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<h2 class="hd hd-2 unit-title">Intro to Circular Motion </h2>
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<p> As you go through the lesson, you will find this list of reading from the textbook complementary to the videos and exercises, and on many occasions these readings will contain relevant material to the presentation:</p><ul><li> Circular Motion, Velocity and Angular velocity: <a href="/courses/course-v1:MITx+8.01.1x+3T2018/pdfbook/0/chapter/6/3"><i class="it">chapter 6, section 2</i></a></li></ul>
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<h2 class="hd hd-2 unit-title">L8v1: Polar Coordinates</h2>
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<h2 class="hd hd-2 unit-title">L8v2: Position and Unit Vectors in Polar Coordinates</h2>
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<p><h2> Chain Rule of Differentiation </h2></p><p> Recall that when taking derivatives of a differentiable function \(f = f(\theta)\) whose argument is also a differentiable function \(\theta=g(t)\) then \(f=f(g(t))=h(t)\) is a differentiable function of \(t\) and </p><p> \(\frac{df}{dt}=\frac{df}{d\theta}\frac{d\theta}{dt}\) </p><p> Note that this is only in the case of circular motion, where \(r\) is a constant in time. Otherwise, our derivative of \(f\) would also have a \(\frac{df}{dr}\frac{dr}{dt}\) term. </p><p><h2> Radians </h2></p><p> One way to measure an angle is in radians. A full circle has \(2\pi\) radians. This week, we will use radians to measure the angles, so all angles will have units of radians, angular velocity will have units of radians/s, and angular acceleration will have units of radians/s\(^2\). If we multiply these by a distance, such as \(r\), the units will be m, m/s, or m/s\(^2\).</p><p><b>External References </b></p><p><a href="http://mathworld.wolfram.com/Radian.html" target="_blank">Wolfram: Radian</a></p>
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<h3 class="hd hd-2">L8v2: Position and Unit Vectors in Polar Coordinates</h3>
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L8Q1: Polar Coordinates
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In a two dimensional space, how do the polar coordinates unit vectors [mathjaxinline](\hat{r},\hat{\theta })[/mathjaxinline] differ from the Cartesian unit vectors [mathjaxinline](\hat{i},\hat{j})[/mathjaxinline]? </p>
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<text> [mathjaxinline](\hat{r},\hat{\theta })[/mathjaxinline] are not orthogonal.</text>
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<text> The magnitude of [mathjaxinline]\hat{r}[/mathjaxinline] and [mathjaxinline]\hat{\theta }[/mathjaxinline] is different from the magnitude of [mathjaxinline]\hat{i}[/mathjaxinline] and [mathjaxinline]\hat{j}[/mathjaxinline].</text>
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<text> The direction of [mathjaxinline]\hat{r}[/mathjaxinline] and [mathjaxinline]\hat{\theta }[/mathjaxinline] changes at different points in space.</text>
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<text> [mathjaxinline](\hat{r},\hat{\theta })[/mathjaxinline] do not differ from [mathjaxinline](\hat{i},\hat{j})[/mathjaxinline] in any fundamental way.</text>
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<h2 class="hd hd-2 unit-title">L8Q2: Position Vector for Circular Motion</h2>
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L8Q2: Position Vector for Circular Motion
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A point-like object undergoes circular motion at a constant speed. The position vector of the object measured from the center of the circle </p>
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<text> has constant magnitude and hence is constant in time.</text>
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<text> has constant magnitude but is changing direction so is not constant in time.</text>
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<text> is changing in magnitude and hence is not constant in time.</text>
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A vector [mathjaxinline]\vec{B}[/mathjaxinline] of constant magnitude rotates counterclockwise. At the moment shown in the left figure, which arrow best describes the direction of [mathjaxinline]\frac{d\vec{B}}{dt}[/mathjaxinline]? <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_ls_ls03_ls03_22_2_1">
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<input type="radio" name="input_ls_ls03_ls03_22_2_1" id="input_ls_ls03_ls03_22_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="ls_ls03_ls03_22_2_1-choice_4-label" for="input_ls_ls03_ls03_22_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_ls_ls03_ls03_22_2_1"> <text> 4</text>
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<input type="radio" name="input_ls_ls03_ls03_22_2_1" id="input_ls_ls03_ls03_22_2_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="ls_ls03_ls03_22_2_1-choice_5-label" for="input_ls_ls03_ls03_22_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_ls_ls03_ls03_22_2_1"> <text> None of the above because the magnitude of [mathjaxinline]\vec{B}[/mathjaxinline] is constant.</text>
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<h2 class="hd hd-2 unit-title">L8Q4: Position and Velocity</h2>
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L8Q4: Position and Velocity
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A particle is moving clockwise with a speed of 2 m/s in a circle of radius 3 m contained in the [mathjaxinline]xy[/mathjaxinline]-plane. The angle [mathjaxinline]\theta[/mathjaxinline] is measured from the [mathjaxinline]+x[/mathjaxinline]-axis in a counter-clockwise direction. </p>
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Which of the following statements about [mathjaxinline]\vec{r}[/mathjaxinline] and [mathjaxinline]\vec{v}[/mathjaxinline] are true: </p>
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<text> [mathjaxinline]\vec{r} =+[/mathjaxinline] (3 m) [mathjaxinline]\hat{r}[/mathjaxinline] and [mathjaxinline]\vec{v}=+[/mathjaxinline](2 m/s) [mathjaxinline]\hat{\theta }[/mathjaxinline]</text>
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<text> [mathjaxinline]\vec{r} =+[/mathjaxinline] (3 m) [mathjaxinline]\hat{r}[/mathjaxinline] and [mathjaxinline]\vec{v} =-[/mathjaxinline](2 m/s) [mathjaxinline]\hat{\theta }[/mathjaxinline]</text>
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<text> [mathjaxinline]\vec{r} =-[/mathjaxinline] (3 m) [mathjaxinline]\hat{r}[/mathjaxinline] and [mathjaxinline]\vec{v}=+[/mathjaxinline](2 m/s) [mathjaxinline]\hat{\theta }[/mathjaxinline]</text>
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<text> [mathjaxinline]\vec{r} =-[/mathjaxinline] (3 m) [mathjaxinline]\hat{r}[/mathjaxinline] and [mathjaxinline]\vec{v} =-[/mathjaxinline](2 m/s) [mathjaxinline]\hat{\theta }[/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">L8Q5: Choosing a Coordinate System</h2>
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L8Q5-1: Choosing Coordinate System - part a
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An object is traveling clockwise around a circle of radius [mathjaxinline]r[/mathjaxinline] with a velocity [mathjaxinline]\vec{v}[/mathjaxinline]. In Figure 1, the angle [mathjaxinline]\theta _{1}[/mathjaxinline] is measured with respect to the horizontal, and hence [mathjaxinline]\hat{\theta _{1}}[/mathjaxinline] is counter-clockwise. In Figure 2, the angle [mathjaxinline]\theta _{2}[/mathjaxinline] is measured with respect to the vertical so [mathjaxinline]\hat{\theta _{2}}[/mathjaxinline] is pointing clockwise. </p>
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<p><b class="bfseries">(Part a)</b> In each coordinate system, what is the sign of [mathjaxinline]\frac{d\theta }{dt}[/mathjaxinline]? </p>
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<text> In Figure 1 [mathjaxinline]\frac{d\theta _{1}}{dt} \gt 0[/mathjaxinline] and in Figure 2 [mathjaxinline]\frac{d\theta _{2}}{dt} \gt 0[/mathjaxinline]</text>
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<text> In Figure 1 [mathjaxinline]\frac{d\theta _{1}}{dt} \gt 0[/mathjaxinline] and in Figure 2 [mathjaxinline]\frac{d\theta _{2}}{dt} \lt 0[/mathjaxinline]</text>
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<text> In Figure 1 [mathjaxinline]\frac{d\theta _{1}}{dt} \lt 0[/mathjaxinline] and in Figure 2 [mathjaxinline]\frac{d\theta _{2}}{dt} \gt 0[/mathjaxinline]</text>
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<text> In Figure 1 [mathjaxinline]\frac{d\theta _{1}}{dt} \lt 0[/mathjaxinline] and in Figure 2 [mathjaxinline]\frac{d\theta _{2}}{dt} \lt 0[/mathjaxinline]</text>
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L8Q5-2: Choosing Coordinate System - part b
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<p><b class="bfseries">(Part b)</b> Which of the following correctly describes the velocity vector in each case? </p>
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<text> [mathjaxinline]\vec{v}=r\frac{d\theta _{1}}{dt}\hat{\theta _{1}}[/mathjaxinline] in Figure 1, [mathjaxinline]\vec{v}=-r\frac{d\theta _{2}}{dt}\hat{\theta _{2}}[/mathjaxinline] in Figure 2.</text>
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<text> [mathjaxinline]\vec{v}=r\frac{d\theta _{1}}{dt}\hat{\theta _{1}}[/mathjaxinline] in Figure 1 and [mathjaxinline]\vec{v}=r\frac{d\theta _{2}}{dt}\hat{\theta _{2}}[/mathjaxinline] in Figure 2.</text>
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<text> [mathjaxinline]\vec{v}=-r\frac{d\theta _{1}}{dt}\hat{\theta _{1}}[/mathjaxinline] in Figure 1, [mathjaxinline]\vec{v}=r\frac{d\theta _{2}}{dt}\hat{\theta _{2}}[/mathjaxinline] in Figure 2.</text>
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<text> [mathjaxinline]\vec{v}=-r\frac{d\theta _{1}}{dt}\hat{\theta _{1}}[/mathjaxinline] in Figures 1 and [mathjaxinline]\vec{v}=-r\frac{d\theta _{2}}{dt}\hat{\theta _{2}}[/mathjaxinline] in Figure 2.</text>
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<h2 class="hd hd-2 unit-title">L8v4: Angular Velocity</h2>
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<h3 class="hd hd-2">L8v4: Angular Velocity</h3>
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<h2 class="hd hd-2 unit-title">L8Q6: Magnitude of the Angular Velocity</h2>
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L8Q6: Magnitude of the Angular Velocity
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Object A sits at the outer edge of a rotating disc, while object B sits halfway between the edge and the axis of rotation. The disc makes a complete revolution once every 30 seconds. The magnitude of the angular velocity of Object B is: </p>
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<text> half the magnitude of the angular velocity of Object A.</text>
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<text> the same as the magnitude of the angular velocity of Object A.</text>
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<text> twice the magnitude of the angular velocity of Object A.</text>
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<h2 class="hd hd-2 unit-title">L8Q7: Velocity, Angular Velocity and Angular Speed</h2>
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L8Q7-1: Velocity, Angular Velocity and Angular Speed - part a
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Consider a particle moving in a circle of radius [mathjaxinline]r[/mathjaxinline]. At time [mathjaxinline]t=0[/mathjaxinline], the particle is located on the [mathjaxinline]+x[/mathjaxinline]-axis. The particle's angular position measured with respect to the positive [mathjaxinline]x[/mathjaxinline]-axis is given by: </p>
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[mathjaxinline]\theta (t)=At-Bt^3[/mathjaxinline] </p>
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where [mathjaxinline]A[/mathjaxinline] and [mathjaxinline]B[/mathjaxinline] are positive constants. </p>
<p><b class="bfseries">(Part a)</b> At what instant of time [mathjaxinline]t =t_1[/mathjaxinline] is the particle momentarily at rest? You answer may include the constants [mathjaxinline]A[/mathjaxinline] and [mathjaxinline]B[/mathjaxinline]. <p style="display:inline">[mathjaxinline]t_1 =[/mathjaxinline] </p> <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ls_ls03_ls03_20_2_1" class="text-input-dynamath capa_inputtype inline textline">
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L8Q7-2: Velocity, Angular Velocity and Angular Speed - part b
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Consider the particle from part a. It is moving in a circle of radius [mathjaxinline]r[/mathjaxinline] and its angular position measured with respect to the positive [mathjaxinline]x[/mathjaxinline]-axis is given by: </p>
<p>
[mathjaxinline]\theta (t)=At-Bt^3[/mathjaxinline] </p>
<p>
where [mathjaxinline]A[/mathjaxinline] and [mathjaxinline]B[/mathjaxinline] are positive constants. Let [mathjaxinline]t_1[/mathjaxinline] be the instant of time calculated in part a for which the particle is momentarily at rest. </p>
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<p><b class="bfseries">(Part b)</b> Drag the proper directions [mathjaxinline](\pm \hat{r},\pm \hat{\theta },\pm \hat{k})[/mathjaxinline] to the corresponding vectorial physical quantities. Also, indicate if the components of the given vectorial physical quantities are positive, negative or zero by dragging [mathjaxinline]\gt 0, \lt 0, =0[/mathjaxinline]. Be careful about distinguishing between vectors and scalars in this problem. </p>
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