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<h2 class="hd hd-2 unit-title">Further Exploring Simple Harmonic Motion</h2>
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Now that we understand the basics of simple harmonic motion, let's take it farther! </p><p>
The first piece we will look at is writing the solution to the oscillatory motion using complex numbers. While this initially looks more complicated, we will see throughout this course that it will end up making many calculations simpler and provides interesting insights into what is happening. We will see that the simple harmonic motion is like circular motion in the complex plane, but we only see the real dimension. </p><p>
Next, we will look at energy conservation in this ideal system where there are no drag forces yet involved. All the equations that we have derived so far could also have been found using energy conservation instead. </p><p>
Finally, during this lesson we will look at a physical pendulum as an analogy of the simple spring and mass system. </p>
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<h2 class="hd hd-2 unit-title">L2v1: Complex Notation to Describe Simple Harmonic Motion</h2>
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<h2 class="hd hd-2 unit-title">L2v2: Properties of Complex Numbers</h2>
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<h2 class="hd hd-2 unit-title">Review of Complex Numbers</h2>
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<p><h2>DEFINITIONS</h2></p><p>
We will be working with both real representations and complex notation. Here is a brief review: </p><ol class="enumerate"><li value="1"><p>
A complex number, [mathjaxinline]z[/mathjaxinline], has a real and imaginary part. We can define [mathjaxinline]x = \operatorname {Re}(z)[/mathjaxinline] and [mathjaxinline]y=\operatorname {Im}(z)[/mathjaxinline], such that </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]z=x + iy[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>1</span>)</span></td></tr></table><p>
where [mathjaxinline]i \equiv \sqrt {-1}[/mathjaxinline]. </p></li><li value="2"><p>
A complex number [mathjaxinline]z=x + iy[/mathjaxinline] may be represented by a magnitude and phase: </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]z = |z| e^{i\phi }[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>2</span>)</span></td></tr></table><p>
where [mathjaxinline]|z| = \sqrt {x^2 + y^2}[/mathjaxinline] and [mathjaxinline]\phi =\arctan {y/x}[/mathjaxinline]. </p></li><li value="3"><p>
Using what is called the “Euler Formula" [mathjaxinline]e^{i\phi }=\cos (\phi )+i\sin (\phi )[/mathjaxinline], the real and imaginary parts are: </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]\operatorname {Re}(z) = |z|\cos (\phi )[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>3</span>)</span></td></tr></table><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]\operatorname {Im}(z) = |z|\sin (\phi )[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>4</span>)</span></td></tr></table></li><li value="4"><p>
If [mathjaxinline]z = |z| e^{i\phi }[/mathjaxinline] is a complex number, then its “complex conjugate" is: </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]\tilde{z} = |z| e^{-i\phi }[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>5</span>)</span></td></tr></table></li><li value="5"><p>
The magnitude of a complex number is also defined in terms of its complex conjugate: </p><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]|z|=(z\tilde{z})^{1/2}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>6</span>)</span></td></tr></table></li><li value="6"><p>
The real and imaginary parts of a complex number can also be expressed in terms of its complex conjugate: </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]\operatorname {Re}(z) = \frac{1}{2}(z + \tilde{z})[/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]\operatorname {Im}(z) = -\frac{i}{2}(z - \tilde{z})[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>8</span>)</span></td></tr></table></li></ol><p><h2>COMMON OPERATIONS</h2></p><p>
Now consider the complex harmonic function that we will typically be working with: </p><table id="a0000000010" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]z = A e^{i\omega t}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>9</span>)</span></td></tr></table><p>
Here are some relevant operations for this type of function: </p><ol class="enumerate"><li value="1"><p>
derivative </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]\frac{d}{dt} e^{i\omega t} = i\omega e^{i\omega t}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>10</span>)</span></td></tr></table></li><li value="2"><p>
multiplication </p><table id="a0000000012" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]e^{i\omega _{1} t} \cdot e^{i\omega _{2} t} = e^{i(\omega _{1}+\omega _{2})t}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>11</span>)</span></td></tr></table></li><li value="3"><p>
time translation (let [mathjaxinline]t \rightarrow t + a[/mathjaxinline]) </p><table id="a0000000013" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]e^{i\omega t} \rightarrow e^{i\omega (t + a)}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>12</span>)</span></td></tr></table></li></ol><p>
As a side note, writing the time dependence in the complex form: </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 z[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle = A e^{i\omega t}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000016"><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 =A\cos (\omega t)+iA\sin (\omega t)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr></table><p>
gives an interpretation of the "angle" that is represented by the quantity [mathjaxinline]\omega t[/mathjaxinline]. Written this way, [mathjaxinline]z[/mathjaxinline] can be seen to be an imaginary number that is following a circular trajectory in 2-dimensional imaginary space, where the real part is [mathjaxinline]x[/mathjaxinline] and the imaginary part is [mathjaxinline]y[/mathjaxinline]. In this representation, [mathjaxinline]\omega t[/mathjaxinline] is the angle in imaginary space that a line from the origin to [mathjaxinline]z[/mathjaxinline] makes with the [mathjaxinline]x[/mathjaxinline] (real) axis. </p>
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<h2 class="hd hd-2 unit-title">L2Q1: Complex Numbers</h2>
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Review of Complex Numbers
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<p>
A complex number [mathjaxinline]z=x + iy[/mathjaxinline] may be represented by a magnitude and phase: [mathjaxinline]z = |z| e^{i\phi }[/mathjaxinline], where [mathjaxinline]|z|=\sqrt {z\tilde{z}}[/mathjaxinline]. </p>
<p>
Which of the following representations correctly describes the inverse of a complex number, [mathjaxinline]\dfrac {1}{z}[/mathjaxinline]? </p>
<p>
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<input type="radio" name="input_lect_02_02_2_1" id="input_lect_02_02_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="lect_02_02_2_1-choice_1-label" for="input_lect_02_02_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lect_02_02_2_1">
<text> a) [mathjaxinline]\dfrac {1}{z}=|z| e^{+i\phi }[/mathjaxinline]</text>
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<input type="radio" name="input_lect_02_02_2_1" id="input_lect_02_02_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="lect_02_02_2_1-choice_2-label" for="input_lect_02_02_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_02_02_2_1">
<text> b) [mathjaxinline]\dfrac {1}{z}=|z| e^{-i\phi }[/mathjaxinline]</text>
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<input type="radio" name="input_lect_02_02_2_1" id="input_lect_02_02_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="lect_02_02_2_1-choice_3-label" for="input_lect_02_02_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lect_02_02_2_1">
<text> c) [mathjaxinline]\dfrac {1}{z}=\dfrac {1}{|z|} e^{+i\phi }[/mathjaxinline]</text>
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<text> d) [mathjaxinline]\dfrac {1}{z}=\dfrac {1}{|z|} e^{-i\phi }[/mathjaxinline]</text>
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More Review of Complex Numbers
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<p><b class="bfseries">(Part a)</b> Find the real and imaginary parts of the complex number [mathjaxinline]z_1 = -4 e^{-i 5\pi /6}[/mathjaxinline]. Your answer may include <code>pi</code> for [mathjaxinline]\pi[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]\text {Re}(z_1)=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\text {Im}(z_1)=[/mathjaxinline]</p>
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<p><b class="bfseries">(Part b)</b> Express [mathjaxinline]z_2= 2+2i[/mathjaxinline] in exponential notation using the Euler Formula. Your answer may include <code>e</code>, <code>i</code>, and <code>pi</code> for [mathjaxinline]\pi[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]z_2 =[/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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<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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<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">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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Total Energy of Simple Harmonic Oscillator
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<p><b class="bfseries">(Part a)</b> The time-dependent kinetic energy and potential energy of a simple harmonic oscillator were given in class as [mathjaxinline]K=\frac{1}{2} m \dot{x}^2[/mathjaxinline] and [mathjaxinline]U=\frac{1}{2} k x^2[/mathjaxinline], where [mathjaxinline]x(t)=A\cos (\omega _0 t + \phi )[/mathjaxinline] and [mathjaxinline]\omega _0 = \sqrt {k/m}[/mathjaxinline]. </p>
<p>
Find an expression for the total energy of the system using the variables, <code>A</code>, <code>omega_0</code> for [mathjaxinline]\omega _0[/mathjaxinline], and <code>k</code>. </p>
<p>
<p style="display:inline">[mathjaxinline]E_{\mathrm{tot}} =[/mathjaxinline] </p>
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<h3 class="hd hd-3 problem-header" id="lect_02_03b-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_02_03b-problem-progress" tabindex="-1">
Changing Energy of Simple Harmonic Oscillator
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<p><b class="bfseries">(Part b)</b> A block of mass, [mathjaxinline]m[/mathjaxinline], is on a spring with spring constant, [mathjaxinline]k[/mathjaxinline], oscillating horizontally back and forth around the center equilibrium point. At some point while the mass oscillates, we add a second block, also of mass [mathjaxinline]m[/mathjaxinline], on top of the first so that the new object oscillating has a total mass of [mathjaxinline]2m[/mathjaxinline]. </p>
<p>
If we add the second block when the spring is at its maximum position (as stretched as it gets, i.e., at a turning point in the motion), how do (i) the total energy [mathjaxinline]E_{2m}[/mathjaxinline] and (ii) the amplitude [mathjaxinline]A_{2m}[/mathjaxinline] compare to their values with just one block [mathjaxinline]E_{m}[/mathjaxinline] and [mathjaxinline]A_{m}[/mathjaxinline]? </p>
<p>
<b class="bfseries">(Part i)</b>
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<text> [mathjaxinline]E_{2m}=E_{m}/2[/mathjaxinline]</text>
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<text> [mathjaxinline]E_{2m} = E_{m}/\sqrt {2}[/mathjaxinline]</text>
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<text> [mathjaxinline]E_{2m} = E_{m}[/mathjaxinline]</text>
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<text> [mathjaxinline]E_{2m} = \sqrt {2}E_{m}[/mathjaxinline]</text>
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<text> [mathjaxinline]E_{2m} = 2E_{m}[/mathjaxinline]</text>
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<b class="bfseries">(Part ii)</b>
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<text> [mathjaxinline]A_{2m} = A_{m}[/mathjaxinline]</text>
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<text> [mathjaxinline]A_{2m} = \sqrt {2}A[/mathjaxinline]</text>
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<text> [mathjaxinline]A_{2m} = 2A_{m}[/mathjaxinline]</text>
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<h3 class="hd hd-3 problem-header" id="lect_02_03c-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_02_03c-problem-progress" tabindex="-1">
Changing Energy of Simple Harmonic Oscillator
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<p><b class="bfseries">(Part c)</b> If we add the second block when the spring is right at the central equilibrium point, how do (i) the total energy [mathjaxinline]E_{2m}[/mathjaxinline] and (ii) the amplitude [mathjaxinline]A_{2m}[/mathjaxinline] compare to their values with just one mass [mathjaxinline]E_{m}[/mathjaxinline] and [mathjaxinline]A_{m}[/mathjaxinline]? Recall from classical mechanics that adding an unmoving object of mass [mathjaxinline]{m}[/mathjaxinline] to a moving object of the same mass causes the velocity to go down by a factor of 2. </p>
<p>
<b class="bfseries">(Part i)</b>
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<text> [mathjaxinline]E_{2m} = E_{m}/2[/mathjaxinline]</text>
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<text> [mathjaxinline]E_{2m} = E_{m}/\sqrt {2}[/mathjaxinline]</text>
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<text> [mathjaxinline]E_{2m} = E_{m}[/mathjaxinline]</text>
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<text> [mathjaxinline]E_{2m} = \sqrt {2}E_{m}[/mathjaxinline]</text>
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<text> [mathjaxinline]E_{2m} = 2E_{m}[/mathjaxinline]</text>
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<b class="bfseries">(Part ii)</b>
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<text> [mathjaxinline]A_{2m} = A_{m}/2[/mathjaxinline]</text>
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<text> [mathjaxinline]A_{2m} = A_{m}/\sqrt {2}[/mathjaxinline]</text>
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<input type="radio" name="input_lect_02_03c_3_1" id="input_lect_02_03c_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="lect_02_03c_3_1-choice_3-label" for="input_lect_02_03c_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lect_02_03c_3_1">
<text> [mathjaxinline]A_{2m} = A_{m}[/mathjaxinline]</text>
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<text> [mathjaxinline]A_{2m} = \sqrt {2}A_{m}[/mathjaxinline]</text>
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<text> [mathjaxinline]A_{2m} = 2A_{m}[/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">Review of Rotational Dynamics</h2>
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<p><h2>TRANSLATIONAL MOTION</h2></p><p>
So far, we have focused primarily on translational motion. For instance, we have examined masses on springs, which oscillate along one axis (in the Cartesian coordinate system). The dynamical variables of translational motion are position [mathjaxinline]\vec{x}[/mathjaxinline], velocity [mathjaxinline]\vec{v}[/mathjaxinline], and acceleration [mathjaxinline]\vec{a}[/mathjaxinline]. </p><p>
Let's briefly state the relations between the dynamical variables for translation motion: </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]\vec{v}=\frac{d}{dt}\vec{x}[/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]\vec{a}=\frac{d}{dt}\vec{v}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>2</span>)</span></td></tr></table><p>
Newton's Second Law for translational motion relates linear acceleration to Force [mathjaxinline]\vec{F}[/mathjaxinline]: </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]\sum \vec{F}=m\vec{a}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>3</span>)</span></td></tr></table><p><h2>ROTATIONAL MOTION</h2></p><p>
For rotational motion, the dynamical variables are angular position [mathjaxinline]\theta[/mathjaxinline], angular velocity [mathjaxinline]\vec{\omega }[/mathjaxinline], and angular acceleration [mathjaxinline]\vec{\alpha }[/mathjaxinline]. Furthermore, the rotational analogue of mass is "moment of inertia," [mathjaxinline]I[/mathjaxinline]. </p><p>
Note, in particular, that it will be necessary to carefully distinguish the [mathjaxinline]\omega[/mathjaxinline] that denotes angular velocity from the [mathjaxinline]\omega[/mathjaxinline] that we have been using up to now to denote the frequency of a simple harmonic oscillator. To avoid this possible confusion, the oscillator frequency will often now be written [mathjaxinline]\omega _0[/mathjaxinline]. The "0" in this case has nothing to do with initial conditions. </p><p>
The dynamical relations between these variables are: </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]\vec{\omega }=\frac{\vec{d\theta }}{dt}[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>4</span>)</span></td></tr></table><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]\vec{\alpha }=\frac{d}{dt}\vec{\omega }[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>5</span>)</span></td></tr></table><p>
And Newton's Second Law for rotational motion relates angular acceleration to Torque [mathjaxinline]\vec{\tau }[/mathjaxinline]: </p><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]\sum \vec{\tau } = I \vec{\alpha }[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>6</span>)</span></td></tr></table><p>
In the following examples, we will look at systems that undergo rotational motion. </p><p>
Note that the angle [mathjaxinline]\theta[/mathjaxinline] is often written without a vector sign. However, in considering rotational dynamics, it is as important to consider the direction of an <i class="it">angular</i> displacement (clockwise versus counterclockwise) as it was to consider the direction of a <i class="it">linear</i> displacement (for example whether a spring was stretched or compressed). </p>
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<h2 class="hd hd-2 unit-title">L2v4: Physical Pendulum</h2>
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<h2 class="hd hd-2 unit-title">L2Q3: Physical Pendulum Initial Conditions</h2>
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Physical Pendulum Initial Conditions
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Consider the solution to the equation of motion of a vertical rod, oscillating around a pivot at one end of the rod: </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]\theta (t) = A\cos (\omega _{0}t + \phi )[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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<p>
where [mathjaxinline]\displaystyle \omega _{0}=\sqrt {\frac{3g}{2l}}[/mathjaxinline], [mathjaxinline]l[/mathjaxinline] is the length of the rod, and [mathjaxinline]A[/mathjaxinline] and [mathjaxinline]\phi[/mathjaxinline] are determined by initial conditions. </p>
<p>
Consider the following initial conditions: [mathjaxinline]\theta (t=0)=\theta _{0}[/mathjaxinline] and [mathjaxinline]\omega (t=0)=0[/mathjaxinline]. Note, [mathjaxinline]\omega (t)[/mathjaxinline] is the angular velocity [mathjaxinline]\frac{\partial }{\partial t}\theta (t)[/mathjaxinline], which is different from the natural angular frequency [mathjaxinline]\omega _{0}[/mathjaxinline]. </p>
<p>
Solve for [mathjaxinline]\theta (t)[/mathjaxinline] and [mathjaxinline]\omega (t)[/mathjaxinline] when [mathjaxinline]t=\frac{3}{4}T[/mathjaxinline] (recall that [mathjaxinline]T=2\pi /\omega _{0}[/mathjaxinline]). Express your answer in terms of <code>theta_0</code> for [mathjaxinline]\theta _{0}[/mathjaxinline] <code>omega_0</code> for [mathjaxinline]\omega _{0}[/mathjaxinline], and relevant numerical constants including <code>pi</code> for [mathjaxinline]\pi[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]\theta (t=\frac{3}{4}T) =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]\omega (t=\frac{3}{4}T) =[/mathjaxinline] </p>
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<code>2520</code>
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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>
</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]
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</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>
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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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<h2 class="hd hd-2 unit-title">L2Q4: Simple Pendulum on Exoplanet</h2>
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Simple Pendulum on Exoplanet
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<p>
Along with the mass on a spring, Yen-Jie decided to send a simple pendulum to the surface of the exoplanet (HD 100546b) as well in order to be fair. Recall, that the surface gravity of any planet is given as </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]g = \frac{G M}{R^2}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
The radius of the exoplanet is 70 times larger than the Earth and the mass is 7000 times larger than the Earth. What will be the oscillation frequency ratio [mathjaxinline]\omega _{\rm Exoplanet}/\omega _{\rm Earth}[/mathjaxinline], where [mathjaxinline]\omega _{\rm Exoplanet} (\omega _{\rm Earth})[/mathjaxinline] is the oscillation angular frequency of the mass on the Exoplanet (Earth)? </p>
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<p style="display:inline">[mathjaxinline]\omega _{\rm Exoplanet}/\omega _{\rm Earth} =[/mathjaxinline] </p>
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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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<tr class="formulainput">
<td class="formulainput">fractions</td>
<td class="formulainput">
<code>2/3</code>
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<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>
<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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