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<h2 class="hd hd-2 unit-title">Introduction to Infinite Systems</h2>
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We will use properties of symmetry to begin to tackle the interesting problem of systems with many (approaching an infinite number of) coupled oscillators! Although one might assume that the dynamics of an infinite system are infinitely complex, it turns out that they are exactly solvable. </p><p>
We will look at the specific case of an infinite number of masses attached to springs (<i class="itshape">strings</i> will come later). </p>
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<h2 class="hd hd-2 unit-title">L11v1: Equations of Motion for an Infinite System</h2>
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<h3 class="hd hd-2">L11v1: Equations of Motion for an Infinite System</h3>
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<h2 class="hd hd-2 unit-title">L11Q1: Practice with Subscript Notation</h2>
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Understanding Subscript Notation
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For infinite systems, as you can imagine, it becomes quite onerous (in fact, impossible!) to write down all of the components of a system (e.g., [mathjaxinline]x_{1}[/mathjaxinline], [mathjaxinline]x_{2}[/mathjaxinline], [mathjaxinline]x_{3}[/mathjaxinline], [mathjaxinline]\ldots[/mathjaxinline]). Therefore, we use enumerated subscripts like [mathjaxinline]x_{j}[/mathjaxinline] and [mathjaxinline]x_{j+1}[/mathjaxinline]. Let's get some practice! </p>
<p>
Consider the following equation of motion that we saw in the previous video, corresponding to the [mathjaxinline]j^{\mathrm{th}}[/mathjaxinline] mass: </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]m\ddot{x}_{j}=Kx_{j-1} -2Kx_{j} + Kx_{j+1}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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<p><b class="bfseries">(Part i)</b> Write down the equation of motion for the [mathjaxinline]4^{\mathrm{th}}[/mathjaxinline] mass. Express your answer in terms of <code>K</code> and use the notation <code>x_{j}</code> for [mathjaxinline]x_{j}[/mathjaxinline], and similar terms. </p>
<p style="display:inline">[mathjaxinline]m\ddot{x}_{4}=[/mathjaxinline]</p>
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<p><b class="bfseries">(Part ii)</b> Write down the equation of motion for the [mathjaxinline]-10^{\mathrm{th}}[/mathjaxinline] mass (the indices [mathjaxinline]j[/mathjaxinline] must be both positive and negative running from [mathjaxinline]-\infty[/mathjaxinline] to [mathjaxinline]+\infty[/mathjaxinline]). Express your answer in terms of <code>K</code> and use the notation <code>x_{j}</code> for [mathjaxinline]x_{j}[/mathjaxinline], and similar terms. </p>
<p style="display:inline">[mathjaxinline]m\ddot{x}_{-10}=[/mathjaxinline]</p>
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<td class="formulainput">integers</td>
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<code>2520</code>
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<code>2/3</code>
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<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
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<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>
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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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<code>e, pi</code>
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<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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<code>abs, ln, sqrt</code>
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<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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<code>sin, cos, tan, sec, csc, cot</code>
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<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>
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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>
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<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">L11Q2: Transverse Motion and Translational Symmetry</h2>
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Translational Motion and Symmetry
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An infinite system of identical masses attached by identical springs exhibits translational symmetry. In previous videos, we wrote the equations of motion for the masses of the system, when they are constrained to move horizontally. This generates motion in the longitudinal direction. </p>
<p>
But, as was demonstrated in the previous video, this system can also exhibit motion in the transverse direction (up and down)! </p>
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If the infinite system of masses and springs undergoes transverse motion, does it still exhibit translational symmetry? </p>
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<text> a) yes, because any infinite system of identical masses exhibits translational symmetry</text>
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<text> b) no, because up and down motion breaks the translational symmetry</text>
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<text> c) no, because translational symmetry means something can only move side-to-side</text>
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<text> d) yes, because the symmetry matrix commutes with the [mathjaxinline]\textbf{M}^{-1}\textbf{K}[/mathjaxinline] matrix</text>
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<h2 class="hd hd-2 unit-title">L11Q3: Physically Valid Range of Beta</h2>
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Possible Value of Beta I
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<p>
Consider the example worked out in lecture, where we solved the matrix equation: </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{M}^{-1}\textbf{K}\textbf{A}=\omega ^{2}\textbf{A}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">
<span>(<span>1</span>)</span>
</td>
</tr>
</table>
<p>
We found the following expression for the [mathjaxinline]j^{\mathrm{th}}[/mathjaxinline] component of this expression: </p>
<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]-\frac{K}{m}A_{j-1}+\frac{2K}{m}A_{j}-\frac{K}{m}A_{j+1}=\omega ^{2}A_{j}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">
<span>(<span>2</span>)</span>
</td>
</tr>
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<p>
and used [mathjaxinline]\omega _{0}^{2}=\frac{K}{m}[/mathjaxinline], and the eigenvectors of the symmetry matrix, [mathjaxinline]A_{j}=\beta ^{j}[/mathjaxinline], to reduce the expression further: </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]\omega ^{2}=\omega _{0}^{2}\left(-\frac{1}{\beta } + 2 - \beta \right)[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">
<span>(<span>3</span>)</span>
</td>
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<p>
If we knew the value of [mathjaxinline]\beta[/mathjaxinline], we could solve the equation! The value of [mathjaxinline]\beta[/mathjaxinline] will affect ALL of the eigenmodes, since [mathjaxinline]A_{j}=\beta ^{j}[/mathjaxinline], so we must consider what value of [mathjaxinline]\beta[/mathjaxinline] will yield a physical result. </p>
<p><b class="bfseries">(Part a)</b> Let's consider [mathjaxinline]\beta \lt 1[/mathjaxinline]. For instance, let [mathjaxinline]\beta =0.5[/mathjaxinline] and let [mathjaxinline]A_{0}=1[/mathjaxinline]. Calculate the following, entering either a numerical answer or <code>infty</code> for [mathjaxinline]\infty[/mathjaxinline]: </p>
<p>
<p style="display:inline">[mathjaxinline]A_{2} =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]A_{5} =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]A_{\infty } =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]A_{-2} =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]A_{-5} =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]A_{-\infty } =[/mathjaxinline] </p>
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<code>2520</code>
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<code>2/3</code>
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<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>
</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>
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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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</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]
</td>
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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>
</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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Possible Value of Beta II
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<p><b class="bfseries">(Part b)</b> Let's consider [mathjaxinline]\beta \gt 1[/mathjaxinline]. For instance, let [mathjaxinline]\beta =2[/mathjaxinline] and let [mathjaxinline]A_{0}=1[/mathjaxinline]. Calculate the following, entering either a numerical answer or <code>infty</code> for [mathjaxinline]\infty[/mathjaxinline]: </p>
<p>
<p style="display:inline">[mathjaxinline]A_{2} =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]A_{5} =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]A_{\infty } =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]A_{-2} =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]A_{-5} =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]A_{-\infty } =[/mathjaxinline] </p>
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<div class="formulainput">
<table class="formulainput">
<tbody>
<tr class="fiptitle">
<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
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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>
</tr>
<tr class="formulainput">
<td class="formulainput">fractions</td>
<td class="formulainput">
<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>
</tr>
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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>
</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>
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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]
</td>
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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>
<tr class="formulainput">
<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>
</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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Possible Value of Beta III
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<p><b class="bfseries">(Part c)</b> Finally, let's consider [mathjaxinline]\beta = 1[/mathjaxinline], and let [mathjaxinline]A_{0}=1[/mathjaxinline]. Calculate the following, entering either a numerical answer or <code>infty</code> for [mathjaxinline]\infty[/mathjaxinline]: </p>
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<p style="display:inline">[mathjaxinline]A_{2} =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]A_{5} =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]A_{\infty } =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]A_{-2} =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]A_{-5} =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]A_{-\infty } =[/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>
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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>
</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>
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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>
</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]
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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>
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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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<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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What Beta is Physical
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<p><b class="bfseries">(Part d)</b> Some solutions to this problem are not physically valid. Choose ALL statements that describe behavior that is not physically valid: </p>
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<text>c) parts of the system will exhibit asymptotically large amplitude</text>
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<h2 class="hd hd-2 unit-title">L11Q4: Wavenumber</h2>
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Wavenumber
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<p>
We showed that a solution to the infinite oscillator problem with translational symmetry is [mathjaxinline]\beta =e^{ika}[/mathjaxinline] (where [mathjaxinline]a[/mathjaxinline] is the separation of the masses) for which the amplitude of the [mathjaxinline]j^{\mathrm{th}}[/mathjaxinline] mass in the system was found to be [mathjaxinline]A_{j}=\sin (jka)[/mathjaxinline]. An amplitude that varies sinusoidally as a function of position looks like a sort of wave. </p>
<p>
What is the wavelength of this wave, [mathjaxinline]\lambda[/mathjaxinline]? To answer this, it may be helpful to remember that the location of the [mathjaxinline]j^{\mathrm{th}}[/mathjaxinline] mass in the system is [mathjaxinline]x=ja[/mathjaxinline]. Express your answer in term of any of the variables <code>j</code>, <code>k</code>, or <code>a</code>. </p>
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<p style="display:inline">[mathjaxinline]\lambda =[/mathjaxinline] </p>
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<th class="formulainput" scope="col">Descriptions</th>
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<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">integers</td>
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<code>2520</code>
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<td class="formulainput">fractions</td>
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<code>2/3</code>
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<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
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<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>
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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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enter <code>2*pi </code> for [mathjaxinline] 2\pi [/mathjaxinline]
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<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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<code>sin, cos, tan, sec, csc, cot</code>
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<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>
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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>
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<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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