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<h2 class="hd hd-2 unit-title">Vertical Spring</h2>
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A Vertical Spring
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<p>
A spring of negligible mass, spring constant [mathjaxinline]k[/mathjaxinline] and natural length [mathjaxinline]l_0[/mathjaxinline] is hanging vertically. This is shown in the left figure above where the spring is neither stretched nor compressed. In the central figure, a block of mass [mathjaxinline]m[/mathjaxinline] is attached to the free end. When equilibrium is reached (the block is at rest), the length of the spring has increased by [mathjaxinline]d_1[/mathjaxinline] with respect to [mathjaxinline]l_0[/mathjaxinline]. We now lower the block by an additional distance [mathjaxinline]d_2 &lt;l_0/2[/mathjaxinline] as shown in the right figure. At [mathjaxinline]t=0[/mathjaxinline] we release it (zero speed) and the block starts to oscillate. </p>
<p><b class="bfseries">(Part a)</b> Calculate [mathjaxinline]d_1[/mathjaxinline]. Express your answer in terms of [mathjaxinline]m[/mathjaxinline], [mathjaxinline]g[/mathjaxinline], [mathjaxinline]k[/mathjaxinline], l_0 for [mathjaxinline]l_0[/mathjaxinline], and d_2 for [mathjaxinline]d_2[/mathjaxinline] as needed. <p style="display:inline">[mathjaxinline]d_1=[/mathjaxinline] </p> <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_pset14_3_2_1" class="text-input-dynamath capa_inputtype inline textline">
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<p><b class="bfseries">(Part b)</b> What is [mathjaxinline]L[/mathjaxinline], the length of the spring when the block reaches its highest point during the oscillations? Express your answer in terms of [mathjaxinline]m[/mathjaxinline], [mathjaxinline]g[/mathjaxinline], [mathjaxinline]k[/mathjaxinline], l_0 for [mathjaxinline]l_0[/mathjaxinline], d_1 for \(d_1\), and d_2 for [mathjaxinline]d_2[/mathjaxinline] as needed. <p style="display:inline">[mathjaxinline]L=[/mathjaxinline] </p> <div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_pset14_3_3_1" class="text-input-dynamath capa_inputtype inline textline">
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<p><b class="bfseries">(Part c)</b> What is [mathjaxinline]v_2[/mathjaxinline], the speed that the box when it passes through the equilibrium position found in part (a)? Express your answer in terms of [mathjaxinline]m[/mathjaxinline], [mathjaxinline]g[/mathjaxinline], [mathjaxinline]k[/mathjaxinline], l_0 for [mathjaxinline]l_0[/mathjaxinline], and d_2 for [mathjaxinline]d_2[/mathjaxinline] as needed. </p>
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<p style="display:inline">[mathjaxinline]v_2[/mathjaxinline] </p>
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<h2 class="hd hd-2 unit-title">Spring with Initial Conditions at t=T/4</h2>
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Worked Example: Spring with initial conditions at t = T/4
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A box of mass [mathjaxinline]m[/mathjaxinline] is attached to a horizontal and ideal spring of spring constant [mathjaxinline]k[/mathjaxinline]. The origin of the coordinate system is set at the location of the block when the spring is at its equilibrium length. The position of the box as a function of time is given as: </p>
<table id="a0000000002" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
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<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle x(t)=C\cos (\omega _0 t ) +D\sin (\omega _0 t)[/mathjaxinline]
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<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">&#160;</td>
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where [mathjaxinline]\omega _0=\sqrt {\dfrac {k}{m}}[/mathjaxinline], and [mathjaxinline]C[/mathjaxinline] and [mathjaxinline]D[/mathjaxinline] are constant to be determined in this problem knowing that at a quarter of the period, at [mathjaxinline]t = T/4[/mathjaxinline] the box is at [mathjaxinline]x_1[/mathjaxinline] and moving with [mathjaxinline]v_1[/mathjaxinline]. </p>
<p><b class="bfseries">(Part a)</b> Calculate the constant [mathjaxinline]C[/mathjaxinline]. Express your answer in terms of x_1 for [mathjaxinline]x_1[/mathjaxinline], v_1 for [mathjaxinline]v_1[/mathjaxinline], and omega_0 for [mathjaxinline]\omega _0[/mathjaxinline] as needed. </p>
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<p style="display:inline">[mathjaxinline]C=[/mathjaxinline] </p>
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<p><b class="bfseries">(Part b)</b> Calculate the constant [mathjaxinline]D[/mathjaxinline]. Express your answer in terms of x_1 for [mathjaxinline]x_1[/mathjaxinline], v_1 for [mathjaxinline]v_1[/mathjaxinline], and omega_0 for [mathjaxinline]\omega _0[/mathjaxinline] as needed. </p>
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<p style="display:inline">[mathjaxinline]D=[/mathjaxinline] </p>
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<h3 class="hd hd-2">Worked Example - Spring with Initial Conditions at t=T/4</h3>
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<h2 class="hd hd-2 unit-title">Two Oscillating Springs</h2>
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Two Oscillating Springs
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A small object of mass [mathjaxinline]m[/mathjaxinline] is attached to two horizontal springs of spring constants [mathjaxinline]k[/mathjaxinline] and [mathjaxinline]2k[/mathjaxinline] as shown. The springs are fixed to vertical walls separated a distance [mathjaxinline]l_0[/mathjaxinline]. The equilibrium length of both springs is [mathjaxinline]l_0[/mathjaxinline]. In the figure below the object is at rest. </p>
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<img src="/assets/courseware/v1/8bea017bf6ca4c66362599da0e27564f/asset-v1:MITx+8.01.4x+1T2019+type@asset+block/images_pset14_6.svg" width="550"/>
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<p><b class="bfseries">(Part a)</b> Calculate [mathjaxinline]l_ e[/mathjaxinline], the distance between the object and the left wall when the object is at equilibrium. Express your answer in terms of [mathjaxinline]m[/mathjaxinline], [mathjaxinline]k[/mathjaxinline], and l_0 for [mathjaxinline]l_0[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]l_ e =[/mathjaxinline] </p>
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<b class="bfseries">(Part b)</b>
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<p>
Set the origin of the coordinate system [mathjaxinline]x=0[/mathjaxinline] at the equilibrium position as shown in the figure above. After pushing the object away from the equilibrium point it starts to oscillate. At the instant shown in the figure, the object is at [mathjaxinline]x[/mathjaxinline] and moving with a velocity [mathjaxinline]v_ x = \dfrac {dx}{dt}[/mathjaxinline]. Calculate [mathjaxinline]a_ x= \dfrac {d^2x}{dt^2}[/mathjaxinline], the object's acceleration. Express your answer in terms of [mathjaxinline]k[/mathjaxinline], [mathjaxinline]m[/mathjaxinline], and [mathjaxinline]x[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]a_ x= \dfrac {d^2x}{dt^2} =[/mathjaxinline] </p>
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<p><b class="bfseries">(Part c)</b> At [mathjaxinline]t=0[/mathjaxinline], the object is released at [mathjaxinline]x=x_0[/mathjaxinline] with an initial velocity [mathjaxinline]\vec{v}_0=-v_0\hat i[/mathjaxinline], [mathjaxinline]v_0&gt;0[/mathjaxinline]. Find the subsequent location of the object from equilibrium as a function of time [mathjaxinline]x(t)[/mathjaxinline] and the x-component of the velocity [mathjaxinline]v_ x(t)[/mathjaxinline] as a function of time. Write your answer using some or all of the following: m, k, t, l_0 for [mathjaxinline]l_0[/mathjaxinline], x_0 for [mathjaxinline]x_0[/mathjaxinline], v_0 for [mathjaxinline]v_0[/mathjaxinline], sqrt() for square root and sin() and cos() for trigonometric functions. </p>
<p>
<p style="display:inline">[mathjaxinline]x(t) =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]v(t) =[/mathjaxinline] </p>
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<h2 class="hd hd-2 unit-title">Spring Box System</h2>
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Spring-block System
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<p>
Consider an ideal spring that has an unstretched length [mathjaxinline]l_0[/mathjaxinline] with spring constant [mathjaxinline]k[/mathjaxinline]. The spring is attached to a mass [mathjaxinline]m[/mathjaxinline] that lies on a horizontal frictionless surface. The spring-mass system is compressed a distance [mathjaxinline]x_0[/mathjaxinline] from equilibrium and then released with an initial speed [mathjaxinline]v_0[/mathjaxinline] towards the equilibrium position. </p>
<center>
<img src="/assets/courseware/v1/3cf0a9b6397a9325b140915d50ed7c95/asset-v1:MITx+8.01.4x+1T2019+type@asset+block/images_examf_21_1.png" width="275"/>
</center>
<p><b class="bfseries">(Part a)</b> What is the angular frequency of oscillation for this system? </p>
<p>
<p style="display:inline">[mathjaxinline]\omega =[/mathjaxinline] </p>
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<p><b class="bfseries">(Part b)</b> How long will it take for the mass to first return to the equilibrium position? Express your answer in terms of [mathjaxinline]v_0[/mathjaxinline], [mathjaxinline]x_0[/mathjaxinline], omega_0 for [mathjaxinline]\omega _0[/mathjaxinline] and arctan() for [mathjaxinline]tan^{-1}()[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]t_{1}=[/mathjaxinline] </p>
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<p><b class="bfseries">(Part c)</b> How long will it take for the spring to first become completely extended? </p>
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<p><b class="bfseries">(Part d)</b> What is the maximum amount that the spring is stretched? </p>
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