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<h2 class="hd hd-2 unit-title">L42: Introduction to Small Oscillations</h2>
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<center><p><b> Small Oscillations</b></p></center><p> In the previous chapter, we studied the simple harmonic motion of a particle around an equilibrium position. In particular we examined the force exerted by an ideal spring of spring constant \(k\): \(\vec{\mathbf{F}}=-k(x-x_0)\hat{\mathbf{i}}\). As a result, the potential function associated with this conservative force is a parabola, \(U(x) = \frac{1}{2}k(x-x_0)^2\), with its minimum at the equilibrium point \(x=x_0\). </p><p>
In this lesson we will consider the one dimensional motion of objects around stable equilibrium points. For example, the potential function shown in the figure below has a stable equilibrium point at \(x=x_0\).
</p><p><center><img src="/assets/courseware/v1/ec93b9622894efef62c2821369a10f6b/asset-v1:MITx+8.01.4x+1T2019+type@asset+block/images_html_L43intro.svg" width="350"/></center></p><p>Assume that an object is initially at rest at \(x=x_0\). If we apply a small kick to the object it will start to oscillate around the equilibrium point. Is this oscillatory motion harmonic?</p><p> If the amplitude of the oscillations is <b>small</b>, the motion of the object around the equilibrium point is a simple harmonic motion, with a well defined frequency independent of the amplitude of the oscillations. </p><p>
To show this we express the potential function in terms of its Taylor expansion around \(x=x_0\):
</p><p>
\[\displaystyle U(x) = U(x_0)+\left.\frac{dU}{dx}\right|_{x=x_0}(x-x_0)+\frac{1}{2}\left.\frac{d^2U}{dx^2}\right|_{x=x_0}(x-x_0)^2+...\]
</p><p> We note that at \(x=x_0\) the potential is a minimum, therefore \(\displaystyle\left.\frac{dU}{dx}\right|_{x=x_0}=0\) and \(\displaystyle\left.\frac{d^2U}{dx^2}\right|_{x=x_0}>0\). </p><p> In addition, if the amplitude of the oscillation is small enough that we can neglect the higher order terms of the polynomial, the potential function around \(x \approx x_0\) can be approximted by a parabola:</p><p>
\[U(x) =C+ A(x-x_0)^2 \]
</p><p> where \(C = U(x_0)\) and \(A = \frac{1}{2}\left.\frac{d^2U}{dx^2}\right|_{x=x_0} >0\). </p><p> In this approximation, the potential function is the potential function of an ideal spring if we define the spring constant to be \(k = \left.\frac{d^2U}{dx^2}\right|_{x=x_0}\).</p><p> Therefore for small amplitude of oscillations, the motion of a particle around a stable equilibrium point can be described as a simple harmonic motion, similar to the motion of an object attached to the end of the ideal spring. </p>
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<h2 class="hd hd-2 unit-title">L43Q1: Periodic Motion</h2>
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<p>3 particles have the same potential energy \(U(x)\), but they have different total mechanical energies: \(E_1\), \(E_2\) and \(E_3\). Which particles have an approximate <i>simple harmonic</i> motion? Check all that apply.</p>
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<p>Which of these particles undergo <i>periodic</i> motion? Check all that apply.</p>
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<h2 class="hd hd-2 unit-title">L43v1: Strategy for Small Oscillation Problems</h2>
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<h2 class="hd hd-2 unit-title">L43v2: Worked Example - Small Oscillations Around the Minimum of a Potential</h2>
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Worked Example: Small oscillations in a potential with one minimum.
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A particle of mass [mathjaxinline]m[/mathjaxinline] is moving under the action of a potential represented by: </p>
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<td style="width:40%; border:none">&#160;</td>
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[mathjaxinline]\displaystyle U(r)=\frac{a_1}{r}+\frac{a_2}{r^2}[/mathjaxinline]
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<td style="width:20%; border:none" class="eqnnum">&#160;</td>
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<p>
where[mathjaxinline]a_1&lt;0[/mathjaxinline] and constant, and [mathjaxinline]a_2&gt;0[/mathjaxinline] and constant </p>
<p><b class="bfseries">(Part a)</b> Find [mathjaxinline]r_0[/mathjaxinline], the value of [mathjaxinline]r[/mathjaxinline] for which [mathjaxinline]U(r)[/mathjaxinline] is a minimum. Express your answer in terms of a_1 for [mathjaxinline]a_1[/mathjaxinline] and a_2 for [mathjaxinline]a_2[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]r_0=[/mathjaxinline] </p>
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<p><b class="bfseries">(Part b)</b> Calculate [mathjaxinline]\displaystyle \left[\frac{d^2U}{dr^2}\right]_{r=r_0}[/mathjaxinline], the second derivative of [mathjaxinline]U(r)[/mathjaxinline] at [mathjaxinline]r=r_0[/mathjaxinline]. Express your answer in terms of a_1 for [mathjaxinline]a_1[/mathjaxinline] and a_2 for [mathjaxinline]a_2[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]\displaystyle \left[\frac{d^2U}{dr^2}\right]_{r=r_0}=[/mathjaxinline] </p>
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<p><b class="bfseries">(Part c)</b> If we neglect the higher order terms and approximate [mathjaxinline]U(r)[/mathjaxinline] as a second order polynomial around [mathjaxinline]r_0[/mathjaxinline], the energy of the particle is given by: </p>
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[mathjaxinline]\displaystyle \displaystyle E = \frac{1}{2}mv^2 + U(r_0) + \frac{1}{2}\left[\frac{d^2U}{dr^2}\right]_{r=r_0} (r-r_0)^2[/mathjaxinline]
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<td style="width:20%; border:none" class="eqnnum">&#160;</td>
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<p>
Calculate [mathjaxinline]\omega _0[/mathjaxinline], the angular frequency for the small oscillation of the particle around [mathjaxinline]r_0[/mathjaxinline]. Express your answer in terms of [mathjaxinline]m[/mathjaxinline], a_1 for [mathjaxinline]a_1[/mathjaxinline] and a_2 for [mathjaxinline]a_2[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]\omega _0=[/mathjaxinline] </p>
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<h3 class="hd hd-2">L43v2: Worked Example - Small Oscillations Around the Minimum of a Potential</h3>
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<h2 class="hd hd-2 unit-title">L43Q2: Small Oscillations Around a Stable Equilibrium Point</h2>
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Small Oscillations About Stable Equilibrium Point
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<p>
An object with mass [mathjaxinline]m[/mathjaxinline] is moving in one dimension with a potential energy given by </p>
<table id="a0000000002" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
<tr id="a0000000003">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle U(x)=-Ax^3+Bx[/mathjaxinline]
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<td style="width:20%; border:none" class="eqnnum">&#160;</td>
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<p>
where [mathjaxinline]A[/mathjaxinline] is a positive constant with units [mathjaxinline]J\cdot m^{-3}[/mathjaxinline] and [mathjaxinline]B[/mathjaxinline] is a positive constant with units [mathjaxinline]J\cdot m^{-1}[/mathjaxinline]. Which expressions best describe the frequency of small oscillations [mathjaxinline]\omega _0[/mathjaxinline] about the stable equilibrium point [mathjaxinline]x_0[/mathjaxinline]? </p>
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<text> [mathjaxinline]\omega _0=\sqrt {\frac{(12AB)^{1/2}}{m}},x_0=+\sqrt {\frac{B}{3A}}[/mathjaxinline]</text>
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<text> [mathjaxinline]\omega _0=\sqrt {\frac{(2AB)^{1/2}}{m}},x_0=+\sqrt {\frac{B}{3A}}[/mathjaxinline]</text>
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<text> [mathjaxinline]\omega _0=\sqrt {\frac{(12AB)^{1/2}}{m}},x_0=-\sqrt {\frac{B}{3A}}[/mathjaxinline]</text>
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<text> [mathjaxinline]\omega _0=\sqrt {\frac{(2AB)^{1/2}}{m}},x_0=-\sqrt {\frac{B}{3A}}[/mathjaxinline]</text>
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<text> [mathjaxinline]\omega _0=\sqrt {-\frac{(12AB)^{1/2}}{m}},x_0=+\sqrt {\frac{B}{3A}}[/mathjaxinline]</text>
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<text> [mathjaxinline]\omega _0=\sqrt {-\frac{(2AB)^{1/2}}{m}},x_0=+\sqrt {\frac{B}{3A}}[/mathjaxinline]</text>
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<text> [mathjaxinline]\omega _0=\sqrt {-\frac{(12AB)^{1/2}}{m}},x_0=-\sqrt {\frac{B}{3A}}[/mathjaxinline]</text>
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<text> [mathjaxinline]\omega _0=\sqrt {-\frac{(2AB)^{1/2}}{m}},x_0=-\sqrt {\frac{B}{3A}}[/mathjaxinline]</text>
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