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<h2 class="hd hd-2 unit-title">Pendulum with Sail</h2>
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Pendulum with Sail: Acceleration
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Consider a simple pendulum consisting of a point-like mass [mathjaxinline]m[/mathjaxinline] attached to a massless string of length [mathjaxinline]L[/mathjaxinline] hanging from a fixed support and constrained to move in a vertical plane. Assume gravitational acceleration to be [mathjaxinline]g[/mathjaxinline]. </p>
<p><b class="bfseries">(Part a)</b> Find the approximate equation of motion of the pendulum in terms of the angle [mathjaxinline]\theta[/mathjaxinline] (the pendulum's deviation from the vertical) when [mathjaxinline]\theta[/mathjaxinline] is small and solve for the angular acceleration [mathjaxinline]\alpha =\ddot{\theta }[/mathjaxinline]. </p>
<p>
Write your answer in terms of <code>g</code>, <code>m</code>, <code>L</code>, and <code>theta</code> for [mathjaxinline]\theta[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]\ddot{\theta } \approx[/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>3.14</code>, <code>.98</code></td>
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<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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<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>
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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>
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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>
</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>
</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>
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<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>
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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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Pendulum with Sail: Velocity
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<p><b class="bfseries">(Part b)</b> Assuming that the pendulum is released from rest at a small initial angle [mathjaxinline]\theta _0[/mathjaxinline], what is its angular speed [mathjaxinline]|\dot{\theta }|[/mathjaxinline] when it returns to the equilibrium point? Hint: consider the solution to the equation of motion that you previsouly derived. </p>
<p>
Write your answer in terms of <code>g</code>, <code>m</code>, <code>L</code>, and <code>theta_0</code> for [mathjaxinline]\theta _0[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]|\dot{\theta }|=[/mathjaxinline]</p>
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<td class="formulainput">integers</td>
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<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>
<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>
<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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Damped Pendulum with Sail
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<p><b class="bfseries">(Part c)</b> Now assume that the pendulum is at its equilibrium point and has the speed found in Part (b). At that instant, a sail pops out of the small mass, creating a torque due to air resistance which depends on the angular velocity: </p>
<table id="a0000000011" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]\tau _{air}=-RL\dot{\theta }[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
Solve the equation of motion for the angular acceleration under this new condition (and again assuming that [mathjaxinline]\theta[/mathjaxinline] is small). </p>
<p>
Write your answer in terms of <code>g</code>, <code>m</code>, <code>L</code>, <code>R</code>, <code>theta</code> for [mathjaxinline]\theta[/mathjaxinline], and <code>dottheta</code> for [mathjaxinline]\dot{\theta }[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]\ddot{\theta } \approx[/mathjaxinline]</p>
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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"><code>arcsin, arccos, arctan</code>, etc.</td>
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<td class="formulainput">enter <code>cosh(4*x+y) </code> for [mathjaxinline]\cosh(4x+y) [/mathjaxinline]</td>
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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">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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Pendulum with Sail: Critical Damping
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<p><b class="bfseries">(Part d)</b> Find the value of [mathjaxinline]R[/mathjaxinline] for which the pendulum with the sail is critically damped. </p>
<p>
Write your answer in terms of <code>g</code>, <code>m</code>, <code>L</code>, <code>theta</code> for [mathjaxinline]\theta[/mathjaxinline], and <code>dottheta</code> for [mathjaxinline]\dot{\theta }[/mathjaxinline]. <p style="display:inline">[mathjaxinline]R=[/mathjaxinline]</p> <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_review_02_02d_2_1" class="text-input-dynamath capa_inputtype inline textline">
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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>
</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>
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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>
</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>
<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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<h3 class="hd hd-3 problem-header" id="review_02_02e-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@review_02_02e-problem-progress" tabindex="-1">
Critically Damped Pendulum with Sail
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<p><b class="bfseries">(Part e)</b> If it is critically damped as described in Part (d), what is the maximum angle [mathjaxinline]\theta _{max}[/mathjaxinline] that the pendulum will reach as it swings past the equilibrium point with the sail popped out? Use the initial conditions mentioned previously. </p>
<p>
Write your answer in terms of <code>g</code>, <code>m</code>, <code>L</code>, <code>e</code>, and <code>theta_0</code> for [mathjaxinline]\theta _0[/mathjaxinline]. <p style="display:inline">[mathjaxinline]\theta _{max}=[/mathjaxinline]</p> <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_review_02_02e_2_1" class="text-input-dynamath capa_inputtype inline textline">
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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> (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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<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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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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<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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<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>
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<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>
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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>
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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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Driven Critically Damped Pendulum
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<p><b class="bfseries">(Part f)</b> Consider the same critically damped pendulum as described in Part (d) but now with a driving torque given by [mathjaxinline]\Delta L \sin (\omega _ d t)[/mathjaxinline], where [mathjaxinline]\omega _ d=\omega _0[/mathjaxinline]. What amplitude of driving force [mathjaxinline]\Delta[/mathjaxinline] is required for the steady state oscillation to have a maximum angle [mathjaxinline]\theta _{max}=\theta _0[/mathjaxinline]? </p>
<p>
Write your answer in terms of <code>g</code>, <code>m</code>, <code>L</code>, <code>e</code>, and <code>theta_0</code> for [mathjaxinline]\theta _0[/mathjaxinline]. <p style="display:inline">[mathjaxinline]\Delta =[/mathjaxinline]</p> <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_review_02_02f_2_1" class="text-input-dynamath capa_inputtype inline textline">
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<code>2520</code>
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<td class="formulainput">
<code>2/3</code>
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<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>
<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>
<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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<p><h2>REVIEW OF KIRCHHOFF'S LAWS FOR RESISTOR, INDUCTORS, AND CAPACITORS</h2></p><p>
Consider the simple RLC circuit shown below, with resistor [mathjaxinline]R[/mathjaxinline], inductor [mathjaxinline]L[/mathjaxinline], and capacitor [mathjaxinline]C[/mathjaxinline], all in series. The direction of current flow is shown, moving from the positive plate of the capacitor in a clockwise direction around the circuit. </p><center><img src="/assets/courseware/v1/cde1c95870c53250c20c3762dffe5e78/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_pset_02_01_fig_1.svg" width="385"/></center><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/><p>
We will briefly review Kirchoff's laws for RLC circuits, which states that the sum of voltages over a complete loop is zero: </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]\sum _{\mathrm{loop}}V_{i}=0[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p><h3>Choosing a direction</h3></p><p>
When applying Kirchhoff's laws, one must choose two directions: (1) the direction of current in the loop (if not given), and (2) the direction to sum voltages around the loop. This will affect the sign of each voltage component, which we explain next. </p><p><h3>Resistors</h3> For an <b class="bfseries">resistor</b>, the real physical voltage [mathjaxinline]V_{R}(t)[/mathjaxinline] and the real physical current [mathjaxinline]I(t)[/mathjaxinline] are related by: </p><table id="a0000000003" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000004"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle V_{R}(t) = R\; I(t)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"><span>(<span>1</span>)</span></td></tr></table><p>
The voltage "drops" over a resistor in the direction of current, therefore we would SUBTRACT the term [mathjaxinline]R\; I(t)[/mathjaxinline] when applying Kirchhoff's law. </p><p><h3>Inductors</h3> For an <b class="bfseries">inductor</b>, the real physical voltage [mathjaxinline]V_{L}(t)[/mathjaxinline] and the real physical current [mathjaxinline]I(t)[/mathjaxinline] are related by: </p><table id="a0000000005" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000006"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle V_{L}(t) = L\dfrac {dI(t)}{dt}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"><span>(<span>2</span>)</span></td></tr></table><p>
The voltage "drops" over an inductor in the direction of current, therefore we would SUBTRACT the term [mathjaxinline]L\dfrac {dI(t)}{dt}[/mathjaxinline] when applying Kirchhoff's law. </p><p><h3>Capacitors</h3> Finally, for a <b class="bfseries">capacitor</b>, the real physical voltage [mathjaxinline]V_{C}(t)[/mathjaxinline] and the real physical charge [mathjaxinline]Q(t)[/mathjaxinline] are related by: </p><table id="a0000000007" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000008"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle V_{C}(t)=\dfrac {Q}{C}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"><span>(<span>3</span>)</span></td></tr></table><p>
When "traveling" around a circuit from the negative plate to the positive plate of a capacitor, we would ADD the term [mathjaxinline]\dfrac {Q}{C}[/mathjaxinline] when applying Kirchhoff's law. </p><p>
For a capacitor, we ALSO, have to consider the direction of current flow in order to find the relation between [mathjaxinline]Q(t)[/mathjaxinline] and [mathjaxinline]I(t)[/mathjaxinline]. </p><ul class="itemize"><li><p>
If positive current is FLOWING INTO the positive plate, the charge is increasing and has a positive time derivative, [mathjaxinline]I=\frac{dQ}{dt}[/mathjaxinline].<br/></p></li><li><p>
If positive current is LEAVING the positive plate, the charge is decreasing and so [mathjaxinline]I = -\frac{dQ}{dt}[/mathjaxinline]. </p></li></ul>
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Review of RLC Circuit
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<p><b class="bfseries">(Part a)</b> Express Kirchhoff's law around the loop, going around in a clockwise manner. Write your expression by isolating [mathjaxinline]\frac{Q}{C}[/mathjaxinline] on one side of the equation. </p>
<p>
Write your answer in terms of <code>Q</code>, <code>R</code>, <code>L</code>, <code>C</code>, <code>I</code>, and <code>dotI</code> for [mathjaxinline]\dot{I}[/mathjaxinline] (recall the notation [mathjaxinline]\dot{I}\equiv \frac{dI}{dt}[/mathjaxinline]). </p>
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<p style="display:inline">[mathjaxinline]\frac{Q}{C}=[/mathjaxinline]</p>
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<p><b class="bfseries">(Part b)</b> For the circuit shown above, what is the relation between [mathjaxinline]I(t)[/mathjaxinline] and [mathjaxinline]\frac{dQ}{dt}[/mathjaxinline]? </p>
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<text> a) [mathjaxinline]I=\frac{dQ}{dt}[/mathjaxinline]</text>
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<text> b) [mathjaxinline]I=-\frac{dQ}{dt}[/mathjaxinline]</text>
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<p><b class="bfseries">(Part c)</b> Write down the equation of "motion" for [mathjaxinline]Q(t)[/mathjaxinline], the charge stored in the capacitor. Write your answer in terms of <code>dotQ</code> for [mathjaxinline]\dot{Q}[/mathjaxinline],<code>Q</code>, <code>R</code>, <code>L</code>, and <code>C</code> (recall the notation [mathjaxinline]\dot{Q}\equiv \frac{dQ}{dt}[/mathjaxinline]). </p>
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<p style="display:inline">[mathjaxinline]\ddot{Q}=[/mathjaxinline]</p>
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<td class="formulainput">enter (english) name of letter</td>
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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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<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">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">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">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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<h2 class="hd hd-2 unit-title">Driven RLC Circuits: Part I</h2>
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Impedance for a Series RLC Circuit
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An AC source is connected to a circuit element that can be a resistor, an inductor, a capacitor, or a combination of those. Assume that the voltage source is given by [mathjaxinline]V(t) = V_0\cos (\omega t)[/mathjaxinline], where [mathjaxinline]V_0[/mathjaxinline] is the amplitude in volts, and [mathjaxinline]\omega[/mathjaxinline] is the angular frequency in radians/sec. The resulting current in the circuit is [mathjaxinline]I(t) = I_0\cos (\omega t - \phi )[/mathjaxinline], where [mathjaxinline]I_0[/mathjaxinline] is the current amplitude in amperes, and [mathjaxinline]\phi[/mathjaxinline] is the phase shift of the current with respect to the voltage in radians. </p>
<p>
Consider now the complex function of time [mathjaxinline]V_ c(t) = V_0 e^{i\omega t}[/mathjaxinline] and [mathjaxinline]I_ c(t) = I_0 e^{i(\omega t -\phi )}[/mathjaxinline]. These functions, with the subscript "c", are the <b class="bfseries">complexified</b> versions of the physical real functions [mathjaxinline]V(t)[/mathjaxinline] and [mathjaxinline]I(t)[/mathjaxinline] defined above. </p>
<p><b class="bfseries">(Part a)</b> Calculate the real part of [mathjaxinline]V_ c[/mathjaxinline] and [mathjaxinline]I_ c[/mathjaxinline]. Express your answer in terms of <code>t</code>, <code>V_0</code> for [mathjaxinline]V_0[/mathjaxinline], <code>I_0</code> for [mathjaxinline]I_0[/mathjaxinline], <code>omega</code> for [mathjaxinline]\omega[/mathjaxinline], and <code>phi</code> for [mathjaxinline]\phi[/mathjaxinline], as needed. </p>
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<p style="display:inline">[mathjaxinline]\text {Re}(V_ c)=[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]\text {Re}(I_ c)=[/mathjaxinline] </p>
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<p><b class="bfseries">(Part b)</b> Calculate the imaginary part of [mathjaxinline]V_ c[/mathjaxinline] and [mathjaxinline]I_ c[/mathjaxinline]. Express your answer in terms of <code>t</code>, <code>V_0</code> for [mathjaxinline]V_0[/mathjaxinline], <code>I_0</code> for [mathjaxinline]I_0[/mathjaxinline], <code>omega</code> for [mathjaxinline]\omega[/mathjaxinline], and <code>phi</code> for [mathjaxinline]\phi[/mathjaxinline], as needed. Do not include the [mathjaxinline]i[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]\text {Im}(V_ c)=[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]\text {Im}(I_ c)=[/mathjaxinline] </p>
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<code>2520</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>
</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]
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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>
</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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<p><b class="bfseries">(Part c)</b> We now introduce a complex number [mathjaxinline]z[/mathjaxinline] called <b class="bfseries">impedance</b> such that [mathjaxinline]V_ c = z\; I_ c[/mathjaxinline]. If the Euler expression of the impedance is [mathjaxinline]z = |z| \; e^{i\delta }[/mathjaxinline], find [mathjaxinline]|z|[/mathjaxinline], the modulus of [mathjaxinline]z[/mathjaxinline], and [mathjaxinline]\delta[/mathjaxinline], the phase of [mathjaxinline]z[/mathjaxinline] in terms of <code>V_0</code> for [mathjaxinline]V_0[/mathjaxinline], <code>I_0</code> for [mathjaxinline]I_0[/mathjaxinline], <code>omega</code> for [mathjaxinline]\omega[/mathjaxinline], and <code>phi</code> for [mathjaxinline]\phi[/mathjaxinline], as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]|z|=[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]\delta =[/mathjaxinline] </p>
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<code>2520</code>
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<tr class="formulainput">
<td class="formulainput">fractions</td>
<td class="formulainput">
<code>2/3</code>
</td>
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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>
<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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<p><h2>REVIEW OF IMPEDANCE FOR RESISTOR, INDUCTORS, AND CAPACITORS</h2></p><p>
For an AC circuit, we can write the voltage source as [mathjaxinline]V(t) = V_0\cos (\omega t)[/mathjaxinline] and the current in the circuit is [mathjaxinline]I(t) = I_0\cos (\omega t - \phi )[/mathjaxinline]. </p><p><h3>Resistors</h3> For a <b class="bfseries">resistor</b>, the real physical voltage [mathjaxinline]V(t)[/mathjaxinline] and the real physical current [mathjaxinline]I(t)[/mathjaxinline] are related 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"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle V(t) = R\; I(t)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"><span>(<span>1</span>)</span></td></tr></table><p>
The complexified voltage and current, [mathjaxinline]V_ c[/mathjaxinline] and [mathjaxinline]I_ c[/mathjaxinline] satisfy: </p><table id="a0000000004" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000005"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle V_ c(t) = R\; I_ c(t).[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"><span>(<span>2</span>)</span></td></tr></table><p>
From this we conclude that the impedance is [mathjaxinline]z = |z|\; e^{i\delta } = R[/mathjaxinline]. </p><p><h3>Inductors</h3> For an <b class="bfseries">inductor</b>, the real physical voltage [mathjaxinline]V(t)[/mathjaxinline] and the real physical current [mathjaxinline]I(t)[/mathjaxinline] are related by: </p><table id="a0000000006" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000007"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle V(t) = L\dfrac {dI(t)}{dt}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"><span>(<span>3</span>)</span></td></tr></table><p>
and the complexified voltage and current, [mathjaxinline]V_ c[/mathjaxinline] and [mathjaxinline]I_ c[/mathjaxinline] satisfy: </p><table id="a0000000008" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000009"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle V_ c(t)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = L\dfrac {dI_ c(t)}{dt}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"><span>(<span>4</span>)</span></td></tr><tr id="a0000000010"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =L\left(i\omega I_0 e^{i(\omega t -\phi )}\right)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"><span>(<span>5</span>)</span></td></tr><tr id="a0000000011"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle =\left(i\omega L \right)I_0e^{i(\omega t -\phi )}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"><span>(<span>6</span>)</span></td></tr></table><p>
From this we can see that the impedance is [mathjaxinline]z = |z|\; e^{i\delta } = i \omega L = \omega L\; e^{i \pi /2}[/mathjaxinline], where we have used the equality [mathjaxinline]i=e^{i \pi /2}[/mathjaxinline] </p><p><h3>Capacitors</h3> Finally, for a <b class="bfseries">capacitor</b>, we need to first think about the sign of [mathjaxinline]\frac{dQ}{dt}[/mathjaxinline]. If positive current is flowing into the positive plate, the charge is increasing and has a positive time derivative, [mathjaxinline]I=\frac{dQ}{dt}[/mathjaxinline]. If positive current is leaving the positive plate, the charge is decreasing and so [mathjaxinline]I = -\frac{dQ}{dt}[/mathjaxinline]. </p><p>
The voltage across the capacitor is [mathjaxinline]V(t)=\dfrac {Q}{C}[/mathjaxinline], and the time derivative of this relationship is given by: </p><table id="a0000000012" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000013"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \dfrac {dV(t)}{dt} = \dfrac {1}{C}\frac{dQ}{dt} = \dfrac {1}{C}I(t)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"><span>(<span>8</span>)</span></td></tr></table><p>
where [mathjaxinline]V(t)[/mathjaxinline] and [mathjaxinline]I(t)[/mathjaxinline] are the real physical voltage and current. The complexified voltage and current, [mathjaxinline]V_ c[/mathjaxinline] and [mathjaxinline]I_ c[/mathjaxinline] satisfy: </p><table id="a0000000014" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000015"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \dfrac {dV_ c(t)}{dt}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \dfrac {1}{C}I_ c(t)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"><span>(<span>9</span>)</span></td></tr><tr id="a0000000016"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle i \omega V_{0} e^{i\omega t}[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \dfrac {1}{C}I_ c(t)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"><span>(<span>10</span>)</span></td></tr><tr id="a0000000017"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle i \omega V_{c}(t)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \dfrac {1}{C}I_ c(t)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"><span>(<span>11</span>)</span></td></tr><tr id="a0000000018"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle V_ c(t)[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = \dfrac {1}{i\omega C}I_ c(t)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"><span>(<span>12</span>)</span></td></tr></table><p>
Here we have that the impedance is [mathjaxinline]z = |z|\; e^{i\delta } = \frac{1}{i \omega C}= \left(\frac{1}{\omega C}\right)\; e^{-i \pi /2}[/mathjaxinline], using [mathjaxinline]\frac{1}{i}=-i=e^{-i \pi /2}[/mathjaxinline]. </p>
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Impedance for a Series RLC Circuit
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An AC source is connected to a resistor of resistance [mathjaxinline]R[/mathjaxinline], an inductor of inductance [mathjaxinline]L[/mathjaxinline] , and a capacitor of capacitance [mathjaxinline]C[/mathjaxinline] as shown. The voltage source is given by [mathjaxinline]V(t) = V_0\sin (\omega t)[/mathjaxinline] and the current in the circuit is [mathjaxinline]I(t) = I_0\sin (\omega t - \phi )[/mathjaxinline]. Recall that for a series circuit, complex impedances add&#8212;the "equivalent" impedance is the sum of all the impedances (just like the equivalent resistance of resistors in series is the sum of the individual resistances). </p>
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Calculate the modulus [mathjaxinline]|z_{eq}|[/mathjaxinline], and the phase [mathjaxinline]\delta[/mathjaxinline] of the equivalent impedance [mathjaxinline]z_{eq}[/mathjaxinline]. Express your answer in terms of <code>R</code>, <code>L</code>, <code>C</code>, <code>arctan</code> for [mathjaxinline]\tan ^{-1}[/mathjaxinline], and <code>omega</code> for [mathjaxinline]\omega[/mathjaxinline] as needed. </p>
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<p style="display:inline">[mathjaxinline]|z_{eq}|[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]\delta =[/mathjaxinline] </p>
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