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<h2 class="hd hd-2 unit-title">Make it a Line</h2>
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Make it a Line
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<p>
Consider the function [mathjaxinline]f(x) = \dfrac {1}{(a-x)^{3/2}}[/mathjaxinline], where [mathjaxinline]a[/mathjaxinline] is a positive constant. The goal of the exercise is to express [mathjaxinline]f(x)[/mathjaxinline] as a linear function of [mathjaxinline]x[/mathjaxinline] in the region where [mathjaxinline]|x|&lt;&lt;a[/mathjaxinline]. For this purpose we will write the function [mathjaxinline]f(x)[/mathjaxinline] in terms of a new variable [mathjaxinline]z=\dfrac {x}{a}[/mathjaxinline]. Because [mathjaxinline]|x|&lt;&lt;a[/mathjaxinline] the variable [mathjaxinline]z[/mathjaxinline] is small so we can do a Taylor expansion on [mathjaxinline]z[/mathjaxinline]. </p>
<p><b class="bfseries">(Part a)</b> Write [mathjaxinline](a-x)[/mathjaxinline] in terms of [mathjaxinline]x/a[/mathjaxinline]. Express your answer in terms of [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]z[/mathjaxinline], where [mathjaxinline]z=x/a[/mathjaxinline]. <p style="display:inline">[mathjaxinline]a-x =[/mathjaxinline] </p> <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ls_ls43_ls43_08_2_1" class="text-input-dynamath capa_inputtype inline textline">
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<p>
<div class="solution-span">
<span id="solution_ls_ls43_ls43_08_solution_1"/>
</div></p>
<p><b class="bfseries">(Part b)</b> Write [mathjaxinline]f(x)[/mathjaxinline] in terms of [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]z[/mathjaxinline]. <p style="display:inline">[mathjaxinline]f(z) =[/mathjaxinline] </p> <div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_ls_ls43_ls43_08_3_1" class="text-input-dynamath capa_inputtype inline textline">
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1/(a*(1-z))^(3/2) </p>
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<span id="solution_ls_ls43_ls43_08_solution_2"/>
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<p><b class="bfseries">(Part c)</b> The function [mathjaxinline]f(x)[/mathjaxinline] is now a function of [mathjaxinline]z[/mathjaxinline]. Calculate the first two coefficients of the Taylor series of [mathjaxinline]f(z)[/mathjaxinline] around [mathjaxinline]z=0[/mathjaxinline] (taking derivatives with respect to [mathjaxinline]z[/mathjaxinline]. Express your answer in terms of [mathjaxinline]a[/mathjaxinline]. </p>
<p>
<p style="display:inline"><b class="bfseries">Zero order:</b> [mathjaxinline]f(z=0)=[/mathjaxinline] </p>
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<p style="display:inline"><b class="bfseries">First order:</b> [mathjaxinline]f^{'}(z=0)=[/mathjaxinline] </p>
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<span id="solution_ls_ls43_ls43_08_solution_3"/>
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<p><b class="bfseries">(Part d)</b> Write the function [mathjaxinline]f(z)[/mathjaxinline] as a first order polynomial. Express your answer in terms of [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]z[/mathjaxinline]. </p>
<p>
<p style="display:inline"> [mathjaxinline]f(z)=[/mathjaxinline] </p>
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<div class="solution-span">
<span id="solution_ls_ls43_ls43_08_solution_4"/>
</div></p>
<p><b class="bfseries">(Part e)</b> Write the function [mathjaxinline]f(z)[/mathjaxinline] as a first order polynomial approximation of the original function [mathjaxinline]f(x)[/mathjaxinline]. Express your answer in terms of [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]x[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]f(x)=[/mathjaxinline] </p>
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<h2 class="hd hd-2 unit-title">Small Oscillations</h2>
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Small Oscillations
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<p>
A particle of mass [mathjaxinline]m[/mathjaxinline] moves in one dimension. Its potential energy is given by </p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]U(x)=-U_0 e^{-x^2 /a^2}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
where [mathjaxinline]U_0[/mathjaxinline] and [mathjaxinline]a[/mathjaxinline] are constants. The mechanical energy of the particle is constant. </p>
<p><b class="bfseries">(Part a))</b> Draw an energy diagram showing the potential energy [mathjaxinline]U(x)[/mathjaxinline], the kinetic energy [mathjaxinline]K(x)[/mathjaxinline], and the mechanical energy [mathjaxinline]E&lt;0[/mathjaxinline] for a motion which is bound between turning points [mathjaxinline]+a[/mathjaxinline] and [mathjaxinline]-a[/mathjaxinline]. </p>
<p>
<div class="solution-span">
<span id="solution_exam_final_d1_2_solution_1"/>
</div></p>
<p><b class="bfseries">(Part b))</b> Find the force on the particle, [mathjaxinline]F(x)[/mathjaxinline], as a function of position [mathjaxinline]x[/mathjaxinline]. </p>
<p>
Write your answer using some or all of the following: [mathjaxinline]x[/mathjaxinline], [mathjaxinline]a[/mathjaxinline], U_0 for [mathjaxinline]U_0[/mathjaxinline]. Use e to represent [mathjaxinline]e[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]F_0=[/mathjaxinline] </p>
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<span id="solution_exam_final_d1_2_solution_2"/>
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<p><b class="bfseries">(Part c))</b> At what point(s) is the force zero? Are those point(s) stable or unstable equilibrium point(s)? </p>
<p>
<p style="display:inline">[mathjaxinline]x=[/mathjaxinline] </p>
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<div class="solution-span">
<span id="solution_exam_final_d1_2_solution_3"/>
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<p><b class="bfseries">(Part d))</b> What is the angular frequency of small oscillations around the stable equilibrium point? </p>
<p>
Write your answer using some or all of the following: [mathjaxinline]m[/mathjaxinline], [mathjaxinline]a[/mathjaxinline], U_0 for [mathjaxinline]U_0[/mathjaxinline], sqrt() for square root, and e to represent [mathjaxinline]e[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]\omega _0=[/mathjaxinline] </p>
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<div class="solution-span">
<span id="solution_exam_final_d1_2_solution_4"/>
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<p><b class="bfseries">(Part e))</b> Find the speed at the origin [mathjaxinline]x=0[/mathjaxinline] such that when the particle reaches the positions [mathjaxinline]x=+a[/mathjaxinline] and [mathjaxinline]x=-a[/mathjaxinline] , it will reverse its motion. </p>
<p>
Write your answer using some or all of the following: [mathjaxinline]m[/mathjaxinline], [mathjaxinline]a[/mathjaxinline], U_0 for [mathjaxinline]U_0[/mathjaxinline], sqrt() for square root, and e to represent [mathjaxinline]e[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]v_0=[/mathjaxinline] </p>
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<h2 class="hd hd-2 unit-title">Paricle Moving Along a Straight Line</h2>
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Particle moving along a straight line
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Object 1 and object 2 are fixed to the x-axis as shown. They are separated a distance [mathjaxinline]2a[/mathjaxinline] from each other. A particle of mass [mathjaxinline]m[/mathjaxinline] in the x-axis experience an attractive force towards object 2 and a repulsive force away from object 1. When the particle is at position [mathjaxinline]\vec{r} = x\hat{\mathbf{i}}[/mathjaxinline], the force it experiences due to object 1 is [mathjaxinline]\vec{\mathbf{F}}_1 = \dfrac {B}{x-a}\hat{\mathbf{i}}[/mathjaxinline], and the force it experiences due to object 2 is [mathjaxinline]\vec{\mathbf{F}}_2 =- \dfrac {2B}{x+a}\hat{\mathbf{i}}[/mathjaxinline], where [mathjaxinline]B&gt;0[/mathjaxinline] and constant. </p>
<p><b class="bfseries">(Part a)</b> Calculate [mathjaxinline]x_0[/mathjaxinline], the position where the particle is at equilibrium. Express your answer in terms of [mathjaxinline]m[/mathjaxinline], [mathjaxinline]a[/mathjaxinline], and [mathjaxinline]B[/mathjaxinline] as needed. <p style="display:inline">[mathjaxinline]x_0=[/mathjaxinline] </p> <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_pset_pset15_7_2_1" class="text-input-dynamath capa_inputtype inline textline">
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<b class="bfseries">(Part b)</b>
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If the particle is moved away from the equilibrium point and released, it will start to oscillate around [mathjaxinline]x_0[/mathjaxinline]. At the instant shown in the figure, the particle is at a position [mathjaxinline]y(t)[/mathjaxinline] from [mathjaxinline]x_0[/mathjaxinline]. Calculate [mathjaxinline]\Sigma \vec{\mathbf{F}}[/mathjaxinline], the total force on the particle at the instant of time shown in the figure. Express your answer in terms of [mathjaxinline]B[/mathjaxinline], [mathjaxinline]a[/mathjaxinline], [mathjaxinline]y[/mathjaxinline], and hati for [mathjaxinline]\hat{\mathbf{i}}[/mathjaxinline] as needed. </p>
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<p><b class="bfseries">(Part c)</b> Calculate [mathjaxinline]\dfrac {d^2y}{dt^2}[/mathjaxinline], the x-component of the particle's acceleration at the instant shown in the figure. Express your answer in terms of [mathjaxinline]B[/mathjaxinline], [mathjaxinline]a[/mathjaxinline], [mathjaxinline]y[/mathjaxinline], and [mathjaxinline]m[/mathjaxinline]. </p>
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<p><b class="bfseries">(Part d)</b> Assume that [mathjaxinline]y&lt;&lt;x_0[/mathjaxinline], calculate [mathjaxinline]\omega _0[/mathjaxinline], the angular frequency of the oscillations. Express your answer in terms of [mathjaxinline]B[/mathjaxinline], [mathjaxinline]a[/mathjaxinline], and [mathjaxinline]m[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]\omega _0=[/mathjaxinline] </p>
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<h2 class="hd hd-2 unit-title">Disk and Spring</h2>
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<h3 class="hd hd-3 problem-header" id="pset_pset15_4-problem-title" aria-describedby="block-v1:MITx+8.01.4x+1T2019+type@problem+block@pset_pset15_4-problem-progress" tabindex="-1">
A disk and a spring
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<p>
A disk of mass [mathjaxinline]m[/mathjaxinline] and radius [mathjaxinline]R[/mathjaxinline] is free to rotate around a frictionless fixed axle that goes through its center (point [mathjaxinline]S[/mathjaxinline] in the figure) and perpendicular to the disk's surface. A massless spring of spring constant [mathjaxinline]k[/mathjaxinline] is attached to the edge of the disk, point [mathjaxinline]A[/mathjaxinline] in the figure. The other end of the spring is attached to a ring that goes through a vertical rod. The ring is free to slide up and down along the frictionless rod. At the instant shown in the figure above the spring at equilibrium and the disc is at rest. When point [mathjaxinline]A[/mathjaxinline] is moved away from its equilibrium position, [mathjaxinline]\theta = 0[/mathjaxinline], and then is released the disk starts to oscillate. Assume that the spring remains horizontal during the oscillations. </p>
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<p>
Consider the coordinate system shown in the figure above. In the left figure, when the spring is compressed, point [mathjaxinline]A[/mathjaxinline] is at an angle [mathjaxinline]\theta &lt;0[/mathjaxinline] measured from the vertical. When the spring is expanded, as shown in the right figure, point [mathjaxinline]A[/mathjaxinline] it to the left of the vertical at an angle [mathjaxinline]\theta &gt;0[/mathjaxinline]. </p>
<p>
<b class="bfseries">Torque Approach:</b>
</p>
<p><b class="bfseries">(Part a)</b> Calculate [mathjaxinline]\Sigma \vec{\mathbf{\tau }}_ S[/mathjaxinline], the total torque about point S exerted by the external forces on the disk. Express your answer in terms of [mathjaxinline]m[/mathjaxinline], [mathjaxinline]g[/mathjaxinline], [mathjaxinline]R[/mathjaxinline], [mathjaxinline]k[/mathjaxinline], theta for [mathjaxinline]\theta[/mathjaxinline], and hatk for [mathjaxinline]\hat{\mathbf{k}}[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]\Sigma \vec{\mathbf{\tau }}_ S=[/mathjaxinline] </p>
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<p><b class="bfseries">(Part b)</b> Apply [mathjaxinline]\Sigma \vec{\mathbf{\tau }}_ S = I_ S\vec{\mathbf{\alpha }}[/mathjaxinline] to the disk to obtain an expression for [mathjaxinline]\dfrac {d^2\theta }{dt^2}[/mathjaxinline], the disk's angular acceleration. Express your answer in terms of [mathjaxinline]m[/mathjaxinline], [mathjaxinline]g[/mathjaxinline], [mathjaxinline]R[/mathjaxinline], [mathjaxinline]k[/mathjaxinline], and theta for [mathjaxinline]\theta[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]\dfrac {d^2\theta }{dt^2}=[/mathjaxinline] </p>
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<p><b class="bfseries">(Part c)</b> Assume that the amplitude of the oscillations are small, calculate [mathjaxinline]\omega _0[/mathjaxinline], the angular frequency of oscillations. Express your answer in terms of [mathjaxinline]k[/mathjaxinline], [mathjaxinline]m[/mathjaxinline], [mathjaxinline]R[/mathjaxinline], and [mathjaxinline]g[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]\omega _0=[/mathjaxinline] </p>
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<p>
<b class="bfseries">Energy Approach:</b>
</p>
<p>
We will now solve the problem using the energy approach. Define the zero of elastic potential energy when the spring is at its equilibrium position ([mathjaxinline]\theta = 0[/mathjaxinline]). </p>
<p>
Consider the instant when point [mathjaxinline]A[/mathjaxinline] is at an angle [mathjaxinline]\theta[/mathjaxinline] with respect to the vertical and the disk is rotating about point [mathjaxinline]S[/mathjaxinline] with an angular velocity [mathjaxinline]\omega _ z=\dfrac {d\theta }{dt}[/mathjaxinline], and angular acceleration [mathjaxinline]\alpha _ z =\dfrac {d^2\theta }{dt^2}[/mathjaxinline]. </p>
<p><b class="bfseries">(Part d)</b> Calculate [mathjaxinline]U[/mathjaxinline], the potential energy of the spring in terms of [mathjaxinline]k[/mathjaxinline], [mathjaxinline]R[/mathjaxinline], theta for [mathjaxinline]\theta[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]U=[/mathjaxinline] </p>
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<p><b class="bfseries">(Part e)</b> Calculate [mathjaxinline]\dfrac {dU}{dt}[/mathjaxinline], the time derivative of the potential energy of the spring in terms of [mathjaxinline]k[/mathjaxinline], [mathjaxinline]R[/mathjaxinline], theta for [mathjaxinline]\theta[/mathjaxinline], and omega_z for [mathjaxinline]\omega _ z=\dfrac {d\theta }{dt}[/mathjaxinline] . </p>
<p>
<p style="display:inline">[mathjaxinline]\dfrac {dU}{dt}=[/mathjaxinline] </p>
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<p><b class="bfseries">(Part f)</b> Calculate [mathjaxinline]K[/mathjaxinline], the kinetic energy of the disk. Express your answer in terms of [mathjaxinline]R[/mathjaxinline], [mathjaxinline]m[/mathjaxinline], and omega_z for [mathjaxinline]\omega _ z[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]K=[/mathjaxinline] </p>
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<p><b class="bfseries">(Part g)</b> Calculate [mathjaxinline]\dfrac {dK}{dt}[/mathjaxinline], the time derivative of the kinetic energy. Express your answer in terms of [mathjaxinline]R[/mathjaxinline], [mathjaxinline]m[/mathjaxinline], omega_z for [mathjaxinline]\omega _ z[/mathjaxinline], and alpha_z for [mathjaxinline]\alpha _ z[/mathjaxinline] </p>
<p>
<p style="display:inline">[mathjaxinline]\dfrac {dK}{dt}=[/mathjaxinline] </p>
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<p><b class="bfseries">(Part h)</b> The mechanical energy is constant. Starting with [mathjaxinline]\dfrac {dE}{dt}=0[/mathjaxinline] obtain and expression for [mathjaxinline]\dfrac {d^2\theta }{dt^2}[/mathjaxinline], the disk's angular acceleration. Express your answer in terms of [mathjaxinline]m[/mathjaxinline], [mathjaxinline]k[/mathjaxinline], and theta for [mathjaxinline]\theta[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]\dfrac {d^2\theta }{dt^2}=[/mathjaxinline] </p>
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