<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-runtime-class="LmsRuntime" data-request-token="3815f498685311ef86120214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lesson10intro_b" data-block-type="vertical" data-graded="True" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<h2 class="hd hd-2 unit-title">Introduction to Boundary Conditions and Continuous Systems of Oscillators</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@html+block@lesson10_intro_b">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-has-score="False" data-runtime-class="LmsRuntime" data-request-token="3815f498685311ef86120214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@html+block@lesson10_intro_b" data-block-type="html" data-graded="True" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<p>
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>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-runtime-class="LmsRuntime" data-request-token="3815f498685311ef86120214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-L13v1" data-block-type="vertical" data-graded="True" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<h2 class="hd hd-2 unit-title">L13v1: Finite System with Space-Translation Symmetry: Boundary Conditions</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@video+block@L13v1">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-has-score="False" data-runtime-class="LmsRuntime" data-request-token="3815f498685311ef86120214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@video+block@L13v1" data-block-type="video" data-graded="True" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L13v1: Finite system with space-translation symmetry: boundary conditions</h3>
<div
id="video_L13v1"
class="video closed"
data-metadata='{"prioritizeHls": false, "sources": [], "autoplay": false, "transcriptLanguage": "en", "lmsRootURL": "https://openlearninglibrary.mit.edu", "transcriptTranslationUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L13v1/handler/transcript/translation/__lang__", "speed": null, "generalSpeed": 1.0, "streams": "1.00:uDDYou92zCU", "autoAdvance": false, "ytApiUrl": "https://www.youtube.com/iframe_api", "completionPercentage": 0.95, "start": 0.0, "savedVideoPosition": 0.0, "autohideHtml5": false, "saveStateEnabled": false, "recordedYoutubeIsAvailable": true, "publishCompletionUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L13v1/handler/publish_completion", "saveStateUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L13v1/handler/xmodule_handler/save_user_state", "poster": null, "duration": 0.0, "captionDataDir": null, "showCaptions": "true", "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L13v1/handler/transcript/available_translations", "ytTestTimeout": 1500, "transcriptLanguages": {"en": "English"}, "completionEnabled": false, "ytMetadataEndpoint": "", "end": 0.0}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="L13v1"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_L13v1">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L13v1">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<a class="btn-link external-track" href="/assets/courseware/v1/edca43eea9ee33b03410fe2349fe2a94/asset-v1:MITx+8.03x+1T2020+type@asset+block/subs_L13v1.srt">Download transcript</a>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-runtime-class="LmsRuntime" data-request-token="3815f498685311ef86120214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lect_09_ex04" data-block-type="vertical" data-graded="True" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<h2 class="hd hd-2 unit-title">L13Q1: Boundary Conditions for a Finite String-Mass System</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_09_04">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-runtime-class="LmsRuntime" data-request-token="3815f498685311ef86120214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_09_04" data-block-type="problem" data-graded="True" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_09_04" class="problems-wrapper" role="group"
aria-labelledby="lect_09_04-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_09_04" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_09_04/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="2.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_09_04-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_09_04-problem-progress" tabindex="-1">
Boundary Conditions
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_09_04-problem-progress"></div>
<div class="problem">
<div>
<p>
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>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lect_09_04_2_1">
<fieldset aria-describedby="status_lect_09_04_2_1">
<div class="field">
<input type="radio" name="input_lect_09_04_2_1" id="input_lect_09_04_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="lect_09_04_2_1-choice_1-label" for="input_lect_09_04_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lect_09_04_2_1">
<text> a) [mathjaxinline]k=\frac{n\pi }{a}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<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>
</label>
</div>
<div class="field">
<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>
</label>
</div>
<div class="field">
<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>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_09_04_2_1" id="input_lect_09_04_2_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="lect_09_04_2_1-choice_5-label" for="input_lect_09_04_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_lect_09_04_2_1">
<text> e) [mathjaxinline]k=\frac{(2n +1)\pi }{2a}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<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>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_09_04_2_1" id="input_lect_09_04_2_1_choice_7" class="field-input input-radio" value="choice_7"/><label id="lect_09_04_2_1-choice_7-label" for="input_lect_09_04_2_1_choice_7" class="response-label field-label label-inline" aria-describedby="status_lect_09_04_2_1">
<text> g) [mathjaxinline]k=\frac{(2n +1)\pi }{a}[/mathjaxinline]</text>
</label>
</div>
<span id="answer_lect_09_04_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_lect_09_04_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_lect_09_04_solution_1"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
<p><b class="bfseries">(Part b)</b> [mathjaxinline]\cos {ka}=0[/mathjaxinline] </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lect_09_04_3_1">
<fieldset aria-describedby="status_lect_09_04_3_1">
<div class="field">
<input type="radio" name="input_lect_09_04_3_1" id="input_lect_09_04_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="lect_09_04_3_1-choice_1-label" for="input_lect_09_04_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lect_09_04_3_1">
<text> a) [mathjaxinline]k=\frac{n\pi }{a}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<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>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_09_04_3_1" id="input_lect_09_04_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="lect_09_04_3_1-choice_3-label" for="input_lect_09_04_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lect_09_04_3_1">
<text> c) [mathjaxinline]k=\frac{n\pi }{2a}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<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>
</label>
</div>
<div class="field">
<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>
<div class="field">
<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>
</div>
<div class="field">
<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>
</label>
</div>
<span id="answer_lect_09_04_3_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_lect_09_04_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_lect_09_04_solution_2"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Boundary Conditions" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lect_09_04" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_09_04">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_09_04-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_04-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_04-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>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-runtime-class="LmsRuntime" data-request-token="3815f498685311ef86120214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lect_08_ex07" data-block-type="vertical" data-graded="True" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<h2 class="hd hd-2 unit-title">L13Q2: Normal Modes of Finite String-Mass System [WITH SIMULATION]</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_08_07a">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-runtime-class="LmsRuntime" data-request-token="3815f498685311ef86120214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_08_07a" data-block-type="problem" data-graded="True" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_08_07a" class="problems-wrapper" role="group"
aria-labelledby="lect_08_07a-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_08_07a" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_08_07a/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="2.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_08_07a-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_08_07a-problem-progress" tabindex="-1">
Normal Modes of Finite String-Mass System - part a
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_08_07a-problem-progress"></div>
<div class="problem">
<div>
<p>
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>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<div align="center">
<img src="/assets/courseware/v1/ea64c64c62d548b7c03b71868985a097/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_lect_08_07_image1.png" width="550"/>
</div>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<p><b class="bfseries">(Part i)</b> What is the total number of normal modes in the system? </p>
<p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_lect_08_07a_2_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_lect_08_07a_2_1" id="input_lect_08_07a_2_1" aria-describedby="status_lect_08_07a_2_1" value="" size="30"/>
<span class="trailing_text" id="trailing_text_lect_08_07a_2_1"/>
<span class="status unanswered" id="status_lect_08_07a_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_lect_08_07a_2_1" class="answer"/>
</div>
</div></div>
</p>
<p><b class="bfseries">(Part ii)</b> What is the index of the normal mode that is shown? </p>
<p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_lect_08_07a_3_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_lect_08_07a_3_1" id="input_lect_08_07a_3_1" aria-describedby="status_lect_08_07a_3_1" value="" size="30"/>
<span class="trailing_text" id="trailing_text_lect_08_07a_3_1"/>
<span class="status unanswered" id="status_lect_08_07a_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_lect_08_07a_3_1" class="answer"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_lect_08_07a_solution_1"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Normal Modes of Finite String-Mass System - part a" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lect_08_07a" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_08_07a">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_08_07a-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_08_07a-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_08_07a-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 class="vert vert-1" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_08_07b">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-runtime-class="LmsRuntime" data-request-token="3815f498685311ef86120214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_08_07b" data-block-type="problem" data-graded="True" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_08_07b" class="problems-wrapper" role="group"
aria-labelledby="lect_08_07b-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_08_07b" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_08_07b/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="2.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_08_07b-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_08_07b-problem-progress" tabindex="-1">
Normal Modes of Finite String-Mass System - part b
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_08_07b-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bfseries">(Part b)</b> Plot 2 </p>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<div align="center">
<img src="/assets/courseware/v1/9e96ff7181b6f63c9769bc6a999aed1c/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_lect_08_07_image2.png" width="550"/>
</div>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<p><b class="bfseries">(Part i)</b> What is the total number of normal modes in the system? </p>
<p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_lect_08_07b_2_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_lect_08_07b_2_1" id="input_lect_08_07b_2_1" aria-describedby="status_lect_08_07b_2_1" value="" size="30"/>
<span class="trailing_text" id="trailing_text_lect_08_07b_2_1"/>
<span class="status unanswered" id="status_lect_08_07b_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_lect_08_07b_2_1" class="answer"/>
</div>
</div></div>
</p>
<p><b class="bfseries">(Part ii)</b> What is the index of the normal mode that is shown? </p>
<p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_lect_08_07b_3_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_lect_08_07b_3_1" id="input_lect_08_07b_3_1" aria-describedby="status_lect_08_07b_3_1" value="" size="30"/>
<span class="trailing_text" id="trailing_text_lect_08_07b_3_1"/>
<span class="status unanswered" id="status_lect_08_07b_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_lect_08_07b_3_1" class="answer"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_lect_08_07b_solution_1"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Normal Modes of Finite String-Mass System - part b" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lect_08_07b" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_08_07b">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_08_07b-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_08_07b-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_08_07b-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 class="vert vert-2" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_08_07c">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-runtime-class="LmsRuntime" data-request-token="3815f498685311ef86120214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_08_07c" data-block-type="problem" data-graded="True" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_08_07c" class="problems-wrapper" role="group"
aria-labelledby="lect_08_07c-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_08_07c" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_08_07c/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="2.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_08_07c-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_08_07c-problem-progress" tabindex="-1">
Normal Modes of Finite String-Mass System - part c
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_08_07c-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bfseries">(Part c)</b> Plot 3 </p>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<div align="center">
<img src="/assets/courseware/v1/0f71ebc91aaf2057e54da613ef31eef3/asset-v1:MITx+8.03x+1T2020+type@asset+block/images_lect_08_07_image3.png" width="550"/>
</div>
<p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
<p><b class="bfseries">(Part i)</b> What is the total number of normal modes in the system? </p>
<p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_lect_08_07c_2_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_lect_08_07c_2_1" id="input_lect_08_07c_2_1" aria-describedby="status_lect_08_07c_2_1" value="" size="30"/>
<span class="trailing_text" id="trailing_text_lect_08_07c_2_1"/>
<span class="status unanswered" id="status_lect_08_07c_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_lect_08_07c_2_1" class="answer"/>
</div>
</div></div>
</p>
<p><b class="bfseries">(Part ii)</b> What is the index of the normal mode that is shown? </p>
<p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_lect_08_07c_3_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_lect_08_07c_3_1" id="input_lect_08_07c_3_1" aria-describedby="status_lect_08_07c_3_1" value="" size="30"/>
<span class="trailing_text" id="trailing_text_lect_08_07c_3_1"/>
<span class="status unanswered" id="status_lect_08_07c_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_lect_08_07c_3_1" class="answer"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_lect_08_07c_solution_1"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Normal Modes of Finite String-Mass System - part c" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lect_08_07c" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_08_07c">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_08_07c-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_08_07c-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_08_07c-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 class="vert vert-3" data-id="block-v1:MITx+8.03x+1T2020+type@html+block@SIM_lect_08_07">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-has-score="False" data-runtime-class="LmsRuntime" data-request-token="3815f498685311ef86120214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@html+block@SIM_lect_08_07" data-block-type="html" data-graded="True" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<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/>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-runtime-class="LmsRuntime" data-request-token="3815f498685311ef86120214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-L13v2" data-block-type="vertical" data-graded="True" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<h2 class="hd hd-2 unit-title">L13v2: Continuous Massive String System</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@video+block@L13v2">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-has-score="False" data-runtime-class="LmsRuntime" data-request-token="3815f498685311ef86120214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@video+block@L13v2" data-block-type="video" data-graded="True" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L13v2: Continuous Massive String System</h3>
<div
id="video_L13v2"
class="video closed"
data-metadata='{"prioritizeHls": false, "sources": [], "autoplay": false, "transcriptLanguage": "en", "lmsRootURL": "https://openlearninglibrary.mit.edu", "transcriptTranslationUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L13v2/handler/transcript/translation/__lang__", "speed": null, "generalSpeed": 1.0, "streams": "1.00:emGa5BOFwvg", "autoAdvance": false, "ytApiUrl": "https://www.youtube.com/iframe_api", "completionPercentage": 0.95, "start": 0.0, "savedVideoPosition": 0.0, "autohideHtml5": false, "saveStateEnabled": false, "recordedYoutubeIsAvailable": true, "publishCompletionUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L13v2/handler/publish_completion", "saveStateUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L13v2/handler/xmodule_handler/save_user_state", "poster": null, "duration": 0.0, "captionDataDir": null, "showCaptions": "true", "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L13v2/handler/transcript/available_translations", "ytTestTimeout": 1500, "transcriptLanguages": {"en": "English"}, "completionEnabled": false, "ytMetadataEndpoint": "", "end": 0.0}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="L13v2"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_L13v2">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L13v2">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<a class="btn-link external-track" href="/assets/courseware/v1/80f63dd625b29470b9bc420b83762a58/asset-v1:MITx+8.03x+1T2020+type@asset+block/subs_L13v2.srt">Download transcript</a>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-runtime-class="LmsRuntime" data-request-token="3815f498685311ef86120214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lect_09_ex05_2" data-block-type="vertical" data-graded="True" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<h2 class="hd hd-2 unit-title">L13Q3: Taylor Series Continuous Approximation</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_09_05_2">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-runtime-class="LmsRuntime" data-request-token="3815f498685311ef86120214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_09_05_2" data-block-type="problem" data-graded="True" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_09_05_2" class="problems-wrapper" role="group"
aria-labelledby="lect_09_05_2-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_09_05_2" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_09_05_2/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_09_05_2-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_09_05_2-problem-progress" tabindex="-1">
Taylor Series Continuous Approximation
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_09_05_2-problem-progress"></div>
<div class="problem">
<div>
<p>
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>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_lect_09_05_2_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/3c6b846f79f362682b770fc1967d7a10/asset-v1:MITx+8.03x+1T2020+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_lect_09_05_2_2_1" id="input_lect_09_05_2_2_1" aria-describedby="status_lect_09_05_2_2_1" value="" class="math"/>
<span class="trailing_text" id="trailing_text_lect_09_05_2_2_1"/>
<span class="status unanswered" id="status_lect_09_05_2_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_lect_09_05_2_2_1" class="answer"/>
<div id="display_lect_09_05_2_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_lect_09_05_2_2_1_dynamath" name="input_lect_09_05_2_2_1_dynamath"/>
</div>
</div></div>
<span>
<link href="/assets/courseware/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+8.03x+1T2020+type@asset+block/css_mymodal.css" rel="stylesheet" type="text/css"/>
<div align="right">
<a href="#mymodal-one" class="btn btn-default">Input Help
</a>
</div>
<div class="mymodal-positioner">
<a href="#" class="mymodal" id="mymodal-one" aria-hidden="true"/>
<div class="mymodal-dialog">
<div class="mymodal-header">
<h4>Input Help</h4>
<a href="#" class="mymodal-btn-close">&#215;</a>
</div>
<div class="formulainput">
<table class="formulainput">
<tbody>
<tr class="fiptitle">
<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<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>
</tr>
</tbody>
</table>
</div>
<div class="mymodal-footer">
<a href="#" class="btn btn-primary" style="color:#FFFFFF;">Close</a>
</div>
</div>
</div>
</span>
<p>
<div class="solution-span">
<span id="solution_lect_09_05_2_solution_1"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Taylor Series Continuous Approximation" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lect_09_05_2" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_09_05_2">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_09_05_2-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_05_2-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_05_2-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>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-runtime-class="LmsRuntime" data-request-token="3815f498685311ef86120214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-L13v3" data-block-type="vertical" data-graded="True" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<h2 class="hd hd-2 unit-title">L13v3: Wave Equation in the Continuous Massive String System</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@video+block@L13v3">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-has-score="False" data-runtime-class="LmsRuntime" data-request-token="3815f498685311ef86120214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@video+block@L13v3" data-block-type="video" data-graded="True" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L13v3: Wave Equation in the Continuous Massive String System</h3>
<div
id="video_L13v3"
class="video closed"
data-metadata='{"prioritizeHls": false, "sources": [], "autoplay": false, "transcriptLanguage": "en", "lmsRootURL": "https://openlearninglibrary.mit.edu", "transcriptTranslationUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L13v3/handler/transcript/translation/__lang__", "speed": null, "generalSpeed": 1.0, "streams": "1.00:OgAr7lmmsIA", "autoAdvance": false, "ytApiUrl": "https://www.youtube.com/iframe_api", "completionPercentage": 0.95, "start": 0.0, "savedVideoPosition": 0.0, "autohideHtml5": false, "saveStateEnabled": false, "recordedYoutubeIsAvailable": true, "publishCompletionUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L13v3/handler/publish_completion", "saveStateUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L13v3/handler/xmodule_handler/save_user_state", "poster": null, "duration": 0.0, "captionDataDir": null, "showCaptions": "true", "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@video+block@L13v3/handler/transcript/available_translations", "ytTestTimeout": 1500, "transcriptLanguages": {"en": "English"}, "completionEnabled": false, "ytMetadataEndpoint": "", "end": 0.0}'
data-bumper-metadata='null'
data-autoadvance-enabled="False"
data-poster='null'
tabindex="-1"
>
<div class="focus_grabber first"></div>
<div class="tc-wrapper">
<div class="video-wrapper">
<span tabindex="0" class="spinner" aria-hidden="false" aria-label="Loading video player"></span>
<span tabindex="-1" class="btn-play fa fa-youtube-play fa-2x is-hidden" aria-hidden="true" aria-label="Play video"></span>
<div class="video-player-pre"></div>
<div class="video-player">
<div id="L13v3"></div>
<h4 class="hd hd-4 video-error is-hidden">No playable video sources found.</h4>
<h4 class="hd hd-4 video-hls-error is-hidden">
Your browser does not support this video format. Try using a different browser.
</h4>
</div>
<div class="video-player-post"></div>
<div class="closed-captions"></div>
<div class="video-controls is-hidden">
<div>
<div class="vcr"><div class="vidtime">0:00 / 0:00</div></div>
<div class="secondary-controls"></div>
</div>
</div>
</div>
</div>
<div class="focus_grabber last"></div>
<h3 class="hd hd-4 downloads-heading sr" id="video-download-transcripts_L13v3">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L13v3">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<a class="btn-link external-track" href="/assets/courseware/v1/601ba5461d36d00c1064a214ca9c53fd/asset-v1:MITx+8.03x+1T2020+type@asset+block/subs_L13v3.srt">Download transcript</a>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-runtime-class="LmsRuntime" data-request-token="3815f498685311ef86120214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lect_09_ex05_1" data-block-type="vertical" data-graded="True" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<h2 class="hd hd-2 unit-title">L13Q4: Taking the Continuous Limit</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_09_05_1">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-runtime-class="LmsRuntime" data-request-token="3815f498685311ef86120214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_09_05_1" data-block-type="problem" data-graded="True" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_09_05_1" class="problems-wrapper" role="group"
aria-labelledby="lect_09_05_1-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_09_05_1" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_09_05_1/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_09_05_1-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_09_05_1-problem-progress" tabindex="-1">
Defining Mass Density
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_09_05_1-problem-progress"></div>
<div class="problem">
<div>
<p>
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>
<p>
What additional assumption(s) must be made in order to produce a physical result? Check ALL that apply. </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lect_09_05_1_2_1">
<fieldset aria-describedby="status_lect_09_05_1_2_1">
<div class="field">
<input type="checkbox" name="input_lect_09_05_1_2_1[]" id="input_lect_09_05_1_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="lect_09_05_1_2_1-choice_0-label" for="input_lect_09_05_1_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_lect_09_05_1_2_1">
<text>a) The total mass of the system goes to zero.</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_09_05_1_2_1[]" id="input_lect_09_05_1_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="lect_09_05_1_2_1-choice_1-label" for="input_lect_09_05_1_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lect_09_05_1_2_1">
<text>b) The mass of each individual oscillator goes to zero.</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_09_05_1_2_1[]" id="input_lect_09_05_1_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="lect_09_05_1_2_1-choice_2-label" for="input_lect_09_05_1_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_09_05_1_2_1">
<text>c) The linear mass density, [mathjaxinline]\rho _{L}[/mathjaxinline], is a constant quantity.</text>
</label>
</div>
<div class="field">
<input type="checkbox" name="input_lect_09_05_1_2_1[]" id="input_lect_09_05_1_2_1_choice_3" class="field-input input-checkbox" value="choice_3"/><label id="lect_09_05_1_2_1-choice_3-label" for="input_lect_09_05_1_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lect_09_05_1_2_1">
<text>d) The separation, [mathjaxinline]a[/mathjaxinline], goes to a constant, nonzero value.</text>
</label>
</div>
<span id="answer_lect_09_05_1_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_lect_09_05_1_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_lect_09_05_1_solution_1"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Defining Mass Density" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lect_09_05_1" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_09_05_1">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_09_05_1-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_05_1-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_05_1-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>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-runtime-class="LmsRuntime" data-request-token="3815f498685311ef86120214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@vertical+block@vert-lect_09_ex06" data-block-type="vertical" data-graded="True" data-init="VerticalStudentView" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<h2 class="hd hd-2 unit-title">L13Q5: Wave Equation and Velocity</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_09_06_0a">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-runtime-class="LmsRuntime" data-request-token="3815f498685311ef86120214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_09_06_0a" data-block-type="problem" data-graded="True" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_09_06_0a" class="problems-wrapper" role="group"
aria-labelledby="lect_09_06_0a-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_09_06_0a" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_09_06_0a/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_09_06_0a-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_09_06_0a-problem-progress" tabindex="-1">
Wave Equation and Velocity - part a
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_09_06_0a-problem-progress"></div>
<div class="problem">
<div>
<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>
</tr>
</table>
<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>
</tr>
</table>
<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>
<p>
<p style="display:inline">[mathjaxinline]v_{p} =[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_lect_09_06_0a_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_lect_09_06_0a_2_1" id="input_lect_09_06_0a_2_1" aria-describedby="status_lect_09_06_0a_2_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_lect_09_06_0a_2_1"/>
<span class="status unanswered" id="status_lect_09_06_0a_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_lect_09_06_0a_2_1" class="answer"/>
<div id="display_lect_09_06_0a_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_lect_09_06_0a_2_1_dynamath" name="input_lect_09_06_0a_2_1_dynamath"/>
</div>
</div></div>
</p>
<span>
<link href="/assets/courseware/v1/5558929dbdda0f3a399b6940d9ab0281/asset-v1:MITx+8.03x+1T2020+type@asset+block/css_mymodal.css" rel="stylesheet" type="text/css"/>
<div align="right">
<a href="#mymodal-one" class="btn btn-default">Input Help
</a>
</div>
<div class="mymodal-positioner">
<a href="#" class="mymodal" id="mymodal-one" aria-hidden="true"/>
<div class="mymodal-dialog">
<div class="mymodal-header">
<h4>Input Help</h4>
<a href="#" class="mymodal-btn-close">&#215;</a>
</div>
<div class="formulainput">
<table class="formulainput">
<tbody>
<tr class="fiptitle">
<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<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>
</tr>
</tbody>
</table>
</div>
<div class="mymodal-footer">
<a href="#" class="btn btn-primary" style="color:#FFFFFF;">Close</a>
</div>
</div>
</div>
</span>
<p>
<div class="solution-span">
<span id="solution_lect_09_06_0a_solution_1"/>
</div><p style="margin-bottom: 0px; margin-top: 0px; display: block; padding-bottom: 20px;" class="gap"/>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Wave Equation and Velocity - part a" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lect_09_06_0a" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_09_06_0a">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="lect_09_06_0a-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_0a-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_0a-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 class="vert vert-1" data-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_09_06_0b">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-runtime-class="LmsRuntime" data-request-token="3815f498685311ef86120214bd5530e3" data-usage-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_09_06_0b" data-block-type="problem" data-graded="True" data-init="XBlockToXModuleShim" data-course-id="course-v1:MITx+8.03x+1T2020" data-runtime-version="1">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_lect_09_06_0b" class="problems-wrapper" role="group"
aria-labelledby="lect_09_06_0b-problem-title"
data-problem-id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_09_06_0b" data-url="/courses/course-v1:MITx+8.03x+1T2020/xblock/block-v1:MITx+8.03x+1T2020+type@problem+block@lect_09_06_0b/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="1.0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="lect_09_06_0b-problem-title" aria-describedby="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_09_06_0b-problem-progress" tabindex="-1">
Wave Equation and Velocity - part b
</h3>
<div class="problem-progress" id="block-v1:MITx+8.03x+1T2020+type@problem+block@lect_09_06_0b-problem-progress"></div>
<div class="problem">
<div>
<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>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_lect_09_06_0b_2_1">
<fieldset aria-describedby="status_lect_09_06_0b_2_1">
<div class="field">
<input type="radio" name="input_lect_09_06_0b_2_1" id="input_lect_09_06_0b_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="lect_09_06_0b_2_1-choice_1-label" for="input_lect_09_06_0b_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_lect_09_06_0b_2_1">
<text> a) velocity increases by multiplicative factor of [mathjaxinline]2[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_09_06_0b_2_1" id="input_lect_09_06_0b_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="lect_09_06_0b_2_1-choice_2-label" for="input_lect_09_06_0b_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_lect_09_06_0b_2_1">
<text> b) velocity increases by multiplicative factor of [mathjaxinline]\sqrt {2}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_09_06_0b_2_1" id="input_lect_09_06_0b_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="lect_09_06_0b_2_1-choice_3-label" for="input_lect_09_06_0b_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_lect_09_06_0b_2_1">
<text> c) velocity decreases by multiplicative factor of [mathjaxinline]\frac{1}{2}[/mathjaxinline]</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_lect_09_06_0b_2_1" id="input_lect_09_06_0b_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="lect_09_06_0b_2_1-choice_4-label" for="input_lect_09_06_0b_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_lect_09_06_0b_2_1">
<text> d) velocity decreases by multiplicative factor of [mathjaxinline]\frac{1}{\sqrt {2}}[/mathjaxinline]</text>
</label>
</div>
<span id="answer_lect_09_06_0b_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_lect_09_06_0b_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_lect_09_06_0b_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Wave Equation and Velocity - part b" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_lect_09_06_0b" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_lect_09_06_0b">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<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>