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<h2 class="hd hd-2 unit-title">Introduction to Simple Harmonic Motion</h2>
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<p>
We start this first lesson of 8.03x with the foundation of Vibrations and Waves: Simple Harmonic Motion. </p><p>
In this lesson we will:<br/> [mathjaxinline]\bullet[/mathjaxinline] review how to identify the forces acting on an object <br/> [mathjaxinline]\bullet[/mathjaxinline] draw the free body diagram <br/> [mathjaxinline]\bullet[/mathjaxinline] solve Newton's Second Law: [mathjaxinline]\vec{F}=m\vec{a}[/mathjaxinline] <br/></p><p>
For a simple mass on a spring, Newton's Second Law will give a second order differential equation, which we can solve to find oscillatory behavior. </p><p>
We will also explore how the behavior of this simple mass-on-a-spring system is replicated literally EVERYWHERE in nature where there is any kind of restoring force. To see that this is the case, we will examine the potential energy of oscillating systems and show that, when the amplitude of oscillations is small enough, any system will behave like a simple harmonic oscillator! </p>
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<h2 class="hd hd-2 unit-title">L1v1: Equation of Motion for a Horizontal Block and Spring</h2>
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<h2 class="hd hd-2 unit-title">L1Q1: Forces on a Vertical Spring</h2>
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Forces on Vertical Spring System
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<p><b class="bfseries">(Part a)</b> Consider a mass, [mathjaxinline]m[/mathjaxinline], that is attached to a spring with spring constant [mathjaxinline]k[/mathjaxinline]. When resting horizontally, the unstretched length of the spring is [mathjaxinline]x_0[/mathjaxinline]. When hanging vertically, at equilibrium, the mass hangs a distance [mathjaxinline]x_1[/mathjaxinline] from the ceiling (therefore the amount of stretch is [mathjaxinline]x_1 - x_0[/mathjaxinline]). </p>
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What is the magnitude of the force due to gravity, [mathjaxinline]F_ g[/mathjaxinline]? Write your answer in terms of any of the relevant variables <code>m</code>, <code>g</code>, or <code>x_1</code> for [mathjaxinline]x_1[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]F_ g =[/mathjaxinline] </p>
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What is the magnitude of the force due to the spring, [mathjaxinline]F_ s[/mathjaxinline]? Write your answer in terms of any of the relevant variables <code>m</code>, <code>k</code>, <code>x_0</code> for [mathjaxinline]x_0[/mathjaxinline], or <code>x_1</code> for [mathjaxinline]x_1[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]F_ s =[/mathjaxinline] </p>
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Now calculate the distance [mathjaxinline]x_1[/mathjaxinline]. Write your answer in terms of any of the relevant variables <code>m</code>, <code>g</code>, <code>k</code>, or <code>x_0</code> for [mathjaxinline]x_0[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]x_1 =[/mathjaxinline] </p>
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<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
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<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
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<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>
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<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
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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>
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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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Forces on Vertical Spring System
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<p><b class="bfseries">(Part b)</b> Now the mass is displaced downward from equilibrium. Let's move our coordinate system so we now have the [mathjaxinline]x=0[/mathjaxinline] line at the equilibrium position of the spring from <b class="bfseries">(Part a)</b>, [mathjaxinline]x_1[/mathjaxinline]. </p>
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If the mass is displaced a distance [mathjaxinline]x[/mathjaxinline] from this position what is the magnitude of the force due to gravity, [mathjaxinline]F_ g[/mathjaxinline]? Write your answer in terms of any of the relevant variables <code>m</code>, <code>g</code>, or <code>x</code>. </p>
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<p style="display:inline">[mathjaxinline]F_ g =[/mathjaxinline] </p>
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What is the magnitude of the force due to the spring, [mathjaxinline]F_ s[/mathjaxinline]? Write your answer in terms of any of the relevant variables <code>m</code>, <code>k</code>, <code>x</code>, <code>x_1</code> for [mathjaxinline]x_1[/mathjaxinline], or <code>x_0</code> for [mathjaxinline]x_0[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]F_ s =[/mathjaxinline] </p>
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Now we can write the equation of motion for the system. Use Newton's Second Law to find the acceleration [mathjaxinline]\ddot{x}[/mathjaxinline]. Note that in this case, we are <i class="it">not</i> asking for the magnitude, but rather for a signed quantity. Use a coordinate system in which down is positive so that [mathjaxinline]x[/mathjaxinline] as shown in the figure above is a positive quantity. Write your answer in terms of any of the relevant variables <code>x</code>, <code>x_1</code> for [mathjaxinline]x_1[/mathjaxinline], <code>x_0</code> for [mathjaxinline]x_0[/mathjaxinline], <code>m</code>, <code>g</code>, or <code>k</code>. </p>
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<p style="display:inline">[mathjaxinline]\ddot{x} =[/mathjaxinline] </p>
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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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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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<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>
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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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<h3 class="hd hd-3 problem-header" id="lect_01_02a_c-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_01_02a_c-problem-progress" tabindex="-1">
Vertical Spring System Oscillations
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<p><b class="bfseries">(Part c)</b> Use the equaton of motion that you derived above to determine the frequency of oscillation. Write your answer in terms of any of the relevant variables <code>x</code>, <code>x_1</code> for [mathjaxinline]x_1[/mathjaxinline], <code>x_0</code> for [mathjaxinline]x_0[/mathjaxinline], <code>m</code>, <code>g</code>, or <code>k</code>. </p>
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<p style="display:inline">[mathjaxinline]\omega =[/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"><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>
</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>
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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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<h2 class="hd hd-2 unit-title">L1Q2: Oscillation Frequency and Gravity</h2>
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Oscillation Frequency and Gravity
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Yen-Jie decided to send a mass on a vertical spring to the surface of an exoplanet (HD 100546b) which is 337 light years away from the Earth. 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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<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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enter <code> 2+3*2 </code> for 8 </td>
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<td class="formulainput">enter <code>abs(x+y) </code> for [mathjaxinline] \left|x+y \right| [/mathjaxinline]<br/>
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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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<h2 class="hd hd-2 unit-title">Comments on Units and Terminology</h2>
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<p><b class="bfseries">Units</b></p><p>
When discussing oscillations (and later waves), the fluctuations of many quantities will be found to follow a sinusoidal dependence on time and/or position. It is assumed implicitly that the argument of any trigonometric function is radians, even though in almost all cases in this course, the quantity whose sine or cosine is being taken has no relationship to an actual physical angle. </p><p>
Technically, "radians" is not a unit like meters or seconds since it is defined as the length along the circular arc defined by an angle divided by the radius of the circle and, therefore, is length/length which is dimensionless. Nonetheless, it is common to use "radians" as a "unit" to indicate that what is being referred to is an angle and not some other dimensionless quantity. For this reason, the radian is officially designated as an SI <i class="it">derived</i> unit (as opposed to SI <i class="it">base</i> units like meters and seconds). </p><p>
In this course, we will generally use the notation "radians" only for actual physical angles. Similarly, we will use "radians/second" to indicate the angular velocity for an actual physical rotation. </p><p>
Another quantity that will be occasionally referred to as a value in radians is a "phase shift". When two objects are oscillating and are not perfectly in sync, i.e., they don't reach the maximum of their oscillations at the same time, they are said to be "out of phase". The phase shift is a measure of how far apart the two oscillations are. </p><p>
We will consider very many quantities such as [mathjaxinline]\sin (k x + \omega t)[/mathjaxinline], where [mathjaxinline]x[/mathjaxinline] and [mathjaxinline]t[/mathjaxinline] are position in meters and time in seconds, respectively. In these cases, the units of [mathjaxinline]\omega[/mathjaxinline] will be [mathjaxinline]\mathrm{seconds}^{-1}[/mathjaxinline], or equivalently [mathjaxinline]1/\mathrm{s}[/mathjaxinline], so that the quantity [mathjaxinline]\omega t[/mathjaxinline] is dimensionless. Similarly, the units of [mathjaxinline]k[/mathjaxinline] are [mathjaxinline]\mathrm{meters}^{-1}[/mathjaxinline], or equivalently [mathjaxinline]1/\mathrm{m}[/mathjaxinline]. </p><p><b class="bfseries">Terminology for "Frequency"</b></p><p>
It is common to refer to the "frequency" [mathjaxinline]\omega[/mathjaxinline] in terms such as [mathjaxinline]\sin (k x + \omega t)[/mathjaxinline] as the "oscillation frequency". Frequently, however, this is also somewhat confusingly referred to as the "angular frequency", even though no actual physical angle is involved. To add to the confusion, the same symbol [mathjaxinline]\omega[/mathjaxinline] is also commonly used to denote the angular <i class="it">velocity</i> of an object that is rotating. The text in this course will use both "oscillation" and "angular" frequency, but will attempt to avoid confusion both in the words used to describe these quantities and also in the use of distinguishing notations: [mathjaxinline]1/\mathrm{s}[/mathjaxinline] for oscillations and [mathjaxinline]\mathrm{rad}/\mathrm{s}[/mathjaxinline] for angular velocities. </p><p>
Finally, there is another common usage for the word "frequency", namely the number of full oscillations, or "cycles", per second. To distinguish this quantity, typically denoted [mathjaxinline]f[/mathjaxinline] and related to the oscillation frequency by [mathjaxinline]f=\omega /(2\pi )[/mathjaxinline], we will use the standard notation "Hz". </p>
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<h3 class="hd hd-3 problem-header" id="lect_01_03a-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_01_03a-problem-progress" tabindex="-1">
One Solution to Equations of Motion
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A general solution to [mathjaxinline]\ddot{x} = -\omega ^{2}x[/mathjaxinline], the equation of motion of a simple harmonic oscillator, is [mathjaxinline]x(t)=a\cos (\omega t) + b\sin (\omega t)[/mathjaxinline], where [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]b[/mathjaxinline] are constants that depend on the initial conditions, namely the position and velocity at [mathjaxinline]t=0[/mathjaxinline]. </p>
<p><b class="bfseries">(Part a)</b> Given the initial conditions for the position and velocity, [mathjaxinline]x(0)=x_0[/mathjaxinline] and [mathjaxinline]v(0)=v_0[/mathjaxinline], express the constants [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]b[/mathjaxinline] in terms of <code>x_0</code> for [mathjaxinline]x_0[/mathjaxinline], <code>v_0</code> for [mathjaxinline]v_0[/mathjaxinline], and <code>omega</code> for [mathjaxinline]\omega[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]a =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]b =[/mathjaxinline] </p>
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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">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>
<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="lect_01_03b-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_01_03b-problem-progress" tabindex="-1">
Alternative Solution to Equations of Motion
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<p><b class="bfseries">(Part b)</b> Another, equivalent, way to write the general solution to the equation of motion is [mathjaxinline]x(t)=A\cos (\omega t + \phi )[/mathjaxinline]. What are [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]b[/mathjaxinline] in terms of <code>A</code> and <code>phi</code> for [mathjaxinline]\phi[/mathjaxinline]? </p>
<p>
<p style="display:inline">[mathjaxinline]a =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]b =[/mathjaxinline] </p>
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<code>2520</code>
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</tr>
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<td class="formulainput">fractions</td>
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<code>2/3</code>
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<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
</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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Alternative Solution to Equations of Motion
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<p><b class="bfseries">(Part c)</b> Finally, express [mathjaxinline]\phi[/mathjaxinline] and [mathjaxinline]A[/mathjaxinline] in terms of the parameters given in the problem, i.e. <code>x_0</code> for [mathjaxinline]x_0[/mathjaxinline], <code>v_0</code> for [mathjaxinline]v_0[/mathjaxinline], and <code>omega</code> for [mathjaxinline]\omega[/mathjaxinline]. (HINT: Equate the answers that you obtained in part (a) and part (b).) </p>
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<p style="display:inline">[mathjaxinline]\phi =[/mathjaxinline] </p>
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<p style="display:inline">[mathjaxinline]A =[/mathjaxinline] </p>
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<h2 class="hd hd-2 unit-title">Review of Simple Harmonic Motion</h2>
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So far we have seen that Newton's Second Law for an ideal mass on a spring gives the 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]\ddot{x} = - \omega ^2 x[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>1</span>)</span></td></tr></table><p>
The solution to this second order differential equation can be written in a few different ways, such as </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]x(t) = a \cos (\omega t) + b \sin (\omega t) = A \cos (\omega t + \phi )[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>2</span>)</span></td></tr></table>
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<h2 class="hd hd-2 unit-title">Review of Small Oscillations</h2>
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Hooke's Law, stating that a spring provides a linear restoring force to a mass, leads to the following equation of 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]\ddot{x} = - \frac{k}{m} x[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>1</span>)</span></td></tr></table><p>
This law is ubiquitous throughout nature because, as was shown in the previous video, any restoring force is approximately linear in the regime of small oscillations about an equilibrium point. We will show this again here. </p><p>
Consider an arbitrary potential, [mathjaxinline]V(x)[/mathjaxinline], with a local minimum at [mathjaxinline]x=x_{0}[/mathjaxinline], from which we will derive an approximate relation for the force starting from the exact equation: </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]F(x) = -\frac{d}{dx}V(x)[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>2</span>)</span></td></tr></table><p>
For small oscillations about the minimum position [mathjaxinline]x_0[/mathjaxinline], we can Taylor expand the potential about this point: </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]V(x) \approx V(x_{0}) + V^{\prime }(x_0) (x-x_{0}) + \frac{1}{2}V^{\prime \prime }(x_0) (x-x_{0})^{2} + \frac{1}{6}V^{\prime \prime \prime }(x_0) (x-x_{0})^{3} + \cdots[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>3</span>)</span></td></tr></table><p>
Note that the magnitude of the first derivative is equal to the magnitude of the force and so [mathjaxinline]V^{\prime }(x_0) = 0[/mathjaxinline] because this is an equilibrium position. Thus, the force is approximately: </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]F(x) \approx -\frac{d}{dx}\left[V(x_0) + \frac{1}{2}V^{\prime \prime }(x_0) (x-x_{0})^{2} + \frac{1}{6}V^{\prime \prime \prime }(x_0) (x-x_{0})^{3} + \cdots \right][/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>4</span>)</span></td></tr></table><p>
The first term [mathjaxinline]V(x_0)[/mathjaxinline] is a constant so: </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]F(x) = -V^{\prime \prime }(x_0) (x-x_{0}) - \frac{1}{2}V^{\prime \prime \prime }(x_0) (x-x_{0})^{2} + \cdots[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>5</span>)</span></td></tr></table><p>
Now, if the magnitude of the third order term is much smaller than the magnitude of the second order term, then we can neglect it in the approximation. This condition is: </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 \left| \frac{1}{2}V^{\prime \prime \prime }(x_0) (x-x_{0})^{2} \right|[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \ll \left| V^{\prime \prime }(x_0) (x-x_{0}) \right|[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"><span>(<span>6</span>)</span></td></tr><tr id="a0000000009"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \left| \frac{1}{2}V^{\prime \prime \prime }(x_0) (x-x_{0}) \right|[/mathjaxinline]
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[mathjaxinline]\displaystyle \ll V^{\prime \prime }(x_0)[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"><span>(<span>7</span>)</span></td></tr><tr id="a0000000010"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \left| V^{\prime \prime \prime }(x_0) (x-x_{0}) \right|[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \ll V^{\prime \prime }(x_0)[/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>
Note, we dropped the factor of [mathjaxinline]\frac{1}{2}[/mathjaxinline] in the last expression, as was done in the lesson video. This is because the inequality is <i class="itshape">much much less than</i>, so the factor of [mathjaxinline]\frac{1}{2}[/mathjaxinline] can be neglected. </p><p>
Finally, we have the following familiar expression! We're almost done! </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]F(x) \approx -V^{\prime \prime }(x_0) (x-x_{0})[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>9</span>)</span></td></tr></table><p>
As we have shown, for instance with a hanging mass under the influence of gravity, the constant force term ([mathjaxinline]x_{0}V^{\prime \prime }(x_0)[/mathjaxinline] in this case) has no impact on the equation of motion when considering only oscillations with respect to the equilibrium position. Thus, we have: </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]F(x) \approx -V^{\prime \prime }(x_0) x[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"><span>(<span>10</span>)</span></td></tr></table><p>
Therefore, any arbitrary potential with a local minimum will exhibit a linear restoring force for small amplitudes of oscillation about that equilibrium position. </p>
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<h2 class="hd hd-2 unit-title">L1Q4: Taylor Expansion</h2>
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Force from Potential
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Let's say we have a particle that experiences a potential given by the function [mathjaxinline]V(x)[/mathjaxinline]. In this problem, we are going to walk through why we can find simple harmonic motion at positions near the minimum of such a potential energy function. </p>
<p><b class="bfseries">(Part a)</b> Let's start by thinking about how to write the force as a function of this potential energy. </p>
<p>
Express your answer in terms of any of the following: <code>x</code>, <code>V</code>, <code>V'</code> for [mathjaxinline]\displaystyle \frac{dV}{dx}[/mathjaxinline], <code>V''</code> for [mathjaxinline]\displaystyle \frac{d^{2}V}{dx^{2}}[/mathjaxinline], etc. </p>
<p style="display:inline">[mathjaxinline]F(x)=[/mathjaxinline]</p>
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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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<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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Force at Equilibrium
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<p><b class="bfseries">(Part b)</b> Next, if we are at a minimum, [mathjaxinline]x_{0}[/mathjaxinline], of the potential, [mathjaxinline]V(x)[/mathjaxinline], what is the force the particle experiences here? </p>
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Express your answer in terms of any of the following: <code>x_0</code> for [mathjaxinline]x_0[/mathjaxinline], <code>x</code>, <code>V</code>, <code>V'</code> for [mathjaxinline]\displaystyle \frac{dV}{dx}[/mathjaxinline], <code>V''</code> for [mathjaxinline]\displaystyle \frac{d^{2}V}{dx^{2}}[/mathjaxinline], etc. </p>
<p style="display:inline">[mathjaxinline]F(x)=[/mathjaxinline]</p>
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<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
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<td class="formulainput"><code>^</code> (raise to a power)</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>
</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="lect_01_05c-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_01_05c-problem-progress" tabindex="-1">
Taylor Expansion of the Force
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<p><b class="bfseries">(Part c)</b> Now let's assume we're moving only a small distance away from this minimum point in the potential. Write the force on this particle as a Taylor series. Include only the first non-zero term. </p>
<p>
Express your answer in terms of any of the following: <code>x_0</code> for [mathjaxinline]x_0[/mathjaxinline], <code>x</code>, <code>V</code> for [mathjaxinline]V(x_0)[/mathjaxinline], <code>V'</code> for [mathjaxinline]\displaystyle \frac{dV(x_0)}{dx}[/mathjaxinline], <code>V''</code> for [mathjaxinline]\displaystyle \frac{d^{2}V(x_0)}{dx^{2}}[/mathjaxinline], etc. </p>
<p style="display:inline">[mathjaxinline]F(x)=[/mathjaxinline]</p>
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<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="3">Numbers</th>
<td class="formulainput">integers</td>
<td class="formulainput">
<code>2520</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">fractions</td>
<td class="formulainput">
<code>2/3</code>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
</tr>
<tr class="formulainput">
<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
</tr>
<tr class="formulainput">
<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">enter (english) name of letter</td>
<td class="formulainput">enter <code>alpha </code> for [mathjaxinline] \alpha [/mathjaxinline]<br/>
enter <code>lambda </code> for [mathjaxinline]\lambda [/mathjaxinline]
</td>
</tr>
<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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Equation of Motion from Taylor Expansion
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<p><b class="bfseries">(Part d)</b> We now have an equation of motion that looks like [mathjaxinline]m \ddot{x} = - C^2 (x-x_0)[/mathjaxinline]. What is [mathjaxinline]C[/mathjaxinline]? </p>
<p>
Express your answer in terms of any of the following: <code>x_0</code> for [mathjaxinline]x_0[/mathjaxinline], <code>x</code>, <code>V</code> for [mathjaxinline]V(x_0)[/mathjaxinline], <code>V'</code> for [mathjaxinline]\displaystyle \frac{dV(x_0)}{dx}[/mathjaxinline], <code>V''</code> for [mathjaxinline]\displaystyle \frac{d^{2}V(x_0)}{dx^{2}}[/mathjaxinline], etc. </p>
<p style="display:inline">[mathjaxinline]C=[/mathjaxinline]</p>
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Conditions for Taylor Expansion Validity
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<p><b class="bfseries">(Part e)</b> Finally, let us consider what conditions are required for this approximation to be valid. If we are only keeping the first non-zero in the force expansion, what must be true in order for it to be reasonable to ignore the next term in the force expansion. Keep all numerical factors!!! </p>
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Express your answer in terms of any of the following: <code>x_0</code> for [mathjaxinline]x_0[/mathjaxinline], <code>x</code>, <code>V</code> for [mathjaxinline]V(x_0)[/mathjaxinline], <code>V'</code> for [mathjaxinline]\displaystyle \frac{dV(x_0)}{dx}[/mathjaxinline], <code>V''</code> for [mathjaxinline]\displaystyle \frac{d^{2}V(x_0)}{dx^{2}}[/mathjaxinline], <code>V'''</code> for [mathjaxinline]\displaystyle \frac{d^{3}V(x_0)}{dx^{3}}[/mathjaxinline], etc. Note, you do not have to write the absolute value symbols, we will assume that we should take the magnitude of your answer. </p>
<p style="display:inline">[mathjaxinline]\displaystyle \frac{d^2V(x_0)}{dx^2} \gg [/mathjaxinline]</p>
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