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<h2 class="hd hd-2 unit-title">Introduction to Boundary Conditions and Continuous Systems of Oscillators</h2>
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Now, we consider a finite system of discrete oscillators, by "fixing" the ends of the system. Importantly, we show that the boundary conditions of the fixed ends make the normal modes discrete. </p><p>
Then, we extend our analysis to continuous systems, as opposed to systems of discrete oscillators. We do this by taking a continuous limit of the discrete-mass system, wherein the "mass" is substituted by a "mass per unit length" or "linear mass density." </p><p>
Thus, for the first time in this course, we are pushing our understanding beyond simple mass-like objects! Next stop, waves! </p>
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<h2 class="hd hd-2 unit-title">L13v1: Finite System with Space-Translation Symmetry: Boundary Conditions</h2>
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<h3 class="hd hd-2">L13v1: Finite system with space-translation symmetry: boundary conditions</h3>
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<h2 class="hd hd-2 unit-title">L13Q1: Boundary Conditions for a Finite String-Mass System</h2>
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Boundary Conditions
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Select the condition that gives all the positive values of [mathjaxinline]k[/mathjaxinline] that satisfy the following relations (Note: let [mathjaxinline]n=1,2,3,\ldots[/mathjaxinline]): </p>
<p><b class="bfseries">(Part a)</b> [mathjaxinline]\sin {ka}=0[/mathjaxinline] </p>
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<text> a) [mathjaxinline]k=\frac{n\pi }{a}[/mathjaxinline]</text>
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<input type="radio" name="input_lect_09_04_2_1" id="input_lect_09_04_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="lect_09_04_2_1-choice_2-label" for="input_lect_09_04_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_09_04_2_1">
<text> b) [mathjaxinline]k=\frac{2n\pi }{a}[/mathjaxinline]</text>
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<input type="radio" name="input_lect_09_04_2_1" id="input_lect_09_04_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="lect_09_04_2_1-choice_3-label" for="input_lect_09_04_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lect_09_04_2_1">
<text> c) [mathjaxinline]k=\frac{n\pi }{2a}[/mathjaxinline]</text>
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<input type="radio" name="input_lect_09_04_2_1" id="input_lect_09_04_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="lect_09_04_2_1-choice_4-label" for="input_lect_09_04_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_lect_09_04_2_1">
<text> d) [mathjaxinline]k=\frac{(2n -1)\pi }{2a}[/mathjaxinline]</text>
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<text> e) [mathjaxinline]k=\frac{(2n +1)\pi }{2a}[/mathjaxinline]</text>
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<input type="radio" name="input_lect_09_04_2_1" id="input_lect_09_04_2_1_choice_6" class="field-input input-radio" value="choice_6"/><label id="lect_09_04_2_1-choice_6-label" for="input_lect_09_04_2_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_lect_09_04_2_1">
<text> f) [mathjaxinline]k=\frac{(2n -1)\pi }{a}[/mathjaxinline]</text>
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<text> g) [mathjaxinline]k=\frac{(2n +1)\pi }{a}[/mathjaxinline]</text>
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<p><b class="bfseries">(Part b)</b> [mathjaxinline]\cos {ka}=0[/mathjaxinline] </p>
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<text> a) [mathjaxinline]k=\frac{n\pi }{a}[/mathjaxinline]</text>
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<input type="radio" name="input_lect_09_04_3_1" id="input_lect_09_04_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="lect_09_04_3_1-choice_2-label" for="input_lect_09_04_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_09_04_3_1">
<text> b) [mathjaxinline]k=\frac{2n\pi }{a}[/mathjaxinline]</text>
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<text> c) [mathjaxinline]k=\frac{n\pi }{2a}[/mathjaxinline]</text>
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<input type="radio" name="input_lect_09_04_3_1" id="input_lect_09_04_3_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="lect_09_04_3_1-choice_4-label" for="input_lect_09_04_3_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_lect_09_04_3_1">
<text> d) [mathjaxinline]k=\frac{(2n -1)\pi }{2a}[/mathjaxinline]</text>
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<input type="radio" name="input_lect_09_04_3_1" id="input_lect_09_04_3_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="lect_09_04_3_1-choice_5-label" for="input_lect_09_04_3_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_lect_09_04_3_1">
<text> e) [mathjaxinline]k=\frac{(2n +1)\pi }{2a}[/mathjaxinline]</text>
</label>
</div>
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<input type="radio" name="input_lect_09_04_3_1" id="input_lect_09_04_3_1_choice_6" class="field-input input-radio" value="choice_6"/><label id="lect_09_04_3_1-choice_6-label" for="input_lect_09_04_3_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_lect_09_04_3_1">
<text> f) [mathjaxinline]k=\frac{(2n -1)\pi }{a}[/mathjaxinline]</text>
</label>
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<input type="radio" name="input_lect_09_04_3_1" id="input_lect_09_04_3_1_choice_7" class="field-input input-radio" value="choice_7"/><label id="lect_09_04_3_1-choice_7-label" for="input_lect_09_04_3_1_choice_7" class="response-label field-label label-inline" aria-describedby="status_lect_09_04_3_1">
<text> g) [mathjaxinline]k=\frac{(2n +1)\pi }{a}[/mathjaxinline]</text>
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<h2 class="hd hd-2 unit-title">L13Q2: Normal Modes of Finite String-Mass System [WITH SIMULATION]</h2>
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Normal Modes of Finite String-Mass System - part a
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Consider the following plots, each of which depicts a finite array of masses connected by strings, oscillating in a certain normal mode of the system. For each plot, identify: (i) the number of normal modes in the system, and (ii) the index of the normal mode that is shown. </p>
<p><b class="bfseries">(Part a)</b> Plot 1 </p>
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<p><b class="bfseries">(Part i)</b> What is the total number of normal modes in the system? </p>
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<p><b class="bfseries">(Part ii)</b> What is the index of the normal mode that is shown? </p>
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Normal Modes of Finite String-Mass System - part b
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<p><b class="bfseries">(Part b)</b> Plot 2 </p>
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<img src="/assets/courseware/v1/9e96ff7181b6f63c9769bc6a999aed1c/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_lect_08_07_image2.png" width="550"/>
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<p><b class="bfseries">(Part i)</b> What is the total number of normal modes in the system? </p>
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<p><b class="bfseries">(Part ii)</b> What is the index of the normal mode that is shown? </p>
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Normal Modes of Finite String-Mass System - part c
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<p><b class="bfseries">(Part c)</b> Plot 3 </p>
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<img src="/assets/courseware/v1/0f71ebc91aaf2057e54da613ef31eef3/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_lect_08_07_image3.png" width="550"/>
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<p><b class="bfseries">(Part i)</b> What is the total number of normal modes in the system? </p>
<p>
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<p><b class="bfseries">(Part ii)</b> What is the index of the normal mode that is shown? </p>
<p>
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<h2>Run the Interactive Python Visualization that Generated the Plots Above!</h2><p>The widget is run in a Jupyter notebook, accessible through the button below. <b>NOTE: The notebook may take up to 3 mintues to load! Please be patient!</b></p><p><div align="center"><a href="https://mybinder.org/v2/gh/mitx-803/vis/master?filepath=many_coupled_masses_1.ipynb" class="btn btn-primary" target="_blank" style="color:#FFFFFF;">ACCESS JUPYTER NOTEBOOK HERE</a></div></p><p><div class="hideshowbox"><h4 onclick="hideshow(this);" style="margin: 0px">How to Run Jupyter Notebooks (expand this section if you need a reminder!)<span class="icon-caret-down toggleimage"/></h4><div class="hideshowcontent"><p><h3>Running Notebooks on an External Server</h3></p><p>To access a simulation, click the "ACCESS JUPYTER NOTEBOOK HERE" button. This will bring you to a loading page, hosted by <i class="itshape">mybinder.org</i> (the loading time is anywhere from 20 seconds to 3 minutes). The Jupyter notebooks are run externally to the course, on a server which runs an instance of Python. There is no need to install Python or related dependencies!</p><div align="center"><iframe src="https://mitx-803.github.io/gifs/python_06.html" width="720" height="590" scrolling="no" frameborder="0"/></div><p><h3>Initializing the Program</h3></p><p>Once loaded, you will see a Jupyter notebook in your browser! You will have to click a button to initialize the program. The button is indicated in the instructions within the notebook, and also shown below.</p><div align="center"><img width="700" src="/assets/courseware/v1/3f6c044fc06f79d82bb2e8a97f7dd11a/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_binder_initialize_button.png"/></div><p/><div align="center"><iframe src="https://mitx-803.github.io/gifs/python_07.html" width="720" height="602" scrolling="no" frameborder="0"/></div><p><h3>Instructions and Source Code</h3></p><p>Each notebook has self-contained instructions on how to use the Python simulation. Additionally, you may toggle the button at the bottom of the notebook to view/augment the source code.</p><div align="center"><iframe src="https://mitx-803.github.io/gifs/python_08.html" width="720" height="608" scrolling="no" frameborder="0"/></div><p><h3>Saving/Running Notebooks Locally</h3></p><p>Finally, you can dowload each notebook to run locally. Additionally, you can visit the git repository to download all notebooks in the course. In order to run notebooks locally, you must install Python and its dependencies. We cannot help with this process, but we encourage you to look at the resources below, if you are interested.</p><div align="center"><iframe src="https://mitx-803.github.io/gifs/python_09.html" width="720" height="609" scrolling="no" frameborder="0"/></div><p><h3>External Links</h3><br/>   [mathjaxinline]\bullet[/mathjaxinline]  git repository: <a href="https://github.com/mitx-803/vis" target="blank">github.com/mitx-803/vis</a><br/>   [mathjaxinline]\bullet[/mathjaxinline]  information on Jupyter notebooks: <a href="https://jupyter.org/" target="blank">Jupyter Notebooks</a><br/>   [mathjaxinline]\bullet[/mathjaxinline]  information on installing Python through Anaconda: <a href="https://www.anaconda.com/distribution/" target="blank">Anaconda</a><br/>   [mathjaxinline]\bullet[/mathjaxinline]  information on the Binder community: <a href="https://mybinder.readthedocs.io/en/latest/" target="blank">Binder</a><br/></p><p><h3>Dependencies</h3></p><p>
The visualizations run on Python 3. Dependencies for running Python code locally (not through Binder) are stated in the git repository, and include (but are not limited to) the following:
<br/>   [mathjaxinline]\bullet[/mathjaxinline]  scipy
<br/>   [mathjaxinline]\bullet[/mathjaxinline]  numpy
<br/>   [mathjaxinline]\bullet[/mathjaxinline]  ipywidgets
<br/>   [mathjaxinline]\bullet[/mathjaxinline]  nbinteract
<br/>   [mathjaxinline]\bullet[/mathjaxinline]  matplotlib
<br/>   [mathjaxinline]\bullet[/mathjaxinline]  pandas
<br/>   [mathjaxinline]\bullet[/mathjaxinline]  IPython
<br/>   [mathjaxinline]\bullet[/mathjaxinline]  ffmpeg
<br/>   [mathjaxinline]\bullet[/mathjaxinline]  jupyter-contrib-nbextensions
<br/>
</p><p>
You will have to find resources that explain how to install these appropriately for your system, if they are not already installed with your Python package.
</p></div><p class="hideshowbottom" onclick="hideshow(this);" style="margin: 0px"><a href="javascript: {return false;}">Show</a></p></div></p><SCRIPT src="/assets/courseware/v1/631e447105fca1b243137b21b9ed6f90/asset-v1:MITx+8.03x+1T2020+type@asset+block/latex2edx.js" type="text/javascript"/><LINK href="/assets/courseware/v1/daf81af0af57b85a105e0ed27b7873a0/asset-v1:MITx+8.03x+1T2020+type@asset+block/latex2edx.css" rel="stylesheet" type="text/css"/><h2>What You Should See</h2><p>When the notebook is initialized, you will see the following visualization. Follow question prompts within the notebook.</p><div align="center"><img width="800" src="/assets/courseware/v1/e8b3930dc732ddfa09bacf75f33474e8/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_many_coupled_masses_1.png"/></div><p/>
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<h3 class="hd hd-2">L13v2: Continuous Massive String System</h3>
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<h2 class="hd hd-2 unit-title">L13Q3: Taylor Series Continuous Approximation</h2>
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Taylor Series Continuous Approximation
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The continuous limit of a system of discrete masses connected by strings is equivalent to [mathjaxinline]a \rightarrow 0[/mathjaxinline], where the distance between the masses becomes infinitesimal. </p>
<p>
In taking this limit, we defined [mathjaxinline]A_{j} \equiv A(x)[/mathjaxinline], where [mathjaxinline]x[/mathjaxinline] is the position of the [mathjaxinline]j^{\mathrm{th}}[/mathjaxinline] mass. Thus, [mathjaxinline]x-a[/mathjaxinline] is the position of the [mathjaxinline](j-1)^{\mathrm{st}}[/mathjaxinline] mass, and [mathjaxinline]x+a[/mathjaxinline] is the position of the [mathjaxinline](j+1)^{\mathrm{st}}[/mathjaxinline] mass, etc. </p>
<p>
Explicitly write the expression [mathjaxinline]A_{j-1} + A_{j+1}[/mathjaxinline] in terms of [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]A(x)[/mathjaxinline], keeping only terms up to [mathjaxinline]\mathcal{O}(a^2)[/mathjaxinline]. Use <code>A</code> for [mathjaxinline]A(x)[/mathjaxinline], <code>A'</code> for [mathjaxinline]A^{\prime }(x)[/mathjaxinline], and <code>A''</code> for [mathjaxinline]A^{\prime \prime }(x)[/mathjaxinline], etc. </p>
<p style="display:inline">[mathjaxinline]A_{j-1}+A_{j+1}=[/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">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">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>
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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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<h2 class="hd hd-2 unit-title">L13Q4: Taking the Continuous Limit</h2>
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Defining Mass Density
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Going from a system of discrete masses to the continuous limit, we made the identification [mathjaxinline]\rho _{L} \equiv \dfrac {m}{a}[/mathjaxinline]. However, since the continuous limit is equivalent to [mathjaxinline]a \rightarrow 0[/mathjaxinline], the ratio [mathjaxinline]\dfrac {m}{a} \rightarrow \infty[/mathjaxinline] for fixed mass. </p>
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What additional assumption(s) must be made in order to produce a physical result? Check ALL that apply. </p>
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<text>a) The total mass of the system goes to zero.</text>
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<text>b) The mass of each individual oscillator goes to zero.</text>
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<text>c) The linear mass density, [mathjaxinline]\rho _{L}[/mathjaxinline], is a constant quantity.</text>
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<text>d) The separation, [mathjaxinline]a[/mathjaxinline], goes to a constant, nonzero value.</text>
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<h2 class="hd hd-2 unit-title">L13Q5: Wave Equation and Velocity</h2>
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Wave Equation and Velocity - part a
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<p><b class="bfseries">(Part a)</b> The wave equation from the previous lecture segment is defined 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]\frac{\partial ^{2} A}{\partial t^{2}} = v_{p}^{2}\frac{\partial ^{2} A}{\partial x^{2}}[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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<p>
Next week, we will discuss one possible solution to this wave equation, which has the form: </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]A(x,t)=A_{0}\cos (kx-\omega t)[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
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<p>
Using this solution, determine how is [mathjaxinline]v_{p}[/mathjaxinline] related to [mathjaxinline]k[/mathjaxinline], [mathjaxinline]x[/mathjaxinline], [mathjaxinline]\omega[/mathjaxinline], and/or [mathjaxinline]t[/mathjaxinline]? Express your answer in terms of <code>k</code>, <code>x</code>, <code>t</code>, or <code>omega</code> for [mathjaxinline]\omega[/mathjaxinline] as needed. </p>
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<p style="display:inline">[mathjaxinline]v_{p} =[/mathjaxinline] </p>
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<th class="formulainput" scope="col">Descriptions</th>
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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">decimals </td>
<td class="formulainput"><code>3.14</code>, <code>.98</code></td>
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<th class="formulainput" scope="row" rowspan="4">Operators</th>
<td class="formulainput"><code>+ - * /</code> (add, subtract, multiply, divide)</td>
<td class="formulainput">enter <code> (x+2*y)/(x-1)</code> for [mathjaxinline] \displaystyle \frac{x+2y}{x-1} [/mathjaxinline] </td>
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<td class="formulainput"><code>^</code> (raise to a power)</td>
<td class="formulainput">enter <code> x^(n+1) </code> for [mathjaxinline] x^{n+1} [/mathjaxinline]</td>
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<td class="formulainput"><code>_</code> (add a subscript)</td>
<td class="formulainput">enter <code> v_0 </code> for [mathjaxinline] v_0 [/mathjaxinline] </td>
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<td class="formulainput">use <code>( )</code> to clarify order of operations</td>
<td class="formulainput"> enter <code>(2+3)*2 </code> for 10 <br/>
enter <code> 2+3*2 </code> for 8 </td>
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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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<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">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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Wave Equation and Velocity - part b
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<p><b class="bfseries">(Part b)</b> We have also seen that [mathjaxinline]v_{p}[/mathjaxinline] is related to the tension [mathjaxinline]T[/mathjaxinline] and linear mass density [mathjaxinline]\rho _{L}[/mathjaxinline] of the string. If we double the tension in the string, how does velocity of waves in the system change? </p>
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<text> a) velocity increases by multiplicative factor of [mathjaxinline]2[/mathjaxinline]</text>
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<text> b) velocity increases by multiplicative factor of [mathjaxinline]\sqrt {2}[/mathjaxinline]</text>
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<text> c) velocity decreases by multiplicative factor of [mathjaxinline]\frac{1}{2}[/mathjaxinline]</text>
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<text> d) velocity decreases by multiplicative factor of [mathjaxinline]\frac{1}{\sqrt {2}}[/mathjaxinline]</text>
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<span class="notification-message" aria-describedby="lect_09_06_0b-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_09_06_0b-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_09_06_0b-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>