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<h2 class="hd hd-2 unit-title">Introduction to Beat Phenomena</h2>
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Now we will go through a worked example of a coupled oscillator system, solving for the normal mode frequencies and amplitudes. This will give us good practice with the matrix methods that were introduced in the last lesson. </p><p>
The general solution for the motion of a system of coupled oscillators can be expressed as the sum of the normal modes of the system. In fact, the general motion of a system of two coupled oscillators will, itself, oscillate back and forth between the normal modes of the system, at a frequency called the "beat" frequency. </p><p>
We will discuss "beat" frequency, which is a type of interference phenomena, and also look at some examples. </p>
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<h2 class="hd hd-2 unit-title">L8v1: Coupled Pendula System</h2>
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Understanding Force Diagrams
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<b class="bfseries">NOTE: We will work through this ungraded example in the lesson video below, but you are encouraged to try it on your own first!</b>
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<p>
Consider the following system of two identical coupled pendula, both of mass [mathjaxinline]m[/mathjaxinline] and length [mathjaxinline]\ell[/mathjaxinline], and attached by a spring: </p>
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<p>
Assume that the spring is unstretched when the two pendula are vertical. Assume that the stretch of the spring depends only on the horizontal displacement of the pendulum masses and that the spring force is purely horizontal. Define forces and displacements that point to the right as positive (i.e. [mathjaxinline]+\hat{x}[/mathjaxinline] points to the right). </p>
<p><b class="bfseries">(Part a)</b> Consider the following pair of force diagrams, which show ALL of the forces acting on each hanging mass. Each mass experiences three forces: (1) the tension force along the direction of the string [mathjaxinline]T[/mathjaxinline], (2) the force due to gravity [mathjaxinline]F_{g}[/mathjaxinline], and (3) the spring force [mathjaxinline]F_{s}[/mathjaxinline]. </p>
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<p>
In the small angle limit, [mathjaxinline]\theta _{1} \ll 1[/mathjaxinline], what is the [mathjaxinline]\hat{x}[/mathjaxinline]-component of the string tension force on the left mass? </p>
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<text> a) [mathjaxinline]\vec{T}_{1,x}=mg\hat{x}[/mathjaxinline]</text>
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<text> b) [mathjaxinline]\vec{T}_{1,x}=-mg\hat{x}[/mathjaxinline]</text>
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<text> c) [mathjaxinline]\vec{T}_{1,x}=mg\frac{x_1}{\ell }\hat{x}[/mathjaxinline]</text>
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<text> d) [mathjaxinline]\vec{T}_{1,x}=-mg\frac{x_1}{\ell }\hat{x}[/mathjaxinline]</text>
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Finding Forces
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<p><b class="bfseries">(Part b)</b> Which are the correct expressions for the spring force on each pendulum (magnitude and direction)? Think carefully about the signs of the various quantities. </p>
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<text> a) [mathjaxinline]\vec{F}_{s,1} = k(x_{1}-x_{2})\hat{x}[/mathjaxinline]; [mathjaxinline]\vec{F}_{s,2} = k(x_{2}-x_{1})\hat{x}[/mathjaxinline]</text>
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<text> b) [mathjaxinline]\vec{F}_{s,1} = k(x_{2}-x_{1})\hat{x}[/mathjaxinline]; [mathjaxinline]\vec{F}_{s,2} = k(x_{1}-x_{2})\hat{x}[/mathjaxinline]</text>
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<text> c) [mathjaxinline]\vec{F}_{s,1} = k(x_{1}+x_{2})\hat{x}[/mathjaxinline]; [mathjaxinline]\vec{F}_{s,2} = -k(x_{2}+x_{1})\hat{x}[/mathjaxinline]</text>
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Equations of Motion
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<p><b class="bfseries">(Part c)</b> What are the equations of motion for each mass in the system? Assume that the angular amplitudes of the motion are small. Express your answers in terms of<code>x_1</code> for [mathjaxinline]x_1[/mathjaxinline], <code>x_2</code> for [mathjaxinline]x_2[/mathjaxinline], <code>m</code>, <code>g</code>, <code>l</code> for [mathjaxinline]\ell[/mathjaxinline], and <code>k</code>. </p>
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<p style="display:inline">[mathjaxinline]m\ddot{x}_1=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]m\ddot{x}_2=[/mathjaxinline]</p>
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<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">integers</td>
<td class="formulainput">
<code>2520</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">fractions</td>
<td class="formulainput">
<code>2/3</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
</td>
</tr>
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<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">
<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
</td>
</tr>
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<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
</td>
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<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
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<h3 class="hd hd-2">L8v1: Coupled Pendula System</h3>
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<h2 class="hd hd-2 unit-title">L8v2: Matrix Solution to Coupled Pendula System</h2>
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Finding the Matrices
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<b class="bfseries">NOTE: We will work through this example in the lesson video below, but you are encouraged to try it on your own first!</b>
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Consider the same system of two coupled pendula, attached by a spring: </p>
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<p>
Identify the matrices [mathjaxinline]\textbf{M}[/mathjaxinline] and [mathjaxinline]\textbf{K}[/mathjaxinline], based on the equations of motion that you previously derived: </p>
<table id="a0000000012" 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 m\ddot{x}_1[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =-\left(k+\frac{mg}{\ell }\right)x_1+kx_2[/mathjaxinline]
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[mathjaxinline]\displaystyle m\ddot{x}_2[/mathjaxinline]
</td>
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[mathjaxinline]\displaystyle =kx_1-\left(k+\frac{mg}{\ell }\right)x_2[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">&#160;</td>
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<p>
Express your answers in terms <code>m</code>, <code>g</code>, <code>l</code> for [mathjaxinline]\ell[/mathjaxinline], and <code>k</code>. </p>
<p style="display:inline">[mathjaxinline]\textbf{M}=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\textbf{K}=[/mathjaxinline]</p>
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<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">integers</td>
<td class="formulainput">
<code>2520</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">fractions</td>
<td class="formulainput">
<code>2/3</code>
</td>
</tr>
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<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">
<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
</td>
</tr>
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<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
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<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
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<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
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<h2 class="hd hd-2 unit-title">L8v3: Solving Determinant for Eigenfrequencies</h2>
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Combining Matrices
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<p>
<b class="bfseries">NOTE: We will work through this example in the lesson video below, but you are encouraged to try it on your own first!</b>
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<p>
Consider the same system of two coupled pendula, attached by a spring: </p>
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<p><b class="bfseries">(Part a)</b> Identify the [mathjaxinline]\textbf{M}^{-1}\textbf{K}[/mathjaxinline] matrix. Express your answers in terms <code>m</code>, <code>g</code>, <code>l</code> for [mathjaxinline]\ell[/mathjaxinline], and <code>k</code>. </p>
<p style="display:inline">[mathjaxinline]\textbf{M}^{-1}\textbf{K}=[/mathjaxinline]</p>
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<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
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<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">integers</td>
<td class="formulainput">
<code>2520</code>
</td>
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<td class="formulainput">fractions</td>
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<code>2/3</code>
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<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
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<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
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<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
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<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
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<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
</tr>
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<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
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<th class="formulainput" scope="row">Mathematical <br/> constants</th>
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<code>e, pi</code>
</td>
<td class="formulainput">enter <code>e^x </code> for [mathjaxinline] e^x [/mathjaxinline]<br/>
enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
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<td class="formulainput">
<code>abs, ln, sqrt</code>
</td>
<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
enter <code>sqrt(x^2-y) </code> for [mathjaxinline] \sqrt{x^2-y} [/mathjaxinline]
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<th class="formulainput" scope="row" rowspan="3">Trigonometric <br/> functions</th>
<td class="formulainput">
<code>sin, cos, tan, sec, csc, cot</code>
</td>
<td class="formulainput">enter <code>sin(4*x+y)^2 </code> for [mathjaxinline]\sin^2(4x+y) [/mathjaxinline]</td>
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<td class="formulainput"><code>arcsin, arccos, arctan</code>, etc.</td>
<td class="formulainput">enter <code>arctan(x^2/3) </code> for [mathjaxinline]\tan^{-1}\left(\frac{x^2}{3}\right) [/mathjaxinline]</td>
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<td class="formulainput"><code>sinh, cosh, arcsinh</code>, etc.</td>
<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Matrices<br/>&amp; Vectors</th>
<td class="formulainput">matrix</td>
<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput">row vector</td>
<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
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Finding Eigenfrequencies
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<p><b class="bfseries">(Part b)</b> Solve for the eigenfrequencies of the system. To do so, find the solutions of the equation [mathjaxinline]|\textbf{M}^{-1}\textbf{K} - \omega ^{2}\textbf{I}|=0[/mathjaxinline], using the [mathjaxinline]\textbf{M}^{-1}\textbf{K}[/mathjaxinline] matrix that you previously identified. </p>
<p>
Note, your answer will take the form of a quadratic equation with [mathjaxinline]\pm[/mathjaxinline] roots. Therefore, report the eigenfrequencies with [mathjaxinline]\omega _{+}[/mathjaxinline] indicating the larger of the two solutions and [mathjaxinline]\omega _{-}[/mathjaxinline] indicating the smaller one. Express your answers in terms <code>m</code>, <code>g</code>, <code>l</code> for [mathjaxinline]\ell[/mathjaxinline], and <code>k</code>. </p>
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<p style="display:inline">[mathjaxinline]\omega _{-} =[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\omega _{+} =[/mathjaxinline]</p>
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<td class="formulainput">enter <code>[[1,0],[0,-1]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 0 \\ 0 &amp; &amp; -1 \end{pmatrix}[/mathjaxinline]</td>
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<td class="formulainput">column vector</td>
<td class="formulainput">enter <code>[[1],[2],[3]]</code> for [mathjaxinline]\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}[/mathjaxinline]</td>
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<td class="formulainput">enter <code>[[1,2,3]]</code> for [mathjaxinline]\begin{pmatrix} 1 &amp; &amp; 2 &amp; &amp; 3 \end{pmatrix}[/mathjaxinline]</td>
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Finding Amplitudes
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<b class="bfseries">NOTE: We will work through this example in the lesson video below, but you are encouraged to try it on your own first!</b>
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After addressing this question, the video goes on to discuss the time dependence of the motion in more detail. </p>
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Consider the same system of two coupled pendula, attached by a spring: </p>
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You found earlier that matrix equation: </p>
<table id="a0000000032" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]|\textbf{M}^{-1}\textbf{K} - \omega ^{2}\textbf{I}|=0[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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where </p>
<table id="a0000000033" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]|\textbf{M}^{-1}\textbf{K} - \omega ^{2}\textbf{I}|=\mathrm{det} \begin{pmatrix} (k/m + g/\ell )-\omega ^2 &amp; &amp; - k/m \\ -k/m &amp; &amp; (k/m + g/\ell )-\omega ^2 \end{pmatrix}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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<p>
gave two solutions for [mathjaxinline]\omega[/mathjaxinline] </p>
<table id="a0000000034" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
<tr id="a0000000035">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \omega _{+}[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =\sqrt {g/\ell +2k/m}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">&#160;</td>
</tr>
<tr id="a0000000036">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \omega _{-}[/mathjaxinline]
</td>
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[mathjaxinline]\displaystyle =\sqrt {g/\ell }[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">&#160;</td>
</tr>
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<p>
Recall that the original equation we were trying to solve was: </p>
<table id="a0000000037" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
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<td class="equation" style="width:80%; border:none">[mathjax]\left(\textbf{M}^{-1}\textbf{K} - \omega ^{2}\textbf{I}\right)\textbf{A} = 0[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
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<p>
Now that you know the two eigenfrequencies [mathjaxinline]\omega _{+}[/mathjaxinline] and [mathjaxinline]\omega _{-}[/mathjaxinline], find the two vectors </p>
<table id="a0000000038" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\textbf{A}= \begin{pmatrix} A_1 \\ A_2 \end{pmatrix}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
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<p>
that correspond to the two eigenfrequencies. Since multiplying any answer [mathjaxinline]\textbf{A}[/mathjaxinline] by a constant will give another equally valid answer, normalize your answers so that the first entry is unity: </p>
<table id="a0000000039" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\left(\frac{1}{A_1}\right) \begin{pmatrix} A_1 \\ A_2 \end{pmatrix} = \begin{pmatrix} 1 \\ A_2/A_1 \end{pmatrix}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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<p>
Your answers should all be pure numbers (with [mathjaxinline]\textbf{A}_{+}[/mathjaxinline] and [mathjaxinline]\textbf{A}_{-}[/mathjaxinline] corresponding to [mathjaxinline]\omega _{+}[/mathjaxinline] and [mathjaxinline]\omega _{-}[/mathjaxinline], respectively). </p>
<p style="display:inline">[mathjaxinline]\textbf{A}_{+}=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\textbf{A}_{-}=[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">L8Q1: More Practice with Finding Eigenvectors</h2>
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In the previous video, we derived normal mode solutions to the system of equations. The solutions for the two normal modes are: </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]\textbf{x}^{(1)}(t) = C_{1} \cos (\omega _{1}t + \phi _{1}) \textbf{A}^{(1)}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>1</span>)</span></td></tr></table><table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\textbf{x}^{(2)}(t) = C_{2} \cos (\omega _{2}t + \phi _{2}) \textbf{A}^{(2)}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>2</span>)</span></td></tr></table><p>
where [mathjaxinline]\omega _{1}[/mathjaxinline] and [mathjaxinline]\omega _{2}[/mathjaxinline] are the normal mode frequencies (eigenvalues), and [mathjaxinline]\textbf{A}^{(1)}[/mathjaxinline] and [mathjaxinline]\textbf{A}^{(2)}[/mathjaxinline] are the normal mode amplitudes (eigenvectors). </p><p>
The general solution is a linear combination of the normal mode solutions, and can be expressed as: </p><table id="a0000000004" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\textbf{x}(t) = D_{1} \textbf{x}^{(1)}(t) + D_{2} \textbf{x}^{(2)}(t)[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>3</span>)</span></td></tr></table><p>
Furthermore, by redefining constants such that [mathjaxinline]\alpha =D_{1}C_{1}[/mathjaxinline] and [mathjaxinline]\beta =D_{2}C_{2}[/mathjaxinline], we can express the general solution as: </p><table id="a0000000005" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\textbf{x}(t) = \alpha \cos (\omega _{1}t + \phi _{1}) \textbf{A}^{(1)} + \beta \cos (\omega _{2}t + \phi _{2}) \textbf{A}^{(2)}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>4</span>)</span></td></tr></table><p>
Thus far, we have expressed everything very generally. For the specific example that we were working with, the eigenvectors were: </p><table id="a0000000006" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\textbf{A}^{(1)} = \begin{pmatrix} 1\\ 1 \end{pmatrix}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>5</span>)</span></td></tr></table><table id="a0000000007" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\textbf{A}^{(2)} = \begin{pmatrix} 1\\ -1 \end{pmatrix}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>6</span>)</span></td></tr></table><p>
The motion of each pendulum mass was then found to be: </p><table id="a0000000008" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]x_{1}(t) = \alpha \cos (\omega _{1}t + \phi _{1}) + \beta \cos (\omega _{2}t + \phi _{2})[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>7</span>)</span></td></tr></table><table id="a0000000009" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]x_{2}(t) = \alpha \cos (\omega _{1}t + \phi _{1}) - \beta \cos (\omega _{2}t + \phi _{2})[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>8</span>)</span></td></tr></table>
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Finding Position versus Time
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Consider a system of three identical masses, attached to springs, as shown. </p>
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<p>
The normal mode frequencies and amplitudes associated with the system are: </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]\textbf{A}^{(1)} = \frac{1}{2} \begin{pmatrix} 1\\ \sqrt {2}\\ 1 \end{pmatrix}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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<table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\textbf{A}^{(2)} = \frac{1}{\sqrt {2}} \begin{pmatrix} 1\\ 0\\ -1 \end{pmatrix}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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<td class="equation" style="width:80%; border:none">[mathjax]\textbf{A}^{(3)} = \frac{1}{2} \begin{pmatrix} 1\\ -\sqrt {2}\\ 1 \end{pmatrix}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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<p>
where, [mathjaxinline]\omega _{1}=\sqrt {\frac{k}{m} \left( 2 - \sqrt {2}\right)}[/mathjaxinline], [mathjaxinline]\omega _{2}=\sqrt {\frac{2k}{m}}[/mathjaxinline], and [mathjaxinline]\omega _{3}=\sqrt {\frac{k}{m} \left( 2 + \sqrt {2}\right)}[/mathjaxinline]. </p>
<p>
We can express the general motion of the system in the following, compact form: </p>
<table id="a0000000005" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\textbf{x}(t) = \alpha \cos (\omega _{1}t + \phi _{1}) \textbf{A}^{(1)} + \beta \cos (\omega _{2}t + \phi _{2}) \textbf{A}^{(2)} + \gamma \cos (\omega _{3}t + \phi _{3}) \textbf{A}^{(3)}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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</table>
<p>
Now, write the solution for the general motion of each mass, separately. Express your answer using <code>alpha</code> for [mathjaxinline]\alpha[/mathjaxinline], <code>beta</code> for [mathjaxinline]\beta[/mathjaxinline], <code>gamma</code> for [mathjaxinline]\gamma[/mathjaxinline], <code>omega_1</code> for [mathjaxinline]\omega _{1}[/mathjaxinline], <code>omega_2</code> for [mathjaxinline]\omega _{2}[/mathjaxinline], <code>omega_3</code> for [mathjaxinline]\omega _{3}[/mathjaxinline], <code>phi_1</code> for [mathjaxinline]\phi _{1}[/mathjaxinline], <code>phi_2</code> for [mathjaxinline]\phi _{2}[/mathjaxinline], <code>phi_3</code> for [mathjaxinline]\phi _{3}[/mathjaxinline], and <code>t</code>. </p>
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<p style="display:inline">[mathjaxinline]x_{1} =[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">L8Q2: Coupled Pendula [WITH SIMULATION]</h2>
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Coupled Pendula with Initial Conditions
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<p>
Consider the system of two coupled pendula of length [mathjaxinline]\ell[/mathjaxinline], attached by a spring with spring constant [mathjaxinline]k[/mathjaxinline], which we have been investigating during this lesson. Recall that the system has two normal modes, and the general motion of each oscillator is a superposition of those modes. We denote the motion of the left oscillator at [mathjaxinline]x_{1}(t)[/mathjaxinline] and the motion of the right oscillator as [mathjaxinline]x_{2}(t)[/mathjaxinline]: </p>
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<td class="equation" style="width:80%; border:none">[mathjax]x_{1}(t) = \alpha \cos (\omega _{1}t + \phi _{1}) + \beta \cos (\omega _{2}t + \phi _{2})[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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<td class="equation" style="width:80%; border:none">[mathjax]x_{2}(t) = \alpha \cos (\omega _{1}t + \phi _{1}) - \beta \cos (\omega _{2}t + \phi _{2})[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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The parameters [mathjaxinline]\alpha[/mathjaxinline], [mathjaxinline]\beta[/mathjaxinline], [mathjaxinline]\phi _{1}[/mathjaxinline], [mathjaxinline]\phi _{2}[/mathjaxinline] are determined by the initial conditions [mathjaxinline]x_{1}(0)[/mathjaxinline], [mathjaxinline]x_{2}(0)[/mathjaxinline], [mathjaxinline]\dot{x}_{1}(0)[/mathjaxinline], and [mathjaxinline]\dot{x}_{2}(0)[/mathjaxinline]. </p>
<p>
Now, examine the following plots, which represent the motion of each pendulum as a function of time, with different sets of initial conditions. The colors in the plots represent the motion of either the left (blue) or right (red) pendulum. The top two plots show the contribution from each normal mode to the overall motion, which is shown in the bottom plot. </p>
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<p>
Answer the following questions about these motions: </p>
<p><b class="bfseries">(Part a)</b> Let [mathjaxinline]x_{0}[/mathjaxinline] be a displacement from equilibrium in the positive [mathjaxinline]\hat{x}[/mathjaxinline] direction. Which plot was created from the initial conditions [mathjaxinline]x_{1}(0) = x_0[/mathjaxinline], [mathjaxinline]x_{2}(0) = 0[/mathjaxinline], [mathjaxinline]\dot{x}_{1}(0)[/mathjaxinline] = 0, and [mathjaxinline]\dot{x}_{2}(0) = 0[/mathjaxinline]? </p>
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<p><b class="bfseries">(Part b)</b> Which plot was created from the initial conditions [mathjaxinline]x_{1}(0) = x_0[/mathjaxinline], [mathjaxinline]x_{2}(0) = -x_0[/mathjaxinline], [mathjaxinline]\dot{x}_{1}(0)[/mathjaxinline] = 0, and [mathjaxinline]\dot{x}_{2}(0) = 0[/mathjaxinline]? </p>
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<p><b class="bfseries">(Part c)</b> Which plot was created from the initial conditions [mathjaxinline]x_{1}(0) = x_0[/mathjaxinline], [mathjaxinline]x_{2}(0) = x_0[/mathjaxinline], [mathjaxinline]\dot{x}_{1}(0)[/mathjaxinline] = 0, and [mathjaxinline]\dot{x}_{2}(0) = 0[/mathjaxinline]? </p>
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<h2>Run the Interactive Python Visualization that Generated the Plots Above!</h2><p>The widget is run in a Jupyter notebook, accessible through the button below. <b>NOTE: The notebook may take up to 3 mintues to load! Please be patient!</b></p><p><div align="center"><a href="https://mybinder.org/v2/gh/mitx-803/vis/master?filepath=coupled_pendula_2.ipynb" class="btn btn-primary" target="_blank" style="color:#FFFFFF;">ACCESS JUPYTER NOTEBOOK HERE</a></div></p><p><div class="hideshowbox"><h4 onclick="hideshow(this);" style="margin: 0px">How to Run Jupyter Notebooks (expand this section if you need a reminder!)<span class="icon-caret-down toggleimage"/></h4><div class="hideshowcontent"><p><h3>Running Notebooks on an External Server</h3></p><p>To access a simulation, click the "ACCESS JUPYTER NOTEBOOK HERE" button. This will bring you to a loading page, hosted by <i class="itshape">mybinder.org</i> (the loading time is anywhere from 20 seconds to 3 minutes). The Jupyter notebooks are run externally to the course, on a server which runs an instance of Python. There is no need to install Python or related dependencies!</p><div align="center"><iframe src="https://mitx-803.github.io/gifs/python_06.html" width="720" height="590" scrolling="no" frameborder="0"/></div><p><h3>Initializing the Program</h3></p><p>Once loaded, you will see a Jupyter notebook in your browser! You will have to click a button to initialize the program. The button is indicated in the instructions within the notebook, and also shown below.</p><div align="center"><img width="700" src="/assets/courseware/v1/3f6c044fc06f79d82bb2e8a97f7dd11a/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_binder_initialize_button.png"/></div><p/><div align="center"><iframe src="https://mitx-803.github.io/gifs/python_07.html" width="720" height="602" scrolling="no" frameborder="0"/></div><p><h3>Instructions and Source Code</h3></p><p>Each notebook has self-contained instructions on how to use the Python simulation. Additionally, you may toggle the button at the bottom of the notebook to view/augment the source code.</p><div align="center"><iframe src="https://mitx-803.github.io/gifs/python_08.html" width="720" height="608" scrolling="no" frameborder="0"/></div><p><h3>Saving/Running Notebooks Locally</h3></p><p>Finally, you can dowload each notebook to run locally. Additionally, you can visit the git repository to download all notebooks in the course. In order to run notebooks locally, you must install Python and its dependencies. We cannot help with this process, but we encourage you to look at the resources below, if you are interested.</p><div align="center"><iframe src="https://mitx-803.github.io/gifs/python_09.html" width="720" height="609" scrolling="no" frameborder="0"/></div><p><h3>External Links</h3><br/>   [mathjaxinline]\bullet[/mathjaxinline]  git repository: <a href="https://github.com/mitx-803/vis" target="blank">github.com/mitx-803/vis</a><br/>   [mathjaxinline]\bullet[/mathjaxinline]  information on Jupyter notebooks: <a href="https://jupyter.org/" target="blank">Jupyter Notebooks</a><br/>   [mathjaxinline]\bullet[/mathjaxinline]  information on installing Python through Anaconda: <a href="https://www.anaconda.com/distribution/" target="blank">Anaconda</a><br/>   [mathjaxinline]\bullet[/mathjaxinline]  information on the Binder community: <a href="https://mybinder.readthedocs.io/en/latest/" target="blank">Binder</a><br/></p><p><h3>Dependencies</h3></p><p>
The visualizations run on Python 3. Dependencies for running Python code locally (not through Binder) are stated in the git repository, and include (but are not limited to) the following:
<br/>   [mathjaxinline]\bullet[/mathjaxinline]  scipy
<br/>   [mathjaxinline]\bullet[/mathjaxinline]  numpy
<br/>   [mathjaxinline]\bullet[/mathjaxinline]  ipywidgets
<br/>   [mathjaxinline]\bullet[/mathjaxinline]  nbinteract
<br/>   [mathjaxinline]\bullet[/mathjaxinline]  matplotlib
<br/>   [mathjaxinline]\bullet[/mathjaxinline]  pandas
<br/>   [mathjaxinline]\bullet[/mathjaxinline]  IPython
<br/>   [mathjaxinline]\bullet[/mathjaxinline]  ffmpeg
<br/>   [mathjaxinline]\bullet[/mathjaxinline]  jupyter-contrib-nbextensions
<br/>
</p><p>
You will have to find resources that explain how to install these appropriately for your system, if they are not already installed with your Python package.
</p></div><p class="hideshowbottom" onclick="hideshow(this);" style="margin: 0px"><a href="javascript: {return false;}">Show</a></p></div></p><SCRIPT src="/assets/courseware/v1/631e447105fca1b243137b21b9ed6f90/asset-v1:MITx+8.03x+1T2020+type@asset+block/latex2edx.js" type="text/javascript"/><LINK href="/assets/courseware/v1/daf81af0af57b85a105e0ed27b7873a0/asset-v1:MITx+8.03x+1T2020+type@asset+block/latex2edx.css" rel="stylesheet" type="text/css"/><h2>What You Should See</h2><p>When the notebook is initialized, you will see the following visualization. Follow question prompts within the notebook.</p><div align="center"><img width="800" src="/assets/courseware/v1/74d4d19130256049aedfa1dd226fb7c8/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_coupled_pendula_2.png"/></div><p/>
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Coupled Oscillator Properties
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These questions refer to the general properties of a system of two coupled oscillators, but you may consider the specific system that we have been investigating, two coupled pendula of length [mathjaxinline]\ell[/mathjaxinline], attached by a spring with spring constant [mathjaxinline]k[/mathjaxinline], if it is helpful. </p>
<p><b class="bfseries">(Part a)</b> When the difference between two normal mode frequencies is <i class="itshape">very small</i>, the coupling of the two oscillators must be: </p>
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<text> b) very weak</text>
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<text> c) the normal mode frequencies do not depend on the strength of the coupling</text>
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<p><b class="bfseries">(Part b)</b> Consider a system of two coupled oscillators that undergoes motion which is a superposition of two normal modes&#8212;that is, a system that is NOT oscillating purely in a single normal mode. When the difference between the two normal mode frequencies is <i class="itshape">very small</i>, the rate of energy transfer between the two oscillators of the system is: </p>
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<text> b) very slow</text>
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<text> c) the rate of energy transfer is not related to the difference in the normal mode frequencies</text>
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<h2 class="hd hd-2 unit-title">L8v7: Beat Frequency</h2>
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<h2 class="hd hd-2 unit-title">L8Q4: Identifying the Beat Frequency</h2>
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Beat Frequency
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Let's return to the system of two coupled masses, attached to springs, constrained to move vertically like pistons. The masses are attached to each other by a massless string, which provides approximately constant tension with magnitude [mathjaxinline]T=\kappa L[/mathjaxinline] in the direction of the string. To simplify things, let [mathjaxinline]m_{1} = m_{2} = m[/mathjaxinline] and [mathjaxinline]k_{1} = k_{2} = k[/mathjaxinline]. </p>
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In the limit where the coupling is very small compared to the spring constant ([mathjaxinline]\kappa &lt;&lt; k[/mathjaxinline]), what is the beat frequency of the system, [mathjaxinline]\Delta \omega = |\omega _{+} - \omega _{-}|[/mathjaxinline]? HINT: you may use the convenient approximation [mathjaxinline](1+x)^{n} \approx (1+nx)[/mathjaxinline] for [mathjaxinline]x \ll 1[/mathjaxinline]. Express your answer to first order in the variable [mathjaxinline]\kappa[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]\Delta \omega =[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">L8v10: Beat Phenomena [DEMO]</h2>
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<h3 class="hd hd-2">L8v10: Beat Phenomena [DEMO]</h3>
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