\">In lacrosse, a ball is thrown from a net on the end of a stick by rotating the stick and forearm about the elbow. If the angular velocity of the ball about the elbow joint is 30.0 rad\/s and the ball is 1.30 m from the elbow joint, what is the velocity of the ball?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2678694\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1576562\">\n<p id=\"fs-id2979343\">A truck with 0.420-m-radius tires travels at 32.0 m\/s. What is the angular velocity of the rotating tires in radians per second? What is this in rev\/min?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1095106\" data-element-type=\"problmes-exercises\">\n<p id=\"fs-id1816633\">76.2 rad\/s<\/p>\n<p id=\"eip-id2409910\">728 rpm<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1429548\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1926043\">\n<p id=\"fs-id2001055\"><strong>Integrated Concepts<\/strong> When kicking a football, the kicker rotates his leg about the hip joint.<\/p>\n<p id=\"fs-id1997458\">(a) If the velocity of the tip of the kicker\u2019s shoe is 35.0 m\/s and the hip joint is 1.05 m from the tip of the shoe, what is the shoe tip\u2019s angular velocity?<\/p>\n<p id=\"fs-id1934232\">(b) The shoe is in contact with the initially stationary 0.500 kg football for 20.0 ms. What average force is exerted on the football to give it a velocity of 20.0 m\/s?<\/p>\n<p id=\"fs-id2057369\"> (c) Find the maximum range of the football, neglecting air resistance.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2401011\">\n<p id=\"eip-id3377877\">(a) 33.3 rad\/s<\/p>\n<p id=\"eip-id1533441\">(b) 500 N<\/p>\n<p id=\"eip-id3252463\">(c) 40.8 m<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2578682\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1545891\">\n<p><strong>Construct Your Own Problem<\/strong><\/p>\n<p id=\"eip-id2601357\">Consider an amusement park ride in which participants are rotated about a vertical axis in a cylinder with vertical walls. Once the angular velocity reaches its full value, the floor drops away and friction between the walls and the riders prevents them from sliding down. Construct a problem in which you calculate the necessary angular velocity that assures the riders will not slide down the wall. Include a free body diagram of a single rider. Among the variables to consider are the radius of the cylinder and the coefficients of friction between the riders\u2019 clothing and the wall.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\n<dl class=\"definition\" id=\"import-auto-id3161313\">\n<dt>arc length<\/dt>\n<dd id=\"fs-id1350787\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d2ea351b54d5015287c3f403c692b563_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#115;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/>, the distance traveled by an object along a circular path<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1818172\">\n<dt>pit<\/dt>\n<dd id=\"fs-id3191636\">a tiny indentation on the spiral track moulded into the top of the polycarbonate layer of CD<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id2010216\">\n<dt>rotation angle<\/dt>\n<dd id=\"fs-id2836750\">the ratio of the arc length to the radius of curvature on a circular path:<\/p>\n<p id=\"import-auto-id2448776\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-865a6ed87d1522311f44146994262a18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#115;&#125;&#123;&#114;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"41\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id3046840\">\n<dt>radius of curvature<\/dt>\n<dd id=\"fs-id1927592\">radius of a circular path<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id2031924\">\n<dt>radians<\/dt>\n<dd id=\"fs-id3400298\">a unit of angle measurement<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1019372\">\n<dt>angular velocity<\/dt>\n<dd id=\"fs-id3010164\"><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;\" \/>,  the rate of change of the angle with which an object moves on a circular path<\/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-311","chapter","type-chapter","status-publish","hentry","license-all-rights-reserved"],"part":302,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/311","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\/311\/revisions"}],"predecessor-version":[{"id":312,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/311\/revisions\/312"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/parts\/302"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/311\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/media?parent=311"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapter-type?post=311"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/contributor?post=311"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/license?post=311"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}