{"id":520,"date":"2017-10-27T16:30:08","date_gmt":"2017-10-27T16:30:08","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/kinematics-of-rotational-motion\/"},"modified":"2017-11-08T03:24:48","modified_gmt":"2017-11-08T03:24:48","slug":"kinematics-of-rotational-motion","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/kinematics-of-rotational-motion\/","title":{"raw":"Kinematics of Rotational Motion","rendered":"Kinematics of Rotational Motion"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Observe the kinematics of rotational motion.<\/li>\n<li>Derive rotational kinematic equations.<\/li>\n<li>Evaluate problem solving strategies for rotational kinematics.<\/li>\n<\/ul>\n<\/div>\n<p id=\"import-auto-id3154073\">Just by using our intuition, we can begin to see how rotational quantities like [latex]\\theta [\/latex], [latex]\\omega [\/latex], and [latex]\\alpha [\/latex] are related to one another. For example, if a motorcycle wheel has a large angular acceleration for a fairly long time, it ends up spinning rapidly and rotates through many revolutions. In more technical terms, if the wheel\u2019s angular acceleration [latex]\\alpha [\/latex] is large for a long period of time [latex]t[\/latex], then the final angular velocity [latex]\\omega [\/latex] and angle of rotation [latex]\\theta [\/latex] are large. The wheel\u2019s rotational motion is exactly analogous to the fact that the motorcycle\u2019s large translational acceleration produces a large final velocity, and the distance traveled will also be large.<\/p>\n<p id=\"import-auto-id1412967\">Kinematics is the description of motion. The <span data-type=\"term\" id=\"import-auto-id3025215\">kinematics of rotational motion<\/span> describes the relationships among rotation angle, angular velocity, angular acceleration, and time. Let us start by finding an equation relating [latex]\\omega [\/latex], [latex]\\alpha [\/latex], and [latex]t[\/latex]. To determine this equation, we recall a familiar kinematic equation for translational, or straight-line, motion:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-580\">[latex]v={v}_{0}+\\text{at}\\text{&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}\\left(\\text{constant&nbsp;}a\\right)[\/latex]<\/div>\n<p id=\"import-auto-id2009326\">Note that in rotational motion [latex]a={a}_{\\text{t}}[\/latex], and we shall use the symbol [latex]a[\/latex] for tangential or linear acceleration from now on. As in linear kinematics, we assume [latex]a[\/latex] is constant, which means that angular acceleration [latex]\\alpha [\/latex] is also a constant, because [latex]a=\\mathrm{r\\alpha }[\/latex]. Now, let us substitute [latex]v=\\mathrm{r\\omega }[\/latex] and [latex]a=\\mathrm{r\\alpha }[\/latex] into the linear equation above:<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\mathrm{r\\omega }={\\mathrm{r\\omega }}_{0}+\\mathrm{r\\alpha t}.[\/latex]<\/div>\n<p id=\"import-auto-id1894483\">The radius [latex]r[\/latex] cancels in the equation, yielding<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-323\">[latex]\\omega ={\\omega }_{0}+\\text{at}\\text{&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;}\\left(\\text{constant&nbsp;}a\\right),[\/latex]<\/div>\n<p id=\"import-auto-id1900500\">where [latex]{\\omega }_{0}[\/latex] is the initial angular velocity. This last equation is a <em data-effect=\"italics\">kinematic relationship<\/em> among [latex]\\omega [\/latex], [latex]\\alpha [\/latex], and [latex]t[\/latex] \u2014that is, it describes their relationship without reference to forces or masses that may affect rotation. It is also precisely analogous in form to its translational counterpart. <\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id3285965\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Making Connections<\/div>\n<p id=\"import-auto-id2667542\">Kinematics for rotational motion is completely analogous to translational kinematics, first presented in <a href=\"\/contents\/e12329e4-8d6c-49cf-aa45-6a05b26ebcba@2\">One-Dimensional Kinematics<\/a>. Kinematics is concerned with the description of motion without regard to force or mass. We will find that translational kinematic quantities, such as displacement, velocity, and acceleration have direct analogs in rotational motion.<\/p>\n<\/div>\n<p id=\"import-auto-id2057294\">Starting with the four kinematic equations we developed in <a href=\"\/contents\/e12329e4-8d6c-49cf-aa45-6a05b26ebcba@2\">One-Dimensional Kinematics<\/a>, we can derive the following four rotational kinematic equations (presented together with their translational counterparts):<\/p>\n<table id=\"import-auto-id3400476\" summary=\"The first column lists rotational kinematic equations. The second row lists translational kinematic equations. The third column lists constants for both.\">\n<caption><span data-type=\"title\">Rotational Kinematic Equations<\/span><\/caption>\n<thead>\n<tr>\n<th data-align=\"center\">\n              Rotational\n            <\/th>\n<th data-align=\"center\">\n              Translational\n<\/th>\n<th>\n<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]\\theta =\\overline{\\omega }t[\/latex]<\/td>\n<td>[latex]x=\\stackrel{-}{v}t[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]\\omega ={\\omega }_{0}+\\mathrm{\\alpha t}[\/latex]<\/td>\n<td>[latex]v={v}_{0}+\\text{at}[\/latex]<\/td>\n<td>(constant [latex]\\alpha [\/latex], [latex]a[\/latex]) <\/td>\n<\/tr>\n<tr>\n<td>[latex]\\theta ={\\omega }_{0}t+\\frac{1}{2}{\\mathrm{\\alpha t}}^{2}[\/latex]<\/td>\n<td>[latex]x={v}_{0}t+\\frac{1}{2}{\\text{at}}^{2}[\/latex]<\/td>\n<td>(constant [latex]\\alpha [\/latex], [latex]a[\/latex]) <\/td>\n<\/tr>\n<tr>\n<td>[latex]{\\omega }^{2}={{\\omega }_{0}}^{2}+2\\text{\u03b1\u03b8}[\/latex]<\/td>\n<td>[latex]{v}^{2}={{v}_{0}}^{2}+2\\text{ax}[\/latex]<\/td>\n<td>(constant [latex]\\alpha [\/latex], [latex]a[\/latex]) <\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"import-auto-id1930347\">In these equations, the subscript 0 denotes initial values (<sub>[latex]{\\theta }_{0}[\/latex]<\/sub>, [latex]{x}_{0}[\/latex], and [latex]{t}_{0}[\/latex] are initial values), and the average angular velocity [latex]\\stackrel{-}{\\omega }[\/latex] and average velocity [latex]\\stackrel{-}{v}[\/latex] are defined as follows:<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\overline{\\omega }=\\frac{{\\omega }_{0}+\\omega }{2}\\text{&nbsp;and&nbsp;}\\overline{v}\\phantom{\\rule{0.25em}{0ex}}=\\phantom{\\rule{0.25em}{0ex}}\\frac{{v}_{0}+v}{2}.[\/latex]<\/div>\n<p id=\"import-auto-id1871596\">The equations given above in <a href=\"#import-auto-id3400476\" class=\"autogenerated-content\">(Figure)<\/a> can be used to solve any rotational or translational kinematics problem in which [latex]a[\/latex] and [latex]\\alpha [\/latex] are constant.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id2969137\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Problem-Solving Strategy for Rotational Kinematics<\/div>\n<ol id=\"fs-id3007376\" data-number-style=\"arabic\">\n<li id=\"import-auto-id2688094\"><em data-effect=\"italics\">Examine the situation to determine that rotational kinematics (rotational motion) is involved<\/em>. Rotation must be involved, but without the need to consider forces or masses that affect the motion.<\/li>\n<li id=\"import-auto-id986882\"><em data-effect=\"italics\">Identify exactly what needs to be determined in the problem (identify the unknowns)<\/em>. A sketch of the situation is useful.<\/li>\n<li id=\"import-auto-id3292730\"><em data-effect=\"italics\">Make a list of what is given or can be inferred from the problem as stated (identify the knowns)<\/em>.<\/li>\n<li id=\"import-auto-id2929121\"><em data-effect=\"italics\">Solve the appropriate equation or equations for the quantity to be determined (the unknown)<\/em>. It can be useful to think in terms of a translational analog because by now you are familiar with such motion.<\/li>\n<li id=\"import-auto-id1367619\"><em data-effect=\"italics\">Substitute the known values along with their units into the appropriate equation, and obtain numerical solutions complete with units<\/em>. Be sure to use units of radians for angles.<\/li>\n<li id=\"import-auto-id1348329\"><em data-effect=\"italics\">Check your answer to see if it is reasonable: Does your answer make sense<\/em>?<\/li>\n<\/ol>\n<\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id3120029\">\n<div data-type=\"title\" class=\"title\">Calculating the Acceleration of a Fishing Reel<\/div>\n<p id=\"import-auto-id2428900\">A deep-sea fisherman hooks a big fish that swims away from the boat pulling the fishing line from his fishing reel. The whole system is initially at rest and the fishing line unwinds from the reel at a radius of 4.50 cm from its axis of rotation. The reel is given an angular acceleration of [latex]\\text{110}\\phantom{\\rule{0.25em}{0ex}}{\\text{rad\/s}}^{2}[\/latex] for 2.00 s as seen in <a href=\"#import-auto-id2953309\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<p id=\"fs-id2925025\">(a) What is the final angular velocity of the reel?<\/p>\n<p id=\"fs-id3034120\">(b) At what speed is fishing line leaving the reel after 2.00 s elapses?<\/p>\n<p id=\"fs-id3137559\">(c) How many revolutions does the reel make?<\/p>\n<p id=\"import-auto-id3202953\">(d) How many meters of fishing line come off the reel in this time?<\/p>\n<p id=\"import-auto-id2041697\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id3104433\">In each part of this example, the strategy is the same as it was for solving problems in linear kinematics. In particular, known values are identified and a relationship is then sought that can be used to solve for the unknown.<\/p>\n<p id=\"import-auto-id2602056\"><strong>Solution for (a)<\/strong><\/p>\n<p id=\"import-auto-id2207403\">Here [latex]\\alpha [\/latex] and [latex]t[\/latex] are given and [latex]\\omega [\/latex] needs to be determined. The most straightforward equation to use is [latex]\\omega ={\\omega }_{0}+\\mathrm{\\alpha t}[\/latex] because the unknown is already on one side and all other terms are known. That equation states that<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\omega ={\\omega }_{0}+\\mathrm{\\alpha t}\\text{.}[\/latex]<\/div>\n<p id=\"import-auto-id1409559\">We are also given that [latex]{\\omega }_{0}=0[\/latex] (it starts from rest), so that<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-74\">[latex]\\omega =0+\\left(\\text{110}\\phantom{\\rule{0.25em}{0ex}}{\\text{rad\/s}}^{2}\\right)\\left(2\\text{.}\\text{00}\\text{s}\\right)=\\text{220}\\phantom{\\rule{0.25em}{0ex}}\\text{rad\/s}\\text{.}[\/latex]<\/div>\n<p id=\"import-auto-id1998248\"><strong>Solution for (b) <\/strong><\/p>\n<p id=\"fs-id1608930\">Now that [latex]\\omega [\/latex] is known, the speed [latex]v[\/latex] can most easily be found using the relationship<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-825\">[latex]v=\\mathrm{r\\omega }\\text{,}[\/latex]<\/div>\n<p id=\"import-auto-id2397794\">where the radius [latex]r[\/latex] of the reel is given to be 4.50 cm; thus,<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]v=\\left(0\\text{.}\\text{0450 m}\\right)\\left(\\text{220 rad\/s}\\right)=9\\text{.}\\text{90 m\/s.}[\/latex]<\/div>\n<p id=\"import-auto-id2991952\">Note again that radians must always be used in any calculation relating linear and angular quantities. Also, because radians are dimensionless, we have [latex]\\text{m}\u00d7\\text{rad}=\\text{m}[\/latex].<\/p>\n<p id=\"import-auto-id1856816\"><strong>Solution for (c)<\/strong><\/p>\n<p id=\"import-auto-id3093662\">Here, we are asked to find the number of revolutions. Because [latex]\\text{1 rev}=\\text{2\u03c0 rad}[\/latex], we can find the number of revolutions by finding [latex]\\theta [\/latex] in radians. We are given [latex]\\alpha [\/latex] and [latex]t[\/latex], and we know [latex]{\\omega }_{{}_{0}}[\/latex] is zero, so that [latex]\\theta [\/latex] can be obtained using [latex]\\theta ={\\omega }_{0}t+\\frac{1}{2}{\\mathrm{\\alpha t}}^{2}[\/latex].<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-252\">[latex]\\begin{array}{lll}\\theta &amp; =&amp; {\\omega }_{0}t+\\frac{1}{2}{\\mathrm{\\alpha t}}^{2}\\\\ &amp; =&amp; 0+\\left(\\text{0.500}\\right)\\left(\\text{110}\\phantom{\\rule{0.25em}{0ex}}{\\text{rad\/s}}^{2}\\right){\\left(\\text{2.00 s}\\right)}^{2}=\\text{220 rad}.\\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id2672816\">Converting radians to revolutions gives<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\theta =\\left(\\text{220 rad}\\right)\\frac{\\text{1 rev}}{\\text{2\u03c0 rad}}=\\text{35.0 rev.}[\/latex]<\/div>\n<p id=\"import-auto-id3079605\"><strong>Solution for (d)<\/strong><\/p>\n<p id=\"fs-id2407432\">The number of meters of fishing line is [latex]x[\/latex], which can be obtained through its relationship with [latex]\\theta [\/latex]:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-665\">[latex]x=\\mathrm{r\\theta }=\\left(\\text{0.0450 m}\\right)\\left(\\text{220 rad}\\right)=\\text{9.90 m}.[\/latex]<\/div>\n<p id=\"import-auto-id3027989\"><strong>Discussion<\/strong><\/p>\n<p>This example illustrates that relationships among rotational quantities are highly analogous to those among linear quantities. We also see in this example how linear and rotational quantities are connected. The answers to the questions are realistic. After unwinding for two seconds, the reel is found to spin at 220 rad\/s, which is 2100 rpm. (No wonder reels sometimes make high-pitched sounds.) The amount of fishing line played out is 9.90 m, about right for when the big fish bites.<\/p>\n<\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id2953309\">\n<div class=\"bc-figcaption figcaption\">Fishing line coming off a rotating reel moves linearly. <a href=\"#fs-id3120029\" class=\"autogenerated-content\">(Figure)<\/a> and <a href=\"#fs-id2396484\" class=\"autogenerated-content\">(Figure)<\/a> consider relationships between rotational and linear quantities associated with a fishing reel.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1427207\" data-alt=\"The figure shows a fishing reel, with radius equal to 4.5 centimeters. The direction of rotation of the reel is counterclockwise. The rotational quantities are theta, omega and alpha, and x, v, a are linear or translational quantities. The reel, fishing line, and the direction of motion have been separately indicated by curved arrows pointing toward those parts.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_11_02_01a.jpg\" data-media-type=\"image\/jpg\" alt=\"The figure shows a fishing reel, with radius equal to 4.5 centimeters. The direction of rotation of the reel is counterclockwise. The rotational quantities are theta, omega and alpha, and x, v, a are linear or translational quantities. The reel, fishing line, and the direction of motion have been separately indicated by curved arrows pointing toward those parts.\" width=\"275\"><\/span><\/p><\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id2396484\">\n<div data-type=\"title\" class=\"title\">Calculating the Duration When the Fishing Reel Slows Down and Stops<\/div>\n<p id=\"import-auto-id1824438\">Now let us consider what happens if the fisherman applies a brake to the spinning reel, achieving an angular acceleration of [latex]\u2013\\text{300}\\phantom{\\rule{0.25em}{0ex}}{\\text{rad\/s}}^{2}[\/latex]. How long does it take the reel to come to a stop?<\/p>\n<p id=\"import-auto-id1411290\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id1429758\">We are asked to find the time [latex]t[\/latex] for the reel to come to a stop. The initial and final conditions are different from those in the previous problem, which involved the same fishing reel. Now we see that the initial angular velocity is [latex]{\\omega }_{0}=\\text{220 rad\/s}[\/latex] and the final angular velocity [latex]\\omega [\/latex] is zero. The angular acceleration is given to be [latex]\\alpha =-\\text{300}\\phantom{\\rule{0.25em}{0ex}}{\\text{rad\/s}}^{2}[\/latex]. Examining the available equations, we see all quantities but <em data-effect=\"italics\">t<\/em> are known in [latex]\\omega ={\\omega }_{0}+\\mathrm{\\alpha t},[\/latex] making it easiest to use this equation.<\/p>\n<p id=\"import-auto-id2931320\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id2442493\">The equation states<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\omega ={\\omega }_{0}+\\mathrm{\\alpha t}\\text{.}[\/latex]<\/div>\n<p id=\"import-auto-id1366643\">We solve the equation algebraically for <em data-effect=\"italics\">t<\/em>, and then substitute the known values as usual, yielding<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]t=\\frac{\\omega -{\\omega }_{0}}{\\alpha }=\\frac{0-\\text{220 rad\/s}}{-\\text{300}\\phantom{\\rule{0.25em}{0ex}}{\\text{rad\/s}}^{2}}=0\\text{.}\\text{733 s.}[\/latex]<\/div>\n<p id=\"import-auto-id1578067\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id2659816\">Note that care must be taken with the signs that indicate the directions of various quantities. Also, note that the time to stop the reel is fairly small because the acceleration is rather large. Fishing lines sometimes snap because of the accelerations involved, and fishermen often let the fish swim for a while before applying brakes on the reel. A tired fish will be slower, requiring a smaller acceleration.<\/p>\n<\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1537945\">\n<div data-type=\"title\" class=\"title\">Calculating the Slow Acceleration of Trains and Their Wheels<\/div>\n<p id=\"import-auto-id2025069\">Large freight trains accelerate very slowly. Suppose one such train accelerates from rest, giving its 0.350-m-radius wheels an angular acceleration of [latex]0\\text{.}\\text{250}\\phantom{\\rule{0.25em}{0ex}}{\\text{rad\/s}}^{2}[\/latex]. After the wheels have made 200 revolutions (assume no slippage): (a) How far has the train moved down the track? (b) What are the final angular velocity of the wheels and the linear velocity of the train?<\/p>\n<p id=\"import-auto-id1930941\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id3341027\">In part (a), we are asked to find [latex]x[\/latex], and in (b) we are asked to find [latex]\\omega [\/latex] and [latex]v[\/latex]. We are given the number of revolutions [latex]\\theta [\/latex], the radius of the wheels [latex]r[\/latex], and the angular acceleration [latex]\\alpha [\/latex].<\/p>\n<p id=\"import-auto-id2970156\"><strong>Solution for (a)<\/strong><\/p>\n<p id=\"import-auto-id3063862\">The distance [latex]x[\/latex] is very easily found from the relationship between distance and rotation angle:<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\theta =\\frac{x}{r}.[\/latex]<\/div>\n<p id=\"import-auto-id3103949\">Solving this equation for [latex]x[\/latex] yields<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-53\">[latex]x=\\mathrm{r\\theta .}[\/latex]<\/div>\n<p>Before using this equation, we must convert the number of revolutions into radians, because we are dealing with a relationship between linear and rotational quantities:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-915\">[latex]\\theta =\\left(\\text{200}\\phantom{\\rule{0.25em}{0ex}}\\text{rev}\\right)\\frac{2\\pi \\phantom{\\rule{0.25em}{0ex}}\\text{rad}}{\\text{1 rev}}=\\text{1257}\\phantom{\\rule{0.25em}{0ex}}\\text{rad}.[\/latex]<\/div>\n<p id=\"import-auto-id1859866\">Now we can substitute the known values into [latex]x=\\mathrm{r\\theta }[\/latex] to find the distance the train moved down the track:<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]x=\\mathrm{r\\theta }=\\left(0.350 m\\right)\\left(\\text{1257 rad}\\right)=\\text{440}\\phantom{\\rule{0.25em}{0ex}}\\text{m}.[\/latex]<\/div>\n<p id=\"import-auto-id969001\"><strong>Solution for (b)<\/strong><\/p>\n<p id=\"import-auto-id1613275\">We cannot use any equation that incorporates [latex]t[\/latex] to find [latex]\\omega [\/latex], because the equation would have at least two unknown values. The equation [latex]{\\omega }^{2}={{\\omega }_{0}}^{2}+2\\text{\u03b1\u03b8}[\/latex] will work, because we know the values for all variables except [latex]\\omega [\/latex]: <\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]{\\omega }^{2}={{\\omega }_{0}}^{2}+2\\text{\u03b1\u03b8}[\/latex]<\/div>\n<p id=\"import-auto-id1997458\">Taking the square root of this equation and entering the known values gives<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\begin{array}{lll}\\omega &amp; =&amp; {\\left[0+2\\left(0\\text{.}\\text{250}{\\text{&nbsp;rad\/s}}^{2}\\right)\\left(\\text{1257}\\text{&nbsp;rad}\\right)\\right]}^{1\/2}\\\\ &amp; =&amp; \\text{25.1&nbsp;rad\/s.}\\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id1405914\">We can find the linear velocity of the train, [latex]v[\/latex], through its relationship to [latex]\\omega [\/latex]:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-84\">[latex]v=\\mathrm{r\\omega }=\\left(0.350 m\\right)\\left(\\text{25.1 rad\/s}\\right)=\\text{8.77 m\/s}.[\/latex]<\/div>\n<p id=\"import-auto-id3154421\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id3165246\">The distance traveled is fairly large and the final velocity is fairly slow (just under 32 km\/h).<\/p>\n<\/div>\n<p id=\"import-auto-id742532\">There is translational motion even for something spinning in place, as the following example illustrates. <a href=\"#import-auto-id2409927\" class=\"autogenerated-content\">(Figure)<\/a> shows a fly on the edge of a rotating microwave oven plate. The example below calculates the total distance it travels.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id2409927\">\n<div class=\"bc-figcaption figcaption\">The image shows a microwave plate. The fly makes revolutions while the food is heated (along with the fly).<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2973616\" data-alt=\"The figure shows a fly that has landed on the rotating plate of the microwave. The direction of rotation of the plate, omega, is counterclockwise and is shown with an arrow.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_11_02_02a.jpg\" data-media-type=\"image\/jpg\" alt=\"The figure shows a fly that has landed on the rotating plate of the microwave. The direction of rotation of the plate, omega, is counterclockwise and is shown with an arrow.\" width=\"250\"><\/span><\/p><\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id2666600\">\n<div data-type=\"title\" class=\"title\">Calculating the Distance Traveled by a Fly on the Edge of a Microwave Oven Plate<\/div>\n<p id=\"import-auto-id2671466\">A person decides to use a microwave oven to reheat some lunch. In the process, a fly accidentally flies into the microwave and lands on the outer edge of the rotating plate and remains there. If the plate has a radius of 0.15 m and rotates at 6.0 rpm, calculate the total distance traveled by the fly during a 2.0-min cooking period. (Ignore the start-up and slow-down times.)<\/p>\n<p id=\"import-auto-id2407243\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id2000742\">First, find the total number of revolutions [latex]\\theta [\/latex], and then the linear distance [latex]x[\/latex] traveled. [latex]\\theta =\\overline{\\omega }t[\/latex] can be used to find [latex]\\theta [\/latex] because [latex]\\stackrel{-}{\\omega }[\/latex]<br>\n     is given to be 6.0 rpm.<\/p>\n<p id=\"import-auto-id1431233\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id2612992\">Entering known values into [latex]\\theta =\\overline{\\omega }t[\/latex] gives<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-477\">[latex]\\theta =\\stackrel{-}{\\omega }t=\\left(\\text{6.0 rpm}\\right)\\left(\\text{2.0 min}\\right)=\\text{12 rev}.[\/latex]<\/div>\n<p id=\"import-auto-id2604731\">As always, it is necessary to convert revolutions to radians before calculating a linear quantity like [latex]x[\/latex] from an angular quantity like [latex]\\theta [\/latex]:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-745\">[latex]\\theta =\\left(\\text{12 rev}\\right)\\left(\\frac{2\\pi \\phantom{\\rule{0.25em}{0ex}}\\text{rad}}{\\text{1 rev}}\\right)=\\text{75}\\text{.4 rad.}[\/latex]<\/div>\n<p id=\"import-auto-id2932244\">Now, using the relationship between [latex]x[\/latex] and [latex]\\theta [\/latex], we can determine the distance traveled:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-624\">[latex]x=\\mathrm{r\\theta }=\\left(0\\text{.}\\text{15}\\text{&nbsp;m}\\right)\\left(\\text{75}\\text{.}4\\text{&nbsp;rad}\\right)=\\text{11}\\text{&nbsp;m}.[\/latex]<\/div>\n<p id=\"import-auto-id2209635\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id1427658\">Quite a trip (if it survives)! Note that this distance is the total distance traveled by the fly. Displacement is actually zero for complete revolutions because they bring the fly back to its original position. The distinction between total distance traveled and displacement was first noted in <a href=\"\/contents\/e12329e4-8d6c-49cf-aa45-6a05b26ebcba@2\">One-Dimensional Kinematics<\/a>.<\/p>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3076176\" data-element-type=\"Check-Understanding\" data-label=\"\">\n<div data-type=\"title\">Check Your Understanding<\/div>\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1586637\">\n<p id=\"import-auto-id1090698\">Rotational kinematics has many useful relationships, often expressed in equation form. Are these relationships laws of physics or are they simply descriptive? (Hint: the same question applies to linear kinematics.)<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id347681\" data-print-placement=\"here\">\n<p id=\"import-auto-id3449474\">Rotational kinematics (just like linear kinematics) is descriptive and does not represent laws of nature. With kinematics, we can describe many things to great precision but kinematics does not consider causes. For example, a large angular acceleration describes a very rapid change in angular velocity without any consideration of its cause.<\/p>\n<\/div>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id1915886\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul id=\"fs-id2423046\">\n<li id=\"import-auto-id3173347\">Kinematics is the description of motion.<\/li>\n<li id=\"import-auto-id2622794\">The kinematics of rotational motion describes the relationships among rotation angle, angular velocity, angular acceleration, and time.<\/li>\n<li id=\"import-auto-id1963043\">Starting with the four kinematic equations we developed in the <a href=\"\/contents\/e12329e4-8d6c-49cf-aa45-6a05b26ebcba@2\">One-Dimensional Kinematics<\/a>, we can derive the four rotational kinematic equations (presented together with their translational counterparts) seen in <a href=\"#import-auto-id3400476\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li id=\"import-auto-id1374592\">In these equations, the subscript 0 denotes initial values (<sub>[latex]{x}_{0}[\/latex]<\/sub> and <sub>[latex]{t}_{0}[\/latex]<\/sub> are initial values), and the average angular velocity [latex]\\stackrel{-}{\\omega }[\/latex] and average velocity [latex]\\stackrel{-}{v}[\/latex] are defined as follows:\n<div data-type=\"equation\" class=\"equation\">[latex]\\overline{\\omega }=\\frac{{\\omega }_{0}+\\omega }{2}\\text{&nbsp;and&nbsp;}\\overline{v}=\\frac{{v}_{0}+v}{2}.[\/latex]<\/div>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id1335002\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3246708\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1560946\">\n<p id=\"fs-id1405587\">With the aid of a string, a gyroscope is accelerated from rest to 32 rad\/s in 0.40 s.<\/p>\n<p id=\"fs-id2052495\">(a) What is its angular acceleration in rad\/s<sup>2<\/sup>?<\/p>\n<p id=\"fs-id1848559\">(b) How many revolutions does it go through in the process?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2401743\" data-element-type=\"problems-exercises\">\n<p id=\"fs-id1596185\">(a) [latex]80\\phantom{\\rule{0.25em}{0ex}}{\\text{rad\/s}}^{2}[\/latex]<\/p>\n<p id=\"fs-id3076175\">(b) 1.0 rev<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1815900\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2669165\">\n<p id=\"fs-id2669166\">Suppose a piece of dust finds itself on a CD. If the spin rate of the CD is 500 rpm, and the piece of dust is 4.3 cm from the center, what is the total distance traveled by the dust in 3 minutes? (Ignore accelerations due to getting the CD rotating.)<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2673238\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1229369\">\n<p id=\"fs-id3094846\">A gyroscope slows from an initial rate of 32.0 rad\/s at a rate of [latex]0\\text{.}\\text{700}{\\text{&nbsp;rad\/s}}^{2}[\/latex].<\/p>\n<p id=\"fs-id3095627\">(a) How long does it take to come to rest?<\/p>\n<p id=\"fs-id3006987\">(b) How many revolutions does it make before stopping?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id3245914\">\n<p id=\"fs-id3085523\">(a) 45.7 s\n<\/p>\n<p id=\"fs-id2603573\">(b) 116 rev<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3055432\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1374452\">\n<p id=\"fs-id1420807\">During a very quick stop, a car decelerates at [latex]7\\text{.}\\text{00}{\\text{&nbsp;m\/s}}^{2}[\/latex].<\/p>\n<p id=\"fs-id1411004\">(a) What is the angular acceleration of its 0.280-m-radius tires, assuming they do not slip on the pavement?<\/p>\n<p id=\"fs-id3142909\">(b) How many revolutions do the tires make before coming to rest, given their initial angular velocity is [latex]\\text{95}\\text{.}0\\text{&nbsp;rad\/s}[\/latex]?<\/p>\n<p id=\"fs-id1402031\">(c) How long does the car take to stop completely?<\/p>\n<p id=\"fs-id1395221\">(d) What distance does the car travel in this time?<\/p>\n<p id=\"fs-id1859867\">(e) What was the car\u2019s initial velocity?<\/p>\n<p id=\"fs-id2611430\">(f) Do the values obtained seem reasonable, considering that this stop happens very quickly?<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id2398786\">\n<div class=\"bc-figcaption figcaption\">Yo-yos are amusing toys that display significant physics and are engineered to enhance performance based on physical laws. (credit: Beyond Neon, Flickr)<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id3077847\" data-alt=\"The figure shows the left arm of a man with tattoo imprints and wearing a glove. He is circulating a yo-yo toy, which is in mid air and connected by the string to his hand. Some people are standing in the background watching the yo-yo trick.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_11_02_03a.jpg\" data-media-type=\"image\/png\" alt=\"The figure shows the left arm of a man with tattoo imprints and wearing a glove. He is circulating a yo-yo toy, which is in mid air and connected by the string to his hand. Some people are standing in the background watching the yo-yo trick.\" width=\"225\"><\/span><\/p><\/div>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2972882\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3244373\">\n<p id=\"import-auto-id1468670\">Everyday application: Suppose a yo-yo has a center shaft that has a 0.250 cm radius and that its string is being pulled.<\/p>\n<p id=\"import-auto-id3341040\">(a) If the string is stationary and the yo-yo accelerates away from it at a rate of [latex]1\\text{.}\\text{50}{\\text{&nbsp;m\/s}}^{2}[\/latex], what is the angular acceleration of the yo-yo? <\/p>\n<p id=\"import-auto-id3079277\">(b) What is the angular velocity after 0.750 s if it starts from rest?<\/p>\n<p id=\"import-auto-id3079279\">(c) The outside radius of the yo-yo is 3.50 cm. What is the tangential acceleration of a point on its edge?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1575104\">\n<p id=\"import-auto-id2670964\">a) [latex]6{\\text{00 rad\/s}}^{2}[\/latex]<\/p>\n<p id=\"import-auto-id1930763\">b) 450 rad\/s<\/p>\n<p id=\"import-auto-id1930765\">c) 21.0 m\/s<\/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-id1380255\">\n<dt>kinematics of rotational motion<\/dt>\n<dd id=\"fs-id1947566\">describes the relationships among rotation angle, angular velocity, angular acceleration, and time<\/dd>\n<\/dl>\n<\/div>\n\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Observe the kinematics of rotational motion.<\/li>\n<li>Derive rotational kinematic equations.<\/li>\n<li>Evaluate problem solving strategies for rotational kinematics.<\/li>\n<\/ul>\n<\/div>\n<p id=\"import-auto-id3154073\">Just by using our intuition, we can begin to see how rotational quantities like <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-761998727948942ceb1b5763e45f01e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/>, <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-946f8144d4e3d460c8621773145884d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> are related to one another. For example, if a motorcycle wheel has a large angular acceleration for a fairly long time, it ends up spinning rapidly and rotates through many revolutions. In more technical terms, if the wheel\u2019s angular acceleration <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-946f8144d4e3d460c8621773145884d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> is large for a long period of time <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b4e3cbf5d4c5c6d9b702dd139f14c147_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\" \/>, then the final angular velocity <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8ffb415af81ab9c23c1d2e7ec67d29c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#111;&#109;&#101;&#103;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> and angle of rotation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-761998727948942ceb1b5763e45f01e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> are large. The wheel\u2019s rotational motion is exactly analogous to the fact that the motorcycle\u2019s large translational acceleration produces a large final velocity, and the distance traveled will also be large.<\/p>\n<p id=\"import-auto-id1412967\">Kinematics is the description of motion. The <span data-type=\"term\" id=\"import-auto-id3025215\">kinematics of rotational motion<\/span> describes the relationships among rotation angle, angular velocity, angular acceleration, and time. Let us start by finding an equation relating <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;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-946f8144d4e3d460c8621773145884d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/>, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b4e3cbf5d4c5c6d9b702dd139f14c147_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\" \/>. To determine this equation, we recall a familiar kinematic equation for translational, or straight-line, motion:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-580\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-30357da8d2491070fc7bc29841800d96_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#61;&#123;&#118;&#125;&#95;&#123;&#48;&#125;&#43;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#116;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#32;&#32;&#32;&#32;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#110;&#115;&#116;&#97;&#110;&#116;&#32;&#125;&#97;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"189\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"import-auto-id2009326\">Note that in rotational motion <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ae6180dd881bc8269b5aacf7c31eba3a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#123;&#97;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"47\" style=\"vertical-align: -3px;\" \/>, and we shall use the symbol <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> for tangential or linear acceleration from now on. As in linear kinematics, we assume <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> is constant, which means that angular acceleration <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-946f8144d4e3d460c8621773145884d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> is also a constant, because <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-24a71c2c6a089d462c542d32347806f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#114;&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"9\" width=\"51\" style=\"vertical-align: -1px;\" \/>. Now, let us substitute <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;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-24a71c2c6a089d462c542d32347806f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#114;&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"9\" width=\"51\" style=\"vertical-align: -1px;\" \/> into the linear equation above:<\/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-80d1ab867151d8a6aa833ccebfa51792_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#114;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;&#61;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#114;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;&#125;&#95;&#123;&#48;&#125;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#114;&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#116;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"119\" style=\"vertical-align: -3px;\" \/><\/div>\n<p id=\"import-auto-id1894483\">The radius <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c409433a9e2dfcdb83360a974d243f18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/> cancels in the equation, yielding<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-323\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-25d1ec309cba92418acec8f098c3fb15_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#61;&#123;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;&#95;&#123;&#48;&#125;&#43;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#116;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#32;&#32;&#32;&#32;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#110;&#115;&#116;&#97;&#110;&#116;&#32;&#125;&#97;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"202\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"import-auto-id1900500\">where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-540bd6c993fc8c560b2f7c25a7fdd59a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"18\" style=\"vertical-align: -3px;\" \/> is the initial angular velocity. This last equation is a <em data-effect=\"italics\">kinematic relationship<\/em> among <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;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-946f8144d4e3d460c8621773145884d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/>, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b4e3cbf5d4c5c6d9b702dd139f14c147_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\" \/> \u2014that is, it describes their relationship without reference to forces or masses that may affect rotation. It is also precisely analogous in form to its translational counterpart. <\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id3285965\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Making Connections<\/div>\n<p id=\"import-auto-id2667542\">Kinematics for rotational motion is completely analogous to translational kinematics, first presented in <a href=\"\/contents\/e12329e4-8d6c-49cf-aa45-6a05b26ebcba@2\">One-Dimensional Kinematics<\/a>. Kinematics is concerned with the description of motion without regard to force or mass. We will find that translational kinematic quantities, such as displacement, velocity, and acceleration have direct analogs in rotational motion.<\/p>\n<\/div>\n<p id=\"import-auto-id2057294\">Starting with the four kinematic equations we developed in <a href=\"\/contents\/e12329e4-8d6c-49cf-aa45-6a05b26ebcba@2\">One-Dimensional Kinematics<\/a>, we can derive the following four rotational kinematic equations (presented together with their translational counterparts):<\/p>\n<table id=\"import-auto-id3400476\" summary=\"The first column lists rotational kinematic equations. The second row lists translational kinematic equations. The third column lists constants for both.\">\n<caption><span data-type=\"title\">Rotational Kinematic Equations<\/span><\/caption>\n<thead>\n<tr>\n<th data-align=\"center\">\n              Rotational\n            <\/th>\n<th data-align=\"center\">\n              Translational\n<\/th>\n<th>\n<\/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-2017110c19c9a5c2274845eb4e962633_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#61;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"50\" style=\"vertical-align: 0px;\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-794602d0ce51e38a216df545f259e791_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#45;&#125;&#123;&#118;&#125;&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"50\" style=\"vertical-align: 0px;\" \/><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-512ab2bf816d0eaa7590090f5913f746_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#61;&#123;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;&#95;&#123;&#48;&#125;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#116;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"93\" style=\"vertical-align: -3px;\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9fc21b86018864d91df38dcdcb1d62eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#61;&#123;&#118;&#125;&#95;&#123;&#48;&#125;&#43;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#116;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"86\" style=\"vertical-align: -3px;\" \/><\/td>\n<td>(constant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-946f8144d4e3d460c8621773145884d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/>) <\/td>\n<\/tr>\n<tr>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1842c8b4d3cc17f4c1040432d579e1df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#61;&#123;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;&#95;&#123;&#48;&#125;&#116;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#116;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"115\" style=\"vertical-align: -6px;\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3db6b58a390f1709ed67e0db15a94599_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#123;&#118;&#125;&#95;&#123;&#48;&#125;&#116;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#116;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"112\" style=\"vertical-align: -6px;\" \/><\/td>\n<td>(constant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-946f8144d4e3d460c8621773145884d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/>) <\/td>\n<\/tr>\n<tr>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-543bbd951de9ff1e56baa218e0b55b04_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#123;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;&#95;&#123;&#48;&#125;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#92;&#116;&#101;&#120;&#116;&#123;&alpha;&theta;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"99\" style=\"vertical-align: -3px;\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5a93ac2e50f3faa7c329d32c2aee5a01_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#118;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#123;&#118;&#125;&#95;&#123;&#48;&#125;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"114\" style=\"vertical-align: -3px;\" \/><\/td>\n<td>(constant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-946f8144d4e3d460c8621773145884d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/>) <\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"import-auto-id1930347\">In these equations, the subscript 0 denotes initial values (<sub><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-31af75b2dc98204908b7f2afaa5695ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"15\" style=\"vertical-align: -3px;\" \/><\/sub>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-96b224e50cd20bf6c24005d45c5a085c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"17\" style=\"vertical-align: -3px;\" \/>, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-034ebc26f68e2b48f39815dfed7d8a21_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#116;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"13\" style=\"vertical-align: -3px;\" \/> are initial values), and the average angular velocity <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b733f64f61816c8d261853212e4534c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#45;&#125;&#123;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"11\" style=\"vertical-align: 1px;\" \/> and average velocity <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b1204f2cbe796f0c0dadaef5ef056ed0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#45;&#125;&#123;&#118;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"9\" style=\"vertical-align: 1px;\" \/> are defined as follows:<\/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-d754d0cb05458a1c802625f21b017ca1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;&#95;&#123;&#48;&#125;&#43;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;&#123;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#97;&#110;&#100;&#32;&#125;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#118;&#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;&#61;&#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;&#102;&#114;&#97;&#99;&#123;&#123;&#118;&#125;&#95;&#123;&#48;&#125;&#43;&#118;&#125;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"196\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id1871596\">The equations given above in <a href=\"#import-auto-id3400476\" class=\"autogenerated-content\">(Figure)<\/a> can be used to solve any rotational or translational kinematics problem in which <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-946f8144d4e3d460c8621773145884d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> are constant.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id2969137\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Problem-Solving Strategy for Rotational Kinematics<\/div>\n<ol id=\"fs-id3007376\" data-number-style=\"arabic\">\n<li id=\"import-auto-id2688094\"><em data-effect=\"italics\">Examine the situation to determine that rotational kinematics (rotational motion) is involved<\/em>. Rotation must be involved, but without the need to consider forces or masses that affect the motion.<\/li>\n<li id=\"import-auto-id986882\"><em data-effect=\"italics\">Identify exactly what needs to be determined in the problem (identify the unknowns)<\/em>. A sketch of the situation is useful.<\/li>\n<li id=\"import-auto-id3292730\"><em data-effect=\"italics\">Make a list of what is given or can be inferred from the problem as stated (identify the knowns)<\/em>.<\/li>\n<li id=\"import-auto-id2929121\"><em data-effect=\"italics\">Solve the appropriate equation or equations for the quantity to be determined (the unknown)<\/em>. It can be useful to think in terms of a translational analog because by now you are familiar with such motion.<\/li>\n<li id=\"import-auto-id1367619\"><em data-effect=\"italics\">Substitute the known values along with their units into the appropriate equation, and obtain numerical solutions complete with units<\/em>. Be sure to use units of radians for angles.<\/li>\n<li id=\"import-auto-id1348329\"><em data-effect=\"italics\">Check your answer to see if it is reasonable: Does your answer make sense<\/em>?<\/li>\n<\/ol>\n<\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id3120029\">\n<div data-type=\"title\" class=\"title\">Calculating the Acceleration of a Fishing Reel<\/div>\n<p id=\"import-auto-id2428900\">A deep-sea fisherman hooks a big fish that swims away from the boat pulling the fishing line from his fishing reel. The whole system is initially at rest and the fishing line unwinds from the reel at a radius of 4.50 cm from its axis of rotation. The reel is given an angular acceleration of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-da7f1e93acdbfb511e0112aa0953ba71_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#49;&#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;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#97;&#100;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"79\" style=\"vertical-align: -4px;\" \/> for 2.00 s as seen in <a href=\"#import-auto-id2953309\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<p id=\"fs-id2925025\">(a) What is the final angular velocity of the reel?<\/p>\n<p id=\"fs-id3034120\">(b) At what speed is fishing line leaving the reel after 2.00 s elapses?<\/p>\n<p id=\"fs-id3137559\">(c) How many revolutions does the reel make?<\/p>\n<p id=\"import-auto-id3202953\">(d) How many meters of fishing line come off the reel in this time?<\/p>\n<p id=\"import-auto-id2041697\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id3104433\">In each part of this example, the strategy is the same as it was for solving problems in linear kinematics. In particular, known values are identified and a relationship is then sought that can be used to solve for the unknown.<\/p>\n<p id=\"import-auto-id2602056\"><strong>Solution for (a)<\/strong><\/p>\n<p id=\"import-auto-id2207403\">Here <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-946f8144d4e3d460c8621773145884d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b4e3cbf5d4c5c6d9b702dd139f14c147_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\" \/> are given and <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;\" \/> needs to be determined. The most straightforward equation to use is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-512ab2bf816d0eaa7590090f5913f746_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#61;&#123;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;&#95;&#123;&#48;&#125;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#116;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"93\" style=\"vertical-align: -3px;\" \/> because the unknown is already on one side and all other terms are known. That equation states 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-7a3bbe4e4999f416c1475fed57b05834_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#61;&#123;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;&#95;&#123;&#48;&#125;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#116;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"98\" style=\"vertical-align: -3px;\" \/><\/div>\n<p id=\"import-auto-id1409559\">We are also given that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4e725f49672e9c389c2e661120540991_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;&#95;&#123;&#48;&#125;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"51\" style=\"vertical-align: -3px;\" \/> (it starts from rest), so that<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-74\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-37304f22162609f31db23e02ea4a3085_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#61;&#48;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#49;&#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;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#97;&#100;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#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;&#114;&#97;&#100;&#47;&#115;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"323\" style=\"vertical-align: -12px;\" \/><\/div>\n<p id=\"import-auto-id1998248\"><strong>Solution for (b) <\/strong><\/p>\n<p id=\"fs-id1608930\">Now that <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 known, 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;\" \/> can most easily be found using the relationship<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-825\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f625bdfec08de19c17377483b5badd24_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;&#44;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"55\" style=\"vertical-align: -3px;\" \/><\/div>\n<p id=\"import-auto-id2397794\">where the radius <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c409433a9e2dfcdb83360a974d243f18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/> of the reel is given to be 4.50 cm; thus,<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f8e5ffec6baf6236d1985845d2e3e58c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#52;&#53;&#48;&#32;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#50;&#48;&#32;&#114;&#97;&#100;&#47;&#115;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#57;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#48;&#32;&#109;&#47;&#115;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"303\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"import-auto-id2991952\">Note again that radians must always be used in any calculation relating linear and angular quantities. Also, because radians are dimensionless, we have <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3cc0a5e55f332bdcd2f9b7cbc6869f75_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&times;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#97;&#100;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"79\" style=\"vertical-align: -1px;\" \/>.<\/p>\n<p id=\"import-auto-id1856816\"><strong>Solution for (c)<\/strong><\/p>\n<p id=\"import-auto-id3093662\">Here, we are asked to find the number of revolutions. Because <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b7ed084aae130b1fd82b70309d4024e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#32;&#114;&#101;&#118;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&pi;&#32;&#114;&#97;&#100;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"102\" style=\"vertical-align: -1px;\" \/>, we can find the number of revolutions by finding <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-761998727948942ceb1b5763e45f01e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> in radians. We are given <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-946f8144d4e3d460c8621773145884d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b4e3cbf5d4c5c6d9b702dd139f14c147_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\" \/>, and we know <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7a624d19c93c736e60df83b280653f58_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;&#95;&#123;&#123;&#125;&#95;&#123;&#48;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"17\" style=\"vertical-align: -4px;\" \/> is zero, so that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-761998727948942ceb1b5763e45f01e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> can be obtained using <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1842c8b4d3cc17f4c1040432d579e1df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#61;&#123;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;&#95;&#123;&#48;&#125;&#116;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#116;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"115\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-252\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-bed23302cbd4911dc043eeb664ae2d60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#108;&#108;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#38;&#32;&#61;&#38;&#32;&#123;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;&#95;&#123;&#48;&#125;&#116;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#116;&#125;&#125;&#94;&#123;&#50;&#125;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#48;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#46;&#53;&#48;&#48;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#49;&#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;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#97;&#100;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#46;&#48;&#48;&#32;&#115;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#50;&#48;&#32;&#114;&#97;&#100;&#125;&#46;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"55\" width=\"399\" style=\"vertical-align: -23px;\" \/><\/div>\n<p id=\"import-auto-id2672816\">Converting radians to revolutions 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-ca6d61a4b5425c070126df4a00ced478_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#50;&#48;&#32;&#114;&#97;&#100;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#32;&#114;&#101;&#118;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&pi;&#32;&#114;&#97;&#100;&#125;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#53;&#46;&#48;&#32;&#114;&#101;&#118;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"232\" style=\"vertical-align: -7px;\" \/><\/div>\n<p id=\"import-auto-id3079605\"><strong>Solution for (d)<\/strong><\/p>\n<p id=\"fs-id2407432\">The number of meters of fishing line is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/>, which can be obtained through its relationship with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-761998727948942ceb1b5763e45f01e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/>:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-665\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a26df66c55328d0369f021ad529d2536_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#114;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#46;&#48;&#52;&#53;&#48;&#32;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#50;&#48;&#32;&#114;&#97;&#100;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#46;&#57;&#48;&#32;&#109;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"312\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"import-auto-id3027989\"><strong>Discussion<\/strong><\/p>\n<p>This example illustrates that relationships among rotational quantities are highly analogous to those among linear quantities. We also see in this example how linear and rotational quantities are connected. The answers to the questions are realistic. After unwinding for two seconds, the reel is found to spin at 220 rad\/s, which is 2100 rpm. (No wonder reels sometimes make high-pitched sounds.) The amount of fishing line played out is 9.90 m, about right for when the big fish bites.<\/p>\n<\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id2953309\">\n<div class=\"bc-figcaption figcaption\">Fishing line coming off a rotating reel moves linearly. <a href=\"#fs-id3120029\" class=\"autogenerated-content\">(Figure)<\/a> and <a href=\"#fs-id2396484\" class=\"autogenerated-content\">(Figure)<\/a> consider relationships between rotational and linear quantities associated with a fishing reel.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1427207\" data-alt=\"The figure shows a fishing reel, with radius equal to 4.5 centimeters. The direction of rotation of the reel is counterclockwise. The rotational quantities are theta, omega and alpha, and x, v, a are linear or translational quantities. The reel, fishing line, and the direction of motion have been separately indicated by curved arrows pointing toward those parts.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_11_02_01a.jpg\" data-media-type=\"image\/jpg\" alt=\"The figure shows a fishing reel, with radius equal to 4.5 centimeters. The direction of rotation of the reel is counterclockwise. The rotational quantities are theta, omega and alpha, and x, v, a are linear or translational quantities. The reel, fishing line, and the direction of motion have been separately indicated by curved arrows pointing toward those parts.\" width=\"275\" \/><\/span><\/p>\n<\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id2396484\">\n<div data-type=\"title\" class=\"title\">Calculating the Duration When the Fishing Reel Slows Down and Stops<\/div>\n<p id=\"import-auto-id1824438\">Now let us consider what happens if the fisherman applies a brake to the spinning reel, achieving an angular acceleration of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d97fed99971f53cf591492448b5f0711_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#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;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#97;&#100;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"92\" style=\"vertical-align: -4px;\" \/>. How long does it take the reel to come to a stop?<\/p>\n<p id=\"import-auto-id1411290\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id1429758\">We are asked to find the time <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b4e3cbf5d4c5c6d9b702dd139f14c147_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\" \/> for the reel to come to a stop. The initial and final conditions are different from those in the previous problem, which involved the same fishing reel. Now we see that the initial angular velocity is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ef69e4a6e85608ee1c3e96e14d8bf926_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;&#95;&#123;&#48;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#50;&#48;&#32;&#114;&#97;&#100;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"117\" style=\"vertical-align: -4px;\" \/> and the final 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 zero. The angular acceleration is given to be <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3c74c648252f855f1a96108f35d4b338_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#61;&#45;&#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;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#97;&#100;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"128\" style=\"vertical-align: -4px;\" \/>. Examining the available equations, we see all quantities but <em data-effect=\"italics\">t<\/em> are known in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-798943328998b71ee041faf358099297_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#61;&#123;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;&#95;&#123;&#48;&#125;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#116;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"98\" style=\"vertical-align: -4px;\" \/> making it easiest to use this equation.<\/p>\n<p id=\"import-auto-id2931320\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id2442493\">The equation states<\/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-7a3bbe4e4999f416c1475fed57b05834_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#61;&#123;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;&#95;&#123;&#48;&#125;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#116;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"98\" style=\"vertical-align: -3px;\" \/><\/div>\n<p id=\"import-auto-id1366643\">We solve the equation algebraically for <em data-effect=\"italics\">t<\/em>, and then substitute the known values as usual, yielding<\/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-f911dcb4130fd0bed1a0e16ab234750c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#45;&#123;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;&#95;&#123;&#48;&#125;&#125;&#123;&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#48;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#50;&#48;&#32;&#114;&#97;&#100;&#47;&#115;&#125;&#125;&#123;&#45;&#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;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#97;&#100;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;&#125;&#61;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#55;&#51;&#51;&#32;&#115;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"252\" style=\"vertical-align: -12px;\" \/><\/div>\n<p id=\"import-auto-id1578067\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id2659816\">Note that care must be taken with the signs that indicate the directions of various quantities. Also, note that the time to stop the reel is fairly small because the acceleration is rather large. Fishing lines sometimes snap because of the accelerations involved, and fishermen often let the fish swim for a while before applying brakes on the reel. A tired fish will be slower, requiring a smaller acceleration.<\/p>\n<\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1537945\">\n<div data-type=\"title\" class=\"title\">Calculating the Slow Acceleration of Trains and Their Wheels<\/div>\n<p id=\"import-auto-id2025069\">Large freight trains accelerate very slowly. Suppose one such train accelerates from rest, giving its 0.350-m-radius wheels an angular acceleration of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a841c8fb4bfa0a1dabf7e9e1a09e6e39_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#53;&#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;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#97;&#100;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"93\" style=\"vertical-align: -4px;\" \/>. After the wheels have made 200 revolutions (assume no slippage): (a) How far has the train moved down the track? (b) What are the final angular velocity of the wheels and the linear velocity of the train?<\/p>\n<p id=\"import-auto-id1930941\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id3341027\">In part (a), we are asked to find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/>, and in (b) we are asked to find <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;\" \/>. We are given the number of revolutions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-761998727948942ceb1b5763e45f01e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/>, the radius of the wheels <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;\" \/>, and the angular acceleration <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-946f8144d4e3d460c8621773145884d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/>.<\/p>\n<p id=\"import-auto-id2970156\"><strong>Solution for (a)<\/strong><\/p>\n<p id=\"import-auto-id3063862\">The distance <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> is very easily found from the relationship between distance and rotation angle:<\/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-1e0f7ca5f7a056aedf1b0f83cf5f5252_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#114;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"48\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id3103949\">Solving this equation for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> yields<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-53\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dd5094aa5b6e4aa79780945c16386e99_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#114;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"54\" style=\"vertical-align: -1px;\" \/><\/div>\n<p>Before using this equation, we must convert the number of revolutions into radians, because we are dealing with a relationship between linear and rotational quantities:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-915\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0e324f746097a634530036c8f7121ce2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#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;&#114;&#101;&#118;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#102;&#114;&#97;&#99;&#123;&#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;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#32;&#114;&#101;&#118;&#125;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#50;&#53;&#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;&#97;&#100;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"241\" style=\"vertical-align: -7px;\" \/><\/div>\n<p id=\"import-auto-id1859866\">Now we can substitute the known values into <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-bb3d11cb19a40de8704f9c75a0799c6d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#114;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"50\" style=\"vertical-align: -1px;\" \/> to find the distance the train moved down the track:<\/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-234c496fa7548cebe02fff2a7aaeec83_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#114;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#46;&#51;&#53;&#48;&#32;&#109;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#50;&#53;&#55;&#32;&#114;&#97;&#100;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#52;&#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;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"300\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"import-auto-id969001\"><strong>Solution for (b)<\/strong><\/p>\n<p id=\"import-auto-id1613275\">We cannot use any equation that incorporates <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b4e3cbf5d4c5c6d9b702dd139f14c147_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\" \/> to find <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 the equation would have at least two unknown values. The equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-543bbd951de9ff1e56baa218e0b55b04_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#123;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;&#95;&#123;&#48;&#125;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#92;&#116;&#101;&#120;&#116;&#123;&alpha;&theta;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"99\" style=\"vertical-align: -3px;\" \/> will work, because we know the values for all variables except <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;\" \/>: <\/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-543bbd951de9ff1e56baa218e0b55b04_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#123;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;&#95;&#123;&#48;&#125;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#92;&#116;&#101;&#120;&#116;&#123;&alpha;&theta;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"99\" style=\"vertical-align: -3px;\" \/><\/div>\n<p id=\"import-auto-id1997458\">Taking the square root of this equation and entering the known values 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-1bcf283fd8f689833b7e0f323293689f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#108;&#108;&#125;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#38;&#32;&#61;&#38;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#91;&#48;&#43;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#53;&#48;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#114;&#97;&#100;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#50;&#53;&#55;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#114;&#97;&#100;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#125;&#94;&#123;&#49;&#47;&#50;&#125;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#53;&#46;&#49;&#32;&#114;&#97;&#100;&#47;&#115;&#46;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"57\" width=\"338\" style=\"vertical-align: -22px;\" \/><\/div>\n<p id=\"import-auto-id1405914\">We can find the linear velocity of the train, <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;\" \/>, through its relationship to <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;\" \/>:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-84\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0670f5dc2a69952e3e36946f1b6602b0_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;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#46;&#51;&#53;&#48;&#32;&#109;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#53;&#46;&#49;&#32;&#114;&#97;&#100;&#47;&#115;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#46;&#55;&#55;&#32;&#109;&#47;&#115;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"336\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"import-auto-id3154421\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id3165246\">The distance traveled is fairly large and the final velocity is fairly slow (just under 32 km\/h).<\/p>\n<\/div>\n<p id=\"import-auto-id742532\">There is translational motion even for something spinning in place, as the following example illustrates. <a href=\"#import-auto-id2409927\" class=\"autogenerated-content\">(Figure)<\/a> shows a fly on the edge of a rotating microwave oven plate. The example below calculates the total distance it travels.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id2409927\">\n<div class=\"bc-figcaption figcaption\">The image shows a microwave plate. The fly makes revolutions while the food is heated (along with the fly).<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2973616\" data-alt=\"The figure shows a fly that has landed on the rotating plate of the microwave. The direction of rotation of the plate, omega, is counterclockwise and is shown with an arrow.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_11_02_02a.jpg\" data-media-type=\"image\/jpg\" alt=\"The figure shows a fly that has landed on the rotating plate of the microwave. The direction of rotation of the plate, omega, is counterclockwise and is shown with an arrow.\" width=\"250\" \/><\/span><\/p>\n<\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id2666600\">\n<div data-type=\"title\" class=\"title\">Calculating the Distance Traveled by a Fly on the Edge of a Microwave Oven Plate<\/div>\n<p id=\"import-auto-id2671466\">A person decides to use a microwave oven to reheat some lunch. In the process, a fly accidentally flies into the microwave and lands on the outer edge of the rotating plate and remains there. If the plate has a radius of 0.15 m and rotates at 6.0 rpm, calculate the total distance traveled by the fly during a 2.0-min cooking period. (Ignore the start-up and slow-down times.)<\/p>\n<p id=\"import-auto-id2407243\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id2000742\">First, find the total number of revolutions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-761998727948942ceb1b5763e45f01e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/>, and then the linear distance <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> traveled. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2017110c19c9a5c2274845eb4e962633_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#61;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"50\" style=\"vertical-align: 0px;\" \/> can be used to find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-761998727948942ceb1b5763e45f01e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> because <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b733f64f61816c8d261853212e4534c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#45;&#125;&#123;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"11\" style=\"vertical-align: 1px;\" \/><br \/>\n     is given to be 6.0 rpm.<\/p>\n<p id=\"import-auto-id1431233\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id2612992\">Entering known values into <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2017110c19c9a5c2274845eb4e962633_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#61;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"50\" style=\"vertical-align: 0px;\" \/> gives<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-477\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-fe0f3c47e92ad762a164316818b1663b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#61;&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#45;&#125;&#123;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;&#116;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#46;&#48;&#32;&#114;&#112;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#46;&#48;&#32;&#109;&#105;&#110;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#50;&#32;&#114;&#101;&#118;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"298\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"import-auto-id2604731\">As always, it is necessary to convert revolutions to radians before calculating a linear quantity like <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> from an angular quantity like <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-761998727948942ceb1b5763e45f01e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/>:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-745\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-fc99ee54ba9c47e66253bf3da7522063_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#50;&#32;&#114;&#101;&#118;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#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;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#32;&#114;&#101;&#118;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#55;&#53;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#52;&#32;&#114;&#97;&#100;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"248\" style=\"vertical-align: -7px;\" \/><\/div>\n<p id=\"import-auto-id2932244\">Now, using the relationship between <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-761998727948942ceb1b5763e45f01e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/>, we can determine the distance traveled:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-624\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3baa4aa09c549e17e04c9fbd8acc2b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#114;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#53;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#55;&#53;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#52;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#114;&#97;&#100;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#49;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#109;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"285\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"import-auto-id2209635\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id1427658\">Quite a trip (if it survives)! Note that this distance is the total distance traveled by the fly. Displacement is actually zero for complete revolutions because they bring the fly back to its original position. The distinction between total distance traveled and displacement was first noted in <a href=\"\/contents\/e12329e4-8d6c-49cf-aa45-6a05b26ebcba@2\">One-Dimensional Kinematics<\/a>.<\/p>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3076176\" data-element-type=\"Check-Understanding\" data-label=\"\">\n<div data-type=\"title\">Check Your Understanding<\/div>\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1586637\">\n<p id=\"import-auto-id1090698\">Rotational kinematics has many useful relationships, often expressed in equation form. Are these relationships laws of physics or are they simply descriptive? (Hint: the same question applies to linear kinematics.)<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id347681\" data-print-placement=\"here\">\n<p id=\"import-auto-id3449474\">Rotational kinematics (just like linear kinematics) is descriptive and does not represent laws of nature. With kinematics, we can describe many things to great precision but kinematics does not consider causes. For example, a large angular acceleration describes a very rapid change in angular velocity without any consideration of its cause.<\/p>\n<\/div>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id1915886\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul id=\"fs-id2423046\">\n<li id=\"import-auto-id3173347\">Kinematics is the description of motion.<\/li>\n<li id=\"import-auto-id2622794\">The kinematics of rotational motion describes the relationships among rotation angle, angular velocity, angular acceleration, and time.<\/li>\n<li id=\"import-auto-id1963043\">Starting with the four kinematic equations we developed in the <a href=\"\/contents\/e12329e4-8d6c-49cf-aa45-6a05b26ebcba@2\">One-Dimensional Kinematics<\/a>, we can derive the four rotational kinematic equations (presented together with their translational counterparts) seen in <a href=\"#import-auto-id3400476\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li id=\"import-auto-id1374592\">In these equations, the subscript 0 denotes initial values (<sub><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-96b224e50cd20bf6c24005d45c5a085c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"17\" style=\"vertical-align: -3px;\" \/><\/sub> and <sub><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-034ebc26f68e2b48f39815dfed7d8a21_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#116;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"13\" style=\"vertical-align: -3px;\" \/><\/sub> are initial values), and the average angular velocity <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b733f64f61816c8d261853212e4534c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#45;&#125;&#123;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"11\" style=\"vertical-align: 1px;\" \/> and average velocity <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b1204f2cbe796f0c0dadaef5ef056ed0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#116;&#97;&#99;&#107;&#114;&#101;&#108;&#123;&#45;&#125;&#123;&#118;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"9\" style=\"vertical-align: 1px;\" \/> are defined as follows:\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1a9d318494e99242ffc8c02238e5618c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;&#95;&#123;&#48;&#125;&#43;&#92;&#111;&#109;&#101;&#103;&#97;&#32;&#125;&#123;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#97;&#110;&#100;&#32;&#125;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#118;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#118;&#125;&#95;&#123;&#48;&#125;&#43;&#118;&#125;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"187\" style=\"vertical-align: -6px;\" \/><\/div>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id1335002\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3246708\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1560946\">\n<p id=\"fs-id1405587\">With the aid of a string, a gyroscope is accelerated from rest to 32 rad\/s in 0.40 s.<\/p>\n<p id=\"fs-id2052495\">(a) What is its angular acceleration in rad\/s<sup>2<\/sup>?<\/p>\n<p id=\"fs-id1848559\">(b) How many revolutions does it go through in the process?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2401743\" data-element-type=\"problems-exercises\">\n<p id=\"fs-id1596185\">(a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-017e470b2cb7fd0e2dace7504bcd9f61_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#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;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#114;&#97;&#100;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"71\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id3076175\">(b) 1.0 rev<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1815900\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2669165\">\n<p id=\"fs-id2669166\">Suppose a piece of dust finds itself on a CD. If the spin rate of the CD is 500 rpm, and the piece of dust is 4.3 cm from the center, what is the total distance traveled by the dust in 3 minutes? (Ignore accelerations due to getting the CD rotating.)<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2673238\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1229369\">\n<p id=\"fs-id3094846\">A gyroscope slows from an initial rate of 32.0 rad\/s at a rate of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-716056f02d842dc88de660017883cd82_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#55;&#48;&#48;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#114;&#97;&#100;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"95\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<p id=\"fs-id3095627\">(a) How long does it take to come to rest?<\/p>\n<p id=\"fs-id3006987\">(b) How many revolutions does it make before stopping?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id3245914\">\n<p id=\"fs-id3085523\">(a) 45.7 s\n<\/p>\n<p id=\"fs-id2603573\">(b) 116 rev<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3055432\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1374452\">\n<p id=\"fs-id1420807\">During a very quick stop, a car decelerates at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5793c38c14ae53ae15f5f2023b30010c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#109;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"75\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<p id=\"fs-id1411004\">(a) What is the angular acceleration of its 0.280-m-radius tires, assuming they do not slip on the pavement?<\/p>\n<p id=\"fs-id3142909\">(b) How many revolutions do the tires make before coming to rest, given their initial angular velocity is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5d7c5dcf059b2a76be9fbc900146ceea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#53;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#114;&#97;&#100;&#47;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"79\" style=\"vertical-align: -4px;\" \/>?<\/p>\n<p id=\"fs-id1402031\">(c) How long does the car take to stop completely?<\/p>\n<p id=\"fs-id1395221\">(d) What distance does the car travel in this time?<\/p>\n<p id=\"fs-id1859867\">(e) What was the car\u2019s initial velocity?<\/p>\n<p id=\"fs-id2611430\">(f) Do the values obtained seem reasonable, considering that this stop happens very quickly?<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id2398786\">\n<div class=\"bc-figcaption figcaption\">Yo-yos are amusing toys that display significant physics and are engineered to enhance performance based on physical laws. (credit: Beyond Neon, Flickr)<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id3077847\" data-alt=\"The figure shows the left arm of a man with tattoo imprints and wearing a glove. He is circulating a yo-yo toy, which is in mid air and connected by the string to his hand. Some people are standing in the background watching the yo-yo trick.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_11_02_03a.jpg\" data-media-type=\"image\/png\" alt=\"The figure shows the left arm of a man with tattoo imprints and wearing a glove. He is circulating a yo-yo toy, which is in mid air and connected by the string to his hand. Some people are standing in the background watching the yo-yo trick.\" width=\"225\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2972882\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3244373\">\n<p id=\"import-auto-id1468670\">Everyday application: Suppose a yo-yo has a center shaft that has a 0.250 cm radius and that its string is being pulled.<\/p>\n<p id=\"import-auto-id3341040\">(a) If the string is stationary and the yo-yo accelerates away from it at a rate of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f9c7d46e8b018b4c756998cbd88e6460_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#48;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#32;&#109;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"74\" style=\"vertical-align: -4px;\" \/>, what is the angular acceleration of the yo-yo? <\/p>\n<p id=\"import-auto-id3079277\">(b) What is the angular velocity after 0.750 s if it starts from rest?<\/p>\n<p id=\"import-auto-id3079279\">(c) The outside radius of the yo-yo is 3.50 cm. What is the tangential acceleration of a point on its edge?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1575104\">\n<p id=\"import-auto-id2670964\">a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-005e3b0eb18d2bdd2c87f7a27edd24b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#32;&#114;&#97;&#100;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"81\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"import-auto-id1930763\">b) 450 rad\/s<\/p>\n<p id=\"import-auto-id1930765\">c) 21.0 m\/s<\/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-id1380255\">\n<dt>kinematics of rotational motion<\/dt>\n<dd id=\"fs-id1947566\">describes the relationships among rotation angle, angular velocity, angular acceleration, and time<\/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-520","chapter","type-chapter","status-publish","hentry","license-all-rights-reserved"],"part":506,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/520","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\/520\/revisions"}],"predecessor-version":[{"id":521,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/520\/revisions\/521"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/parts\/506"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/520\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/media?parent=520"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapter-type?post=520"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/contributor?post=520"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/license?post=520"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}