<div class="xblock xblock-public_view xblock-public_view-vertical" data-init="VerticalStudentView" data-has-score="False" data-runtime-version="1" data-graded="True" data-usage-id="block-v1:MITx+8.01.4x+1T2019+type@vertical+block@vert-L15_html_intro" data-runtime-class="LmsRuntime" data-block-type="vertical" data-request-token="0a2d133e038811efb62b02329aca76dd" data-course-id="course-v1:MITx+8.01.4x+1T2019">
<h2 class="hd hd-2 unit-title">L42: Introduction to the Pendulum and Small Angle Approximation</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.01.4x+1T2019+type@html+block@vertical:W15:L42intro">
<div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-init="XBlockToXModuleShim" data-has-score="False" data-runtime-version="1" data-graded="True" data-usage-id="block-v1:MITx+8.01.4x+1T2019+type@html+block@vertical:W15:L42intro" data-runtime-class="LmsRuntime" data-block-type="html" data-request-token="0a2d133e038811efb62b02329aca76dd" data-course-id="course-v1:MITx+8.01.4x+1T2019">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "HTMLModule"}
</script>
<center><p><b> Small Angle Approximation</b></p></center><p> In the previous week, we explored Simple Harmonic Motion - which was the motion that resulted from a linear restoring conservative force acting on an object. These exact conditions are only met in very limited situations, but they are approximately met in many more situations. This week we explore situations where we can approximate the motion as coming from a linear restoring force (or a potential energy that is square in the position variable).</p><p>To begin this week, we review the Taylor Formula. This will be a valuable tool for us to approximate the force or potential energy. The Taylor Formula is a way to rewrite a function as an infinite series, constructed by taking derivatives of the function and expanding around a certain point. If the motion stays close to that central point that the Taylor Series is taken around, then the function can be well approximated by only the first few terms. We will see that in these first few terms, the familiar formula for Simple Harmonic Motion will emerge.</p><p>In this lesson we will focus on the motion of a pendulum. We will show how we can approach this problem many different ways, force, energy, or torque, and still arrive at the same equation of motion when the angle of motion is very small. We start with a simple pendulum - a mass at the end of a massless rod - and then move on to the more complicated physical pendulum with mass distributed throughout the length.</p>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-init="VerticalStudentView" data-has-score="False" data-runtime-version="1" data-graded="True" data-usage-id="block-v1:MITx+8.01.4x+1T2019+type@vertical+block@vert-L42v01" data-runtime-class="LmsRuntime" data-block-type="vertical" data-request-token="0a2d133e038811efb62b02329aca76dd" data-course-id="course-v1:MITx+8.01.4x+1T2019">
<h2 class="hd hd-2 unit-title">L42v1: Expand Cosine with the Taylor Formula</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.01.4x+1T2019+type@video+block@L42v01">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-init="XBlockToXModuleShim" data-has-score="False" data-runtime-version="1" data-graded="True" data-usage-id="block-v1:MITx+8.01.4x+1T2019+type@video+block@L42v01" data-runtime-class="LmsRuntime" data-block-type="video" data-request-token="0a2d133e038811efb62b02329aca76dd" data-course-id="course-v1:MITx+8.01.4x+1T2019">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L42v1: Expanding Cosine with the Taylor Formula</h3>
<div
id="video_L42v01"
class="video closed"
data-metadata='{"speed": null, "saveStateEnabled": false, "streams": "1.00:bT6Qb0wJPus", "savedVideoPosition": 0.0, "ytTestTimeout": 1500, "generalSpeed": 1.0, "poster": "https://mitx-edx-video-meta-storage.s3.amazonaws.com/media/video-images/54352893f6064d29b317b1be6340aedb.png?Expires=1714107129&AWSAccessKeyId=ASIA4QDFFSURWFHTV653&Signature=fPADHcNaIiRXjZLtejwyoZ79sj4%3D&x-amz-security-token=IQoJb3JpZ2luX2VjEG0aCXVzLWVhc3QtMSJGMEQCIG9RZAb1J8GI9%2BhQpxrjXiWznrDyfsktvRlNrjme6oztAiA6rhY391LbQm4h2YeXw2MGjMGw3SRXQ%2BfWht7FPe6vzSrDBQi2%2F%2F%2F%2F%2F%2F%2F%2F%2F%2F8BEAMaDDg1OTIwNTYzNzQxMSIMCZiZiNi687vp3dm7KpcF4n9eJc4CVj5q9cGJLyBUWfvRcSEMZLzdCBZPkBa5CAarcNGMD6WS%2BT2JBmGcjukpPKN%2FJjv6bLrn755vjh%2BbFof1%2FRmiJ8Fs8nhOvjvTOznSOIYHPKu42Ea8IUgfwntjQaxSrIasVJQNiPhA6KPOrV67l1HUmFKF4KyidUbuBdDGvD5X7D%2F%2F7aIwOUVjcCBlDlMmuZokl%2Be499k13KrwrkBpY%2FMVJ4uthX7I8quRA2JsqktY86TWlbEa9G9m%2FdFfJbvtZkO%2F%2Fd9q70%2BKU7PZxFgV4hcKjQOulzckD1O5UTRks1xRoOKC%2BZr9WiH5pfe3wa%2FM74%2BCS4pD4MTpOgBGX5xJ4G39nMvjyUguSDGykc8XES4lhUD2%2Fj3sV83nSeNejU6gFo0jxrLRjMuuO86ELZiuIqlxQ%2FXlk%2BEhbBvgGG0DD4Bm5BDJPyKGDHp927NlxsiDsy32eQ%2FR%2F7xORLcB7l2JxmP8ibbDjkpO5ITqExAnlhioh5qQJ4%2BjYHQP6T9X0JfzLiMC2REVn1hVuxIbr3LuJDNHRMMDUx8FRwIro0XnN%2BhE%2BqoyIOSstBKZnHKit4n5%2B0W6JCxKjLA9mKM1rIk06SNqI47U6lwpncwpZCBOErDNKkyMrJ1xxe%2FyCoXs3N5OB2iGB3GZnlyEeNUmn8ZCyTMLRSF5elwA7o84Oe%2FZOmUH8Jwa8Z2yl%2FFcNkkF5TLYwRKLUIqOwbdkGbnZbWANcQEtSEIw%2Fk1MUy%2F5iHOVg%2F3rbIkWW6O325nVXqhq5A87RXkiLqvhtiC0SqyHtNvNgGbGreQjHbHRiyuQRjzstADFOQ3Gn96LxiP%2FNoKK6619WLCFexqdKz%2BJ%2B0ZwUQDm0LNKenqgwIw5DHrlUuKULWjtSjwZMNHYrLEGOrIBiysEq4lkIE3m0NJ3dXmeUq2n4b5Hi6YpOUW3igSQBKSv0S9grSeQ74KDG5EdVMboPuSligPLbgMoA9QGzL02viywFMCvE0JmwoZoXzWExCu7w%2FhTQIsC2sY586BexbxKlAkN06GYNciSJyV0csRMru5JssXZYoWRW6X7JJdqNyEsUS1FUFJD%2BNmKIyNMIkoKVfhxN7%2FzxpRibnviqnHkMTGteKqMzY4c8CfefgsGvQ8vzw%3D%3D", "publishCompletionUrl": "/courses/course-v1:MITx+8.01.4x+1T2019/xblock/block-v1:MITx+8.01.4x+1T2019+type@video+block@L42v01/handler/publish_completion", "completionEnabled": false, "autoplay": false, "completionPercentage": 0.95, "captionDataDir": null, "duration": 432.07, "saveStateUrl": "/courses/course-v1:MITx+8.01.4x+1T2019/xblock/block-v1:MITx+8.01.4x+1T2019+type@video+block@L42v01/handler/xmodule_handler/save_user_state", "ytApiUrl": "https://www.youtube.com/iframe_api", "start": 0.0, "transcriptLanguage": "en", "lmsRootURL": "https://openlearninglibrary.mit.edu", "prioritizeHls": false, "autoAdvance": false, "transcriptLanguages": {"en": "English"}, "autohideHtml5": false, "transcriptTranslationUrl": "/courses/course-v1:MITx+8.01.4x+1T2019/xblock/block-v1:MITx+8.01.4x+1T2019+type@video+block@L42v01/handler/transcript/translation/__lang__", "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.01.4x+1T2019/xblock/block-v1:MITx+8.01.4x+1T2019+type@video+block@L42v01/handler/transcript/available_translations", "ytMetadataEndpoint": "", "recordedYoutubeIsAvailable": true, "sources": ["https://d2f1egay8yehza.cloudfront.net/MITx8.01.4x-V001900_DTH.mp4", "https://d2f1egay8yehza.cloudfront.net/MITx8.01.4x-V001900/MITx8.01.4x-V001900.m3u8"], "showCaptions": "true", "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="L42v01"></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_L42v01">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L42v01">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+8.01.4x+1T2019/xblock/block-v1:MITx+8.01.4x+1T2019+type@video+block@L42v01/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+8.01.4x+1T2019/xblock/block-v1:MITx+8.01.4x+1T2019+type@video+block@L42v01/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-init="VerticalStudentView" data-has-score="False" data-runtime-version="1" data-graded="True" data-usage-id="block-v1:MITx+8.01.4x+1T2019+type@vertical+block@vert-Mathematica_prepset14" data-runtime-class="LmsRuntime" data-block-type="vertical" data-request-token="0a2d133e038811efb62b02329aca76dd" data-course-id="course-v1:MITx+8.01.4x+1T2019">
<h2 class="hd hd-2 unit-title">L42Q1: Taylor Series</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@Mathematica_prepset14">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-init="XBlockToXModuleShim" data-has-score="True" data-runtime-version="1" data-graded="True" data-usage-id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@Mathematica_prepset14" data-runtime-class="LmsRuntime" data-block-type="problem" data-request-token="0a2d133e038811efb62b02329aca76dd" data-course-id="course-v1:MITx+8.01.4x+1T2019">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_Mathematica_prepset14" class="problems-wrapper" role="group"
aria-labelledby="Mathematica_prepset14-problem-title"
data-problem-id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@Mathematica_prepset14" data-url="/courses/course-v1:MITx+8.01.4x+1T2019/xblock/block-v1:MITx+8.01.4x+1T2019+type@problem+block@Mathematica_prepset14/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="Mathematica_prepset14-problem-title" aria-describedby="block-v1:MITx+8.01.4x+1T2019+type@problem+block@Mathematica_prepset14-problem-progress" tabindex="-1">
Approximate a function by a polynomial, part 1
</h3>
<div class="problem-progress" id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@Mathematica_prepset14-problem-progress"></div>
<div class="problem">
<div>
<p>
Approximating a function by a sum of simple polynomials is extremely useful not only in physics but also in many other classes. This is done with the Taylor series. A function [mathjaxinline]f(x)[/mathjaxinline] around [mathjaxinline]x=a[/mathjaxinline] can be expressed as: </p>
<table id="a0000000002" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
<tr id="a0000000003">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle f(x)=f(a)+f^{'}(a)(x-a)+f^{''}(a)\frac{(x-a)^2}{2!}+.....+f^ n(a)\frac{(x-a)^ n}{n!}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">&#160;</td>
</tr>
</table>
<p><b class="bfseries">(Part a)</b> Calculate the first four coefficients of the Taylor series of the function [mathjaxinline]\sin (x)[/mathjaxinline] around [mathjaxinline]x=0[/mathjaxinline]. </p>
<p>
<p style="display:inline"><b class="bfseries">Zero order:</b> [mathjaxinline]f(a)=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_Mathematica_prepset14_2_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_Mathematica_prepset14_2_1" id="input_Mathematica_prepset14_2_1" aria-describedby="status_Mathematica_prepset14_2_1" value="" size="10"/>
<span class="trailing_text" id="trailing_text_Mathematica_prepset14_2_1"/>
<span class="status unanswered" id="status_Mathematica_prepset14_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_Mathematica_prepset14_2_1" class="answer"/>
</div>
</div></div>
</p>
<p>
<p style="display:inline"><b class="bfseries">First order:</b> [mathjaxinline]f^{'}(a)=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_Mathematica_prepset14_3_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_Mathematica_prepset14_3_1" id="input_Mathematica_prepset14_3_1" aria-describedby="status_Mathematica_prepset14_3_1" value="" size="10"/>
<span class="trailing_text" id="trailing_text_Mathematica_prepset14_3_1"/>
<span class="status unanswered" id="status_Mathematica_prepset14_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_Mathematica_prepset14_3_1" class="answer"/>
</div>
</div></div>
</p>
<p>
<p style="display:inline"><b class="bfseries">Second order:</b> [mathjaxinline]\frac{f^{''}(a)}{2!}=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_Mathematica_prepset14_4_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_Mathematica_prepset14_4_1" id="input_Mathematica_prepset14_4_1" aria-describedby="status_Mathematica_prepset14_4_1" value="" size="10"/>
<span class="trailing_text" id="trailing_text_Mathematica_prepset14_4_1"/>
<span class="status unanswered" id="status_Mathematica_prepset14_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_Mathematica_prepset14_4_1" class="answer"/>
</div>
</div></div>
</p>
<p>
<p style="display:inline"><b class="bfseries">Third order:</b> [mathjaxinline]\frac{f^{'''}(a)}{3!}=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 4" role="group"><div id="inputtype_Mathematica_prepset14_5_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_Mathematica_prepset14_5_1" id="input_Mathematica_prepset14_5_1" aria-describedby="status_Mathematica_prepset14_5_1" value="" size="10"/>
<span class="trailing_text" id="trailing_text_Mathematica_prepset14_5_1"/>
<span class="status unanswered" id="status_Mathematica_prepset14_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_Mathematica_prepset14_5_1" class="answer"/>
</div>
</div></div>
</p>
<p>
Approximate the function [mathjaxinline]\sin (x)[/mathjaxinline] around [mathjaxinline]x=0[/mathjaxinline] as a two term polynomial. Express your answer in terms of [mathjaxinline]x[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]\sin (x)=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 5" role="group"><div id="inputtype_Mathematica_prepset14_6_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_Mathematica_prepset14_6_1" id="input_Mathematica_prepset14_6_1" aria-describedby="status_Mathematica_prepset14_6_1" value="" class="math" size="25"/>
<span class="trailing_text" id="trailing_text_Mathematica_prepset14_6_1"/>
<span class="status unanswered" id="status_Mathematica_prepset14_6_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_Mathematica_prepset14_6_1" class="answer"/>
<div id="display_Mathematica_prepset14_6_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_Mathematica_prepset14_6_1_dynamath" name="input_Mathematica_prepset14_6_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_Mathematica_prepset14_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Approximate a function by a polynomial, part 1" />
<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_Mathematica_prepset14" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_Mathematica_prepset14">
<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="Mathematica_prepset14-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="Mathematica_prepset14-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="Mathematica_prepset14-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+8.01.4x+1T2019+type@problem+block@Mathematica_prepset14_2">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-init="XBlockToXModuleShim" data-has-score="True" data-runtime-version="1" data-graded="True" data-usage-id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@Mathematica_prepset14_2" data-runtime-class="LmsRuntime" data-block-type="problem" data-request-token="0a2d133e038811efb62b02329aca76dd" data-course-id="course-v1:MITx+8.01.4x+1T2019">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_Mathematica_prepset14_2" class="problems-wrapper" role="group"
aria-labelledby="Mathematica_prepset14_2-problem-title"
data-problem-id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@Mathematica_prepset14_2" data-url="/courses/course-v1:MITx+8.01.4x+1T2019/xblock/block-v1:MITx+8.01.4x+1T2019+type@problem+block@Mathematica_prepset14_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="Mathematica_prepset14_2-problem-title" aria-describedby="block-v1:MITx+8.01.4x+1T2019+type@problem+block@Mathematica_prepset14_2-problem-progress" tabindex="-1">
Approximate a function by a polynomial, part 2
</h3>
<div class="problem-progress" id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@Mathematica_prepset14_2-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bfseries">(Part b)</b> The number of terms in the Taylor series needed to approximate a function depends on the desired accuracy. One way to measure the accuracy of a variable is using the relative error. Suppose that [mathjaxinline]y_{app}[/mathjaxinline] is the approximation of the variable [mathjaxinline]y[/mathjaxinline], the relative error is defined as: </p>
<table id="a0000000006" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
<tr id="a0000000007">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \epsilon =\left| \frac{y-y_{app}}{y}\right| \times 100[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none" class="eqnnum">&#160;</td>
</tr>
</table>
<p>
For an accuracy of 1%, calculate [mathjaxinline]\theta _{max}[/mathjaxinline], the maximum value of [mathjaxinline]\theta[/mathjaxinline] you can use if the [mathjaxinline]\sin (\theta )[/mathjaxinline] is expressed in terms of a Taylor series with only one term? Express your answer in degrees. </p>
<p>
<p style="display:inline">[mathjaxinline]\theta _{max} =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_Mathematica_prepset14_2_2_1" class=" capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_Mathematica_prepset14_2_2_1" id="input_Mathematica_prepset14_2_2_1" aria-describedby="trailing_text_Mathematica_prepset14_2_2_1 status_Mathematica_prepset14_2_2_1" value="" size="10"/>
<span class="trailing_text" id="trailing_text_Mathematica_prepset14_2_2_1">[mathjaxinline]\mathrm{^ o}[/mathjaxinline]</span>
<span class="status unanswered" id="status_Mathematica_prepset14_2_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_Mathematica_prepset14_2_2_1" class="answer"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_Mathematica_prepset14_2_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Approximate a function by a polynomial, part 2" />
<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_Mathematica_prepset14_2" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_Mathematica_prepset14_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="Mathematica_prepset14_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="Mathematica_prepset14_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="Mathematica_prepset14_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-init="VerticalStudentView" data-has-score="False" data-runtime-version="1" data-graded="True" data-usage-id="block-v1:MITx+8.01.4x+1T2019+type@vertical+block@vert-L42v02" data-runtime-class="LmsRuntime" data-block-type="vertical" data-request-token="0a2d133e038811efb62b02329aca76dd" data-course-id="course-v1:MITx+8.01.4x+1T2019">
<h2 class="hd hd-2 unit-title">L42v2: Simple Pendulum - Force Method</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_05">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-init="XBlockToXModuleShim" data-has-score="True" data-runtime-version="1" data-graded="True" data-usage-id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_05" data-runtime-class="LmsRuntime" data-block-type="problem" data-request-token="0a2d133e038811efb62b02329aca76dd" data-course-id="course-v1:MITx+8.01.4x+1T2019">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_ls_ls43_ls43_05" class="problems-wrapper" role="group"
aria-labelledby="ls_ls43_ls43_05-problem-title"
data-problem-id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_05" data-url="/courses/course-v1:MITx+8.01.4x+1T2019/xblock/block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_05/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="ls_ls43_ls43_05-problem-title" aria-describedby="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_05-problem-progress" tabindex="-1">
Simple pendulum, Force method.
</h3>
<div class="problem-progress" id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_05-problem-progress"></div>
<div class="problem">
<div>
<center>
<img src="/assets/courseware/v1/abb6631a6c19a42047620e1929be530f/asset-v1:MITx+8.01.4x+1T2019+type@asset+block/images_ls43_05.svg" width="330"/>
</center>
<p>
A small ball of mass [mathjaxinline]m[/mathjaxinline] is attached to the free end of an ideal string of length [mathjaxinline]l[/mathjaxinline] that is hanging from the ceiling at point [mathjaxinline]S[/mathjaxinline]. The ball is moved away from the vertical and released. At the instant shown in the figure, the ball is at an angle [mathjaxinline]\theta (t)[/mathjaxinline] with respect to the vertical. Suppose the angle [mathjaxinline]\theta[/mathjaxinline] is small throughout the motion. </p>
<p><b class="bfseries">(Part a)</b> Consider the coordinate system shown in the figure. If [mathjaxinline]T[/mathjaxinline] is the magnitude of the force of tension exerted by the string on the ball, calculate [mathjaxinline]\Sigma F_{\theta }[/mathjaxinline], the [mathjaxinline]\hat{\mathbf{\theta }}[/mathjaxinline]-component of the total force acting on the ball. Express your answer in terms of [mathjaxinline]m[/mathjaxinline], [mathjaxinline]g[/mathjaxinline], [mathjaxinline]l[/mathjaxinline], and theta for [mathjaxinline]\theta[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]\Sigma F_{\theta }=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ls_ls43_ls43_05_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_ls_ls43_ls43_05_2_1" id="input_ls_ls43_ls43_05_2_1" aria-describedby="status_ls_ls43_ls43_05_2_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_ls_ls43_ls43_05_2_1"/>
<span class="status unanswered" id="status_ls_ls43_ls43_05_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ls_ls43_ls43_05_2_1" class="answer"/>
<div id="display_ls_ls43_ls43_05_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ls_ls43_ls43_05_2_1_dynamath" name="input_ls_ls43_ls43_05_2_1_dynamath"/>
</div>
</div></div>
<div class="solution-span">
<span id="solution_ls_ls43_ls43_05_solution_1"/>
</div></p>
<p><b class="bfseries">(Part b)</b> Express [mathjaxinline]a_{\theta }[/mathjaxinline], the tangential component of the acceleration in terms of [mathjaxinline]\alpha = \dfrac {d^2\theta }{dt^2}[/mathjaxinline], the angular acceleration. Write your answer in terms of [mathjaxinline]l[/mathjaxinline] and alpha for [mathjaxinline]\alpha[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]a_{\theta }=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_ls_ls43_ls43_05_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_ls_ls43_ls43_05_3_1" id="input_ls_ls43_ls43_05_3_1" aria-describedby="status_ls_ls43_ls43_05_3_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_ls_ls43_ls43_05_3_1"/>
<span class="status unanswered" id="status_ls_ls43_ls43_05_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ls_ls43_ls43_05_3_1" class="answer"/>
<div id="display_ls_ls43_ls43_05_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ls_ls43_ls43_05_3_1_dynamath" name="input_ls_ls43_ls43_05_3_1_dynamath"/>
</div>
</div></div>
<div class="solution-span">
<span id="solution_ls_ls43_ls43_05_solution_2"/>
</div></p>
<p><b class="bfseries">(Part c)</b> Apply Newton's 2nd law to the ball to obtain an expression for [mathjaxinline]\alpha = \dfrac {d^2\theta }{dt^2}[/mathjaxinline], the angular acceleration of the ball in terms of [mathjaxinline]m[/mathjaxinline], [mathjaxinline]l[/mathjaxinline], [mathjaxinline]g[/mathjaxinline], theta for [mathjaxinline]\theta[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]\dfrac {d^2\theta }{dt^2}=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_ls_ls43_ls43_05_4_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_ls_ls43_ls43_05_4_1" id="input_ls_ls43_ls43_05_4_1" aria-describedby="status_ls_ls43_ls43_05_4_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_ls_ls43_ls43_05_4_1"/>
<span class="status unanswered" id="status_ls_ls43_ls43_05_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ls_ls43_ls43_05_4_1" class="answer"/>
<div id="display_ls_ls43_ls43_05_4_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ls_ls43_ls43_05_4_1_dynamath" name="input_ls_ls43_ls43_05_4_1_dynamath"/>
</div>
</div></div>
<div class="solution-span">
<span id="solution_ls_ls43_ls43_05_solution_3"/>
</div></p>
<p><b class="bfseries">(Part d)</b> Assuming that the amplitude of the oscillation is small, calculate [mathjaxinline]\omega _0[/mathjaxinline], the angular frequency of oscillation. Express your answer in terms of [mathjaxinline]m[/mathjaxinline], [mathjaxinline]g[/mathjaxinline], [mathjaxinline]l[/mathjaxinline], and theta for [mathjaxinline]\theta[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]\omega _0=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 4" role="group"><div id="inputtype_ls_ls43_ls43_05_5_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_ls_ls43_ls43_05_5_1" id="input_ls_ls43_ls43_05_5_1" aria-describedby="status_ls_ls43_ls43_05_5_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_ls_ls43_ls43_05_5_1"/>
<span class="status unanswered" id="status_ls_ls43_ls43_05_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ls_ls43_ls43_05_5_1" class="answer"/>
<div id="display_ls_ls43_ls43_05_5_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ls_ls43_ls43_05_5_1_dynamath" name="input_ls_ls43_ls43_05_5_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_ls_ls43_ls43_05_solution_4"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Simple pendulum, Force method." />
<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_ls_ls43_ls43_05" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_ls_ls43_ls43_05">
<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="ls_ls43_ls43_05-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="ls_ls43_ls43_05-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="ls_ls43_ls43_05-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+8.01.4x+1T2019+type@video+block@L42v02">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-init="XBlockToXModuleShim" data-has-score="False" data-runtime-version="1" data-graded="True" data-usage-id="block-v1:MITx+8.01.4x+1T2019+type@video+block@L42v02" data-runtime-class="LmsRuntime" data-block-type="video" data-request-token="0a2d133e038811efb62b02329aca76dd" data-course-id="course-v1:MITx+8.01.4x+1T2019">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L42v2: Simple Pendulum - Force method</h3>
<div
id="video_L42v02"
class="video closed"
data-metadata='{"speed": null, "saveStateEnabled": false, "streams": "1.00:OxzA1JGOS2U", "savedVideoPosition": 0.0, "ytTestTimeout": 1500, "generalSpeed": 1.0, "poster": "https://mitx-edx-video-meta-storage.s3.amazonaws.com/media/video-images/c1d72c47d9b448239498d5b36dbe3aa7.png?Expires=1714107130&AWSAccessKeyId=ASIA4QDFFSURWFHTV653&Signature=BNrObJ%2BObYv5D1yDt3BoQlT2lSs%3D&x-amz-security-token=IQoJb3JpZ2luX2VjEG0aCXVzLWVhc3QtMSJGMEQCIG9RZAb1J8GI9%2BhQpxrjXiWznrDyfsktvRlNrjme6oztAiA6rhY391LbQm4h2YeXw2MGjMGw3SRXQ%2BfWht7FPe6vzSrDBQi2%2F%2F%2F%2F%2F%2F%2F%2F%2F%2F8BEAMaDDg1OTIwNTYzNzQxMSIMCZiZiNi687vp3dm7KpcF4n9eJc4CVj5q9cGJLyBUWfvRcSEMZLzdCBZPkBa5CAarcNGMD6WS%2BT2JBmGcjukpPKN%2FJjv6bLrn755vjh%2BbFof1%2FRmiJ8Fs8nhOvjvTOznSOIYHPKu42Ea8IUgfwntjQaxSrIasVJQNiPhA6KPOrV67l1HUmFKF4KyidUbuBdDGvD5X7D%2F%2F7aIwOUVjcCBlDlMmuZokl%2Be499k13KrwrkBpY%2FMVJ4uthX7I8quRA2JsqktY86TWlbEa9G9m%2FdFfJbvtZkO%2F%2Fd9q70%2BKU7PZxFgV4hcKjQOulzckD1O5UTRks1xRoOKC%2BZr9WiH5pfe3wa%2FM74%2BCS4pD4MTpOgBGX5xJ4G39nMvjyUguSDGykc8XES4lhUD2%2Fj3sV83nSeNejU6gFo0jxrLRjMuuO86ELZiuIqlxQ%2FXlk%2BEhbBvgGG0DD4Bm5BDJPyKGDHp927NlxsiDsy32eQ%2FR%2F7xORLcB7l2JxmP8ibbDjkpO5ITqExAnlhioh5qQJ4%2BjYHQP6T9X0JfzLiMC2REVn1hVuxIbr3LuJDNHRMMDUx8FRwIro0XnN%2BhE%2BqoyIOSstBKZnHKit4n5%2B0W6JCxKjLA9mKM1rIk06SNqI47U6lwpncwpZCBOErDNKkyMrJ1xxe%2FyCoXs3N5OB2iGB3GZnlyEeNUmn8ZCyTMLRSF5elwA7o84Oe%2FZOmUH8Jwa8Z2yl%2FFcNkkF5TLYwRKLUIqOwbdkGbnZbWANcQEtSEIw%2Fk1MUy%2F5iHOVg%2F3rbIkWW6O325nVXqhq5A87RXkiLqvhtiC0SqyHtNvNgGbGreQjHbHRiyuQRjzstADFOQ3Gn96LxiP%2FNoKK6619WLCFexqdKz%2BJ%2B0ZwUQDm0LNKenqgwIw5DHrlUuKULWjtSjwZMNHYrLEGOrIBiysEq4lkIE3m0NJ3dXmeUq2n4b5Hi6YpOUW3igSQBKSv0S9grSeQ74KDG5EdVMboPuSligPLbgMoA9QGzL02viywFMCvE0JmwoZoXzWExCu7w%2FhTQIsC2sY586BexbxKlAkN06GYNciSJyV0csRMru5JssXZYoWRW6X7JJdqNyEsUS1FUFJD%2BNmKIyNMIkoKVfhxN7%2FzxpRibnviqnHkMTGteKqMzY4c8CfefgsGvQ8vzw%3D%3D", "publishCompletionUrl": "/courses/course-v1:MITx+8.01.4x+1T2019/xblock/block-v1:MITx+8.01.4x+1T2019+type@video+block@L42v02/handler/publish_completion", "completionEnabled": false, "autoplay": false, "completionPercentage": 0.95, "captionDataDir": null, "duration": 467.0, "saveStateUrl": "/courses/course-v1:MITx+8.01.4x+1T2019/xblock/block-v1:MITx+8.01.4x+1T2019+type@video+block@L42v02/handler/xmodule_handler/save_user_state", "ytApiUrl": "https://www.youtube.com/iframe_api", "start": 0.0, "transcriptLanguage": "en", "lmsRootURL": "https://openlearninglibrary.mit.edu", "prioritizeHls": false, "autoAdvance": false, "transcriptLanguages": {"en": "English"}, "autohideHtml5": false, "transcriptTranslationUrl": "/courses/course-v1:MITx+8.01.4x+1T2019/xblock/block-v1:MITx+8.01.4x+1T2019+type@video+block@L42v02/handler/transcript/translation/__lang__", "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.01.4x+1T2019/xblock/block-v1:MITx+8.01.4x+1T2019+type@video+block@L42v02/handler/transcript/available_translations", "ytMetadataEndpoint": "", "recordedYoutubeIsAvailable": true, "sources": ["https://d2f1egay8yehza.cloudfront.net/MITx8.01.4x-V002400_DTH.mp4", "https://d2f1egay8yehza.cloudfront.net/MITx8.01.4x-V002400/MITx8.01.4x-V002400.m3u8"], "showCaptions": "true", "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="L42v02"></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_L42v02">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L42v02">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+8.01.4x+1T2019/xblock/block-v1:MITx+8.01.4x+1T2019+type@video+block@L42v02/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+8.01.4x+1T2019/xblock/block-v1:MITx+8.01.4x+1T2019+type@video+block@L42v02/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-init="VerticalStudentView" data-has-score="False" data-runtime-version="1" data-graded="True" data-usage-id="block-v1:MITx+8.01.4x+1T2019+type@vertical+block@vert-L42v03" data-runtime-class="LmsRuntime" data-block-type="vertical" data-request-token="0a2d133e038811efb62b02329aca76dd" data-course-id="course-v1:MITx+8.01.4x+1T2019">
<h2 class="hd hd-2 unit-title">L42v3: Simple Pendulum - Energy Method</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_06">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-init="XBlockToXModuleShim" data-has-score="True" data-runtime-version="1" data-graded="True" data-usage-id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_06" data-runtime-class="LmsRuntime" data-block-type="problem" data-request-token="0a2d133e038811efb62b02329aca76dd" data-course-id="course-v1:MITx+8.01.4x+1T2019">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_ls_ls43_ls43_06" class="problems-wrapper" role="group"
aria-labelledby="ls_ls43_ls43_06-problem-title"
data-problem-id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_06" data-url="/courses/course-v1:MITx+8.01.4x+1T2019/xblock/block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_06/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="ls_ls43_ls43_06-problem-title" aria-describedby="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_06-problem-progress" tabindex="-1">
Simple pendulum, Energy method.
</h3>
<div class="problem-progress" id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_06-problem-progress"></div>
<div class="problem">
<div>
<center>
<img src="/assets/courseware/v1/0e1d5a117d01d6091f9c6fc0bc1f10d2/asset-v1:MITx+8.01.4x+1T2019+type@asset+block/images_ls43_06.svg" width="385"/>
</center>
<p>
A small ball of mass [mathjaxinline]m[/mathjaxinline] is attached to the free end of an ideal string of length [mathjaxinline]l[/mathjaxinline] that is hanging from the ceiling at point [mathjaxinline]S[/mathjaxinline]. The ball is moved away from the vertical and released. At the instant shown in the figure, the ball is at an angle [mathjaxinline]\theta (t)[/mathjaxinline] with respect to the vertical. Suppose the angle [mathjaxinline]\theta[/mathjaxinline] is small throughout the motion. </p>
<p><b class="bfseries">Note:</b> The derivation of the simple harmonic oscillator equation using the energy energy approach is shown in the video below, where the small angle approximation is applied to the potential energy right from the start. </p>
<p>
This problem will guide you to derive the simple harmonic oscillator equation but working with the exact expression of the potential function all the way until the end of the solution where the small angle approximation is applied. </p>
<p><b class="bfseries">(Part a)</b> Set the zero of gravitational potential energy at the position of the ball when it is at its lowest point of its trajectory. Calculate [mathjaxinline]U[/mathjaxinline], the potential energy of the ball at the instant shown in the figure. Express your answer in terms of [mathjaxinline]m[/mathjaxinline], [mathjaxinline]g[/mathjaxinline], [mathjaxinline]l[/mathjaxinline], and theta for [mathjaxinline]\theta[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]U=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ls_ls43_ls43_06_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_ls_ls43_ls43_06_2_1" id="input_ls_ls43_ls43_06_2_1" aria-describedby="status_ls_ls43_ls43_06_2_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_ls_ls43_ls43_06_2_1"/>
<span class="status unanswered" id="status_ls_ls43_ls43_06_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ls_ls43_ls43_06_2_1" class="answer"/>
<div id="display_ls_ls43_ls43_06_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ls_ls43_ls43_06_2_1_dynamath" name="input_ls_ls43_ls43_06_2_1_dynamath"/>
</div>
</div></div>
<div class="solution-span">
<span id="solution_ls_ls43_ls43_06_solution_1"/>
</div></p>
<p><b class="bfseries">(Part b)</b> At the instant shown in the figure the angular velocity of the ball is [mathjaxinline]\omega _ z = \dfrac {d\theta }{dt}[/mathjaxinline]. Calculate [mathjaxinline]K[/mathjaxinline], the kinetic energy of the ball in terms of [mathjaxinline]m[/mathjaxinline], [mathjaxinline]g[/mathjaxinline], [mathjaxinline]l[/mathjaxinline], theta for [mathjaxinline]\theta[/mathjaxinline], and omega_z for [mathjaxinline]\omega _ z = \dfrac {d\theta }{dt}[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]K=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_ls_ls43_ls43_06_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_ls_ls43_ls43_06_3_1" id="input_ls_ls43_ls43_06_3_1" aria-describedby="status_ls_ls43_ls43_06_3_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_ls_ls43_ls43_06_3_1"/>
<span class="status unanswered" id="status_ls_ls43_ls43_06_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ls_ls43_ls43_06_3_1" class="answer"/>
<div id="display_ls_ls43_ls43_06_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ls_ls43_ls43_06_3_1_dynamath" name="input_ls_ls43_ls43_06_3_1_dynamath"/>
</div>
</div></div>
<div class="solution-span">
<span id="solution_ls_ls43_ls43_06_solution_2"/>
</div></p>
<p><b class="bfseries">(Part c)</b> Calculate [mathjaxinline]\dfrac {dU}{dt}[/mathjaxinline], the time derivative of the potential energy. Express your answer in terms of [mathjaxinline]m[/mathjaxinline], [mathjaxinline]l[/mathjaxinline], [mathjaxinline]g[/mathjaxinline], theta for [mathjaxinline]\theta[/mathjaxinline], omega_z for [mathjaxinline]\omega _ z = \dfrac {d\theta }{dt}[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]\dfrac {dU}{dt}=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_ls_ls43_ls43_06_4_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_ls_ls43_ls43_06_4_1" id="input_ls_ls43_ls43_06_4_1" aria-describedby="status_ls_ls43_ls43_06_4_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_ls_ls43_ls43_06_4_1"/>
<span class="status unanswered" id="status_ls_ls43_ls43_06_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ls_ls43_ls43_06_4_1" class="answer"/>
<div id="display_ls_ls43_ls43_06_4_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ls_ls43_ls43_06_4_1_dynamath" name="input_ls_ls43_ls43_06_4_1_dynamath"/>
</div>
</div></div>
<div class="solution-span">
<span id="solution_ls_ls43_ls43_06_solution_3"/>
</div></p>
<p><b class="bfseries">(Part d)</b> Calculate [mathjaxinline]\dfrac {dK}{dt}[/mathjaxinline], the time derivative of the kinetic energy. Express your answer in terms of [mathjaxinline]m[/mathjaxinline], [mathjaxinline]l[/mathjaxinline], [mathjaxinline]g[/mathjaxinline], theta for [mathjaxinline]\theta[/mathjaxinline], omega_z for [mathjaxinline]\omega _ z = \dfrac {d\theta }{dt}[/mathjaxinline], and alpha for [mathjaxinline]\alpha = \dfrac {d^2\theta }{dt^2}[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]\dfrac {dK}{dt}=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 4" role="group"><div id="inputtype_ls_ls43_ls43_06_5_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_ls_ls43_ls43_06_5_1" id="input_ls_ls43_ls43_06_5_1" aria-describedby="status_ls_ls43_ls43_06_5_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_ls_ls43_ls43_06_5_1"/>
<span class="status unanswered" id="status_ls_ls43_ls43_06_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ls_ls43_ls43_06_5_1" class="answer"/>
<div id="display_ls_ls43_ls43_06_5_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ls_ls43_ls43_06_5_1_dynamath" name="input_ls_ls43_ls43_06_5_1_dynamath"/>
</div>
</div></div>
<div class="solution-span">
<span id="solution_ls_ls43_ls43_06_solution_4"/>
</div></p>
<p><b class="bfseries">(Part e)</b> The mechanical energy of the ball is constant therefore [mathjaxinline]\dfrac {dE}{dt} = 0[/mathjaxinline]. Starting from [mathjaxinline]\dfrac {dE}{dt} = 0[/mathjaxinline] obtain an equation for [mathjaxinline]\dfrac {d^2\theta }{dt^2}[/mathjaxinline], the angular acceleration of the ball. Express your answer in terms of [mathjaxinline]m[/mathjaxinline], [mathjaxinline]l[/mathjaxinline], [mathjaxinline]g[/mathjaxinline], theta for [mathjaxinline]\theta[/mathjaxinline], and omega_z for [mathjaxinline]\omega _ z = \dfrac {d\theta }{dt}[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]\dfrac {d^2\theta }{dt^2}=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 5" role="group"><div id="inputtype_ls_ls43_ls43_06_6_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_ls_ls43_ls43_06_6_1" id="input_ls_ls43_ls43_06_6_1" aria-describedby="status_ls_ls43_ls43_06_6_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_ls_ls43_ls43_06_6_1"/>
<span class="status unanswered" id="status_ls_ls43_ls43_06_6_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ls_ls43_ls43_06_6_1" class="answer"/>
<div id="display_ls_ls43_ls43_06_6_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ls_ls43_ls43_06_6_1_dynamath" name="input_ls_ls43_ls43_06_6_1_dynamath"/>
</div>
</div></div>
<div class="solution-span">
<span id="solution_ls_ls43_ls43_06_solution_5"/>
</div></p>
<p><b class="bfseries">(Part f)</b> Assuming that the amplitude of the oscillation is small, calculate [mathjaxinline]\omega _0[/mathjaxinline], the angular frequency of oscillation. Express your answer in terms of [mathjaxinline]m[/mathjaxinline], [mathjaxinline]g[/mathjaxinline], [mathjaxinline]l[/mathjaxinline], and theta for [mathjaxinline]\theta[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]\omega _0=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 6" role="group"><div id="inputtype_ls_ls43_ls43_06_7_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_ls_ls43_ls43_06_7_1" id="input_ls_ls43_ls43_06_7_1" aria-describedby="status_ls_ls43_ls43_06_7_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_ls_ls43_ls43_06_7_1"/>
<span class="status unanswered" id="status_ls_ls43_ls43_06_7_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ls_ls43_ls43_06_7_1" class="answer"/>
<div id="display_ls_ls43_ls43_06_7_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ls_ls43_ls43_06_7_1_dynamath" name="input_ls_ls43_ls43_06_7_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_ls_ls43_ls43_06_solution_6"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Simple pendulum, Energy method." />
<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_ls_ls43_ls43_06" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_ls_ls43_ls43_06">
<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="ls_ls43_ls43_06-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="ls_ls43_ls43_06-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="ls_ls43_ls43_06-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+8.01.4x+1T2019+type@video+block@L42v03">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-init="XBlockToXModuleShim" data-has-score="False" data-runtime-version="1" data-graded="True" data-usage-id="block-v1:MITx+8.01.4x+1T2019+type@video+block@L42v03" data-runtime-class="LmsRuntime" data-block-type="video" data-request-token="0a2d133e038811efb62b02329aca76dd" data-course-id="course-v1:MITx+8.01.4x+1T2019">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L42v3: Simple Pendulum - Energy method</h3>
<div
id="video_L42v03"
class="video closed"
data-metadata='{"speed": null, "saveStateEnabled": false, "streams": "1.00:HCjGIcV2Ve0", "savedVideoPosition": 0.0, "ytTestTimeout": 1500, "generalSpeed": 1.0, "poster": "https://mitx-edx-video-meta-storage.s3.amazonaws.com/media/video-images/07232ff1ea7e4bd1b1e3b094f321a5a2.png?Expires=1714107131&AWSAccessKeyId=ASIA4QDFFSURWFHTV653&Signature=ZMdwtrE07GlgmCxkOdHbIlySdlI%3D&x-amz-security-token=IQoJb3JpZ2luX2VjEG0aCXVzLWVhc3QtMSJGMEQCIG9RZAb1J8GI9%2BhQpxrjXiWznrDyfsktvRlNrjme6oztAiA6rhY391LbQm4h2YeXw2MGjMGw3SRXQ%2BfWht7FPe6vzSrDBQi2%2F%2F%2F%2F%2F%2F%2F%2F%2F%2F8BEAMaDDg1OTIwNTYzNzQxMSIMCZiZiNi687vp3dm7KpcF4n9eJc4CVj5q9cGJLyBUWfvRcSEMZLzdCBZPkBa5CAarcNGMD6WS%2BT2JBmGcjukpPKN%2FJjv6bLrn755vjh%2BbFof1%2FRmiJ8Fs8nhOvjvTOznSOIYHPKu42Ea8IUgfwntjQaxSrIasVJQNiPhA6KPOrV67l1HUmFKF4KyidUbuBdDGvD5X7D%2F%2F7aIwOUVjcCBlDlMmuZokl%2Be499k13KrwrkBpY%2FMVJ4uthX7I8quRA2JsqktY86TWlbEa9G9m%2FdFfJbvtZkO%2F%2Fd9q70%2BKU7PZxFgV4hcKjQOulzckD1O5UTRks1xRoOKC%2BZr9WiH5pfe3wa%2FM74%2BCS4pD4MTpOgBGX5xJ4G39nMvjyUguSDGykc8XES4lhUD2%2Fj3sV83nSeNejU6gFo0jxrLRjMuuO86ELZiuIqlxQ%2FXlk%2BEhbBvgGG0DD4Bm5BDJPyKGDHp927NlxsiDsy32eQ%2FR%2F7xORLcB7l2JxmP8ibbDjkpO5ITqExAnlhioh5qQJ4%2BjYHQP6T9X0JfzLiMC2REVn1hVuxIbr3LuJDNHRMMDUx8FRwIro0XnN%2BhE%2BqoyIOSstBKZnHKit4n5%2B0W6JCxKjLA9mKM1rIk06SNqI47U6lwpncwpZCBOErDNKkyMrJ1xxe%2FyCoXs3N5OB2iGB3GZnlyEeNUmn8ZCyTMLRSF5elwA7o84Oe%2FZOmUH8Jwa8Z2yl%2FFcNkkF5TLYwRKLUIqOwbdkGbnZbWANcQEtSEIw%2Fk1MUy%2F5iHOVg%2F3rbIkWW6O325nVXqhq5A87RXkiLqvhtiC0SqyHtNvNgGbGreQjHbHRiyuQRjzstADFOQ3Gn96LxiP%2FNoKK6619WLCFexqdKz%2BJ%2B0ZwUQDm0LNKenqgwIw5DHrlUuKULWjtSjwZMNHYrLEGOrIBiysEq4lkIE3m0NJ3dXmeUq2n4b5Hi6YpOUW3igSQBKSv0S9grSeQ74KDG5EdVMboPuSligPLbgMoA9QGzL02viywFMCvE0JmwoZoXzWExCu7w%2FhTQIsC2sY586BexbxKlAkN06GYNciSJyV0csRMru5JssXZYoWRW6X7JJdqNyEsUS1FUFJD%2BNmKIyNMIkoKVfhxN7%2FzxpRibnviqnHkMTGteKqMzY4c8CfefgsGvQ8vzw%3D%3D", "publishCompletionUrl": "/courses/course-v1:MITx+8.01.4x+1T2019/xblock/block-v1:MITx+8.01.4x+1T2019+type@video+block@L42v03/handler/publish_completion", "completionEnabled": false, "autoplay": false, "completionPercentage": 0.95, "captionDataDir": null, "duration": 324.53, "saveStateUrl": "/courses/course-v1:MITx+8.01.4x+1T2019/xblock/block-v1:MITx+8.01.4x+1T2019+type@video+block@L42v03/handler/xmodule_handler/save_user_state", "ytApiUrl": "https://www.youtube.com/iframe_api", "start": 0.0, "transcriptLanguage": "en", "lmsRootURL": "https://openlearninglibrary.mit.edu", "prioritizeHls": false, "autoAdvance": false, "transcriptLanguages": {"en": "English"}, "autohideHtml5": false, "transcriptTranslationUrl": "/courses/course-v1:MITx+8.01.4x+1T2019/xblock/block-v1:MITx+8.01.4x+1T2019+type@video+block@L42v03/handler/transcript/translation/__lang__", "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.01.4x+1T2019/xblock/block-v1:MITx+8.01.4x+1T2019+type@video+block@L42v03/handler/transcript/available_translations", "ytMetadataEndpoint": "", "recordedYoutubeIsAvailable": true, "sources": ["https://d2f1egay8yehza.cloudfront.net/MITx8.01.4x-V002100_DTH.mp4", "https://d2f1egay8yehza.cloudfront.net/MITx8.01.4x-V002100/MITx8.01.4x-V002100.m3u8"], "showCaptions": "true", "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="L42v03"></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_L42v03">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L42v03">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+8.01.4x+1T2019/xblock/block-v1:MITx+8.01.4x+1T2019+type@video+block@L42v03/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+8.01.4x+1T2019/xblock/block-v1:MITx+8.01.4x+1T2019+type@video+block@L42v03/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-init="VerticalStudentView" data-has-score="False" data-runtime-version="1" data-graded="True" data-usage-id="block-v1:MITx+8.01.4x+1T2019+type@vertical+block@vert-L42v04" data-runtime-class="LmsRuntime" data-block-type="vertical" data-request-token="0a2d133e038811efb62b02329aca76dd" data-course-id="course-v1:MITx+8.01.4x+1T2019">
<h2 class="hd hd-2 unit-title">L42v4: Simple Pendulum - Torque Method</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_07">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-init="XBlockToXModuleShim" data-has-score="True" data-runtime-version="1" data-graded="True" data-usage-id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_07" data-runtime-class="LmsRuntime" data-block-type="problem" data-request-token="0a2d133e038811efb62b02329aca76dd" data-course-id="course-v1:MITx+8.01.4x+1T2019">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_ls_ls43_ls43_07" class="problems-wrapper" role="group"
aria-labelledby="ls_ls43_ls43_07-problem-title"
data-problem-id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_07" data-url="/courses/course-v1:MITx+8.01.4x+1T2019/xblock/block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_07/handler/xmodule_handler"
data-problem-score="0"
data-problem-total-possible="0"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="ls_ls43_ls43_07-problem-title" aria-describedby="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_07-problem-progress" tabindex="-1">
Simple pendulum, Torque method
</h3>
<div class="problem-progress" id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_07-problem-progress"></div>
<div class="problem">
<div>
<center>
<img src="/assets/courseware/v1/abb6631a6c19a42047620e1929be530f/asset-v1:MITx+8.01.4x+1T2019+type@asset+block/images_ls43_05.svg" width="385"/>
</center>
<p>
A small ball of mass [mathjaxinline]m[/mathjaxinline] is attached to the free end of an ideal string of length [mathjaxinline]l[/mathjaxinline] that is hanging from the ceiling at point [mathjaxinline]S[/mathjaxinline]. The ball is moved away from the vertical and released. At the instant shown in the figure, the ball is at an angle [mathjaxinline]\theta (t)[/mathjaxinline] with respect to the vertical. Suppose the angle [mathjaxinline]\theta[/mathjaxinline] is small throughout the motion. </p>
<p><b class="bfseries">(Part a)</b> At the instant shown in the figure, the ball is at an angle [mathjaxinline]\theta[/mathjaxinline] with respect to the vertical. At that instant, calculate [mathjaxinline]\Sigma \vec{\mathbf{\tau }}_ S[/mathjaxinline], the total torque about point S exerted by the external forces on the ball. Express your answer in terms of [mathjaxinline]m[/mathjaxinline], [mathjaxinline]g[/mathjaxinline], [mathjaxinline]l[/mathjaxinline], theta for [mathjaxinline]\theta[/mathjaxinline], and hatk for [mathjaxinline]\hat{\mathbf{k}}[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]\Sigma \vec{\mathbf{\tau }}_ S=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ls_ls43_ls43_07_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_ls_ls43_ls43_07_2_1" id="input_ls_ls43_ls43_07_2_1" aria-describedby="status_ls_ls43_ls43_07_2_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_ls_ls43_ls43_07_2_1"/>
<span class="status unanswered" id="status_ls_ls43_ls43_07_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ls_ls43_ls43_07_2_1" class="answer"/>
<div id="display_ls_ls43_ls43_07_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ls_ls43_ls43_07_2_1_dynamath" name="input_ls_ls43_ls43_07_2_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_ls_ls43_ls43_07_solution_1"/>
</div></p>
<p><b class="bfseries">(Part b)</b> Apply [mathjaxinline]\Sigma \vec{\mathbf{\tau }}_ S = I_ S\vec{\mathbf{\alpha }}[/mathjaxinline] to the ball to obtain an expression for [mathjaxinline]\dfrac {d^2\theta }{dt^2}[/mathjaxinline], the ball's angular acceleration. Express your answer in terms of [mathjaxinline]m[/mathjaxinline], [mathjaxinline]l[/mathjaxinline], [mathjaxinline]g[/mathjaxinline] and theta for [mathjaxinline]\theta[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]\displaystyle \frac{d^2\theta }{dt^2}=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_ls_ls43_ls43_07_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_ls_ls43_ls43_07_3_1" id="input_ls_ls43_ls43_07_3_1" aria-describedby="status_ls_ls43_ls43_07_3_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_ls_ls43_ls43_07_3_1"/>
<span class="status unanswered" id="status_ls_ls43_ls43_07_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ls_ls43_ls43_07_3_1" class="answer"/>
<div id="display_ls_ls43_ls43_07_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ls_ls43_ls43_07_3_1_dynamath" name="input_ls_ls43_ls43_07_3_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_ls_ls43_ls43_07_solution_2"/>
</div></p>
<p><b class="bfseries">(Part c)</b> Assume that the amplitude of the oscillations is small, calculate [mathjaxinline]\omega _0[/mathjaxinline], the angular frequency of the oscillations. Express your answer in terms of [mathjaxinline]m[/mathjaxinline], [mathjaxinline]l[/mathjaxinline], and [mathjaxinline]g[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]\omega _0=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_ls_ls43_ls43_07_4_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_ls_ls43_ls43_07_4_1" id="input_ls_ls43_ls43_07_4_1" aria-describedby="status_ls_ls43_ls43_07_4_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_ls_ls43_ls43_07_4_1"/>
<span class="status unanswered" id="status_ls_ls43_ls43_07_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ls_ls43_ls43_07_4_1" class="answer"/>
<div id="display_ls_ls43_ls43_07_4_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ls_ls43_ls43_07_4_1_dynamath" name="input_ls_ls43_ls43_07_4_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_ls_ls43_ls43_07_solution_3"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Simple pendulum, Torque method " />
<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_ls_ls43_ls43_07" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_ls_ls43_ls43_07">
<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="ls_ls43_ls43_07-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="ls_ls43_ls43_07-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="ls_ls43_ls43_07-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+8.01.4x+1T2019+type@video+block@L42v04">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-init="XBlockToXModuleShim" data-has-score="False" data-runtime-version="1" data-graded="True" data-usage-id="block-v1:MITx+8.01.4x+1T2019+type@video+block@L42v04" data-runtime-class="LmsRuntime" data-block-type="video" data-request-token="0a2d133e038811efb62b02329aca76dd" data-course-id="course-v1:MITx+8.01.4x+1T2019">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L42v4: Simple Pendulum - Torque method</h3>
<div
id="video_L42v04"
class="video closed"
data-metadata='{"speed": null, "saveStateEnabled": false, "streams": "1.00:zmnQ1RYXNlE", "savedVideoPosition": 0.0, "ytTestTimeout": 1500, "generalSpeed": 1.0, "poster": "https://mitx-edx-video-meta-storage.s3.amazonaws.com/media/video-images/70fd043130c7485cb1cddd408152a725.png?Expires=1714107131&AWSAccessKeyId=ASIA4QDFFSURWFHTV653&Signature=IKUwuU5yK8ZByBtBxvGdD4G5yhg%3D&x-amz-security-token=IQoJb3JpZ2luX2VjEG0aCXVzLWVhc3QtMSJGMEQCIG9RZAb1J8GI9%2BhQpxrjXiWznrDyfsktvRlNrjme6oztAiA6rhY391LbQm4h2YeXw2MGjMGw3SRXQ%2BfWht7FPe6vzSrDBQi2%2F%2F%2F%2F%2F%2F%2F%2F%2F%2F8BEAMaDDg1OTIwNTYzNzQxMSIMCZiZiNi687vp3dm7KpcF4n9eJc4CVj5q9cGJLyBUWfvRcSEMZLzdCBZPkBa5CAarcNGMD6WS%2BT2JBmGcjukpPKN%2FJjv6bLrn755vjh%2BbFof1%2FRmiJ8Fs8nhOvjvTOznSOIYHPKu42Ea8IUgfwntjQaxSrIasVJQNiPhA6KPOrV67l1HUmFKF4KyidUbuBdDGvD5X7D%2F%2F7aIwOUVjcCBlDlMmuZokl%2Be499k13KrwrkBpY%2FMVJ4uthX7I8quRA2JsqktY86TWlbEa9G9m%2FdFfJbvtZkO%2F%2Fd9q70%2BKU7PZxFgV4hcKjQOulzckD1O5UTRks1xRoOKC%2BZr9WiH5pfe3wa%2FM74%2BCS4pD4MTpOgBGX5xJ4G39nMvjyUguSDGykc8XES4lhUD2%2Fj3sV83nSeNejU6gFo0jxrLRjMuuO86ELZiuIqlxQ%2FXlk%2BEhbBvgGG0DD4Bm5BDJPyKGDHp927NlxsiDsy32eQ%2FR%2F7xORLcB7l2JxmP8ibbDjkpO5ITqExAnlhioh5qQJ4%2BjYHQP6T9X0JfzLiMC2REVn1hVuxIbr3LuJDNHRMMDUx8FRwIro0XnN%2BhE%2BqoyIOSstBKZnHKit4n5%2B0W6JCxKjLA9mKM1rIk06SNqI47U6lwpncwpZCBOErDNKkyMrJ1xxe%2FyCoXs3N5OB2iGB3GZnlyEeNUmn8ZCyTMLRSF5elwA7o84Oe%2FZOmUH8Jwa8Z2yl%2FFcNkkF5TLYwRKLUIqOwbdkGbnZbWANcQEtSEIw%2Fk1MUy%2F5iHOVg%2F3rbIkWW6O325nVXqhq5A87RXkiLqvhtiC0SqyHtNvNgGbGreQjHbHRiyuQRjzstADFOQ3Gn96LxiP%2FNoKK6619WLCFexqdKz%2BJ%2B0ZwUQDm0LNKenqgwIw5DHrlUuKULWjtSjwZMNHYrLEGOrIBiysEq4lkIE3m0NJ3dXmeUq2n4b5Hi6YpOUW3igSQBKSv0S9grSeQ74KDG5EdVMboPuSligPLbgMoA9QGzL02viywFMCvE0JmwoZoXzWExCu7w%2FhTQIsC2sY586BexbxKlAkN06GYNciSJyV0csRMru5JssXZYoWRW6X7JJdqNyEsUS1FUFJD%2BNmKIyNMIkoKVfhxN7%2FzxpRibnviqnHkMTGteKqMzY4c8CfefgsGvQ8vzw%3D%3D", "publishCompletionUrl": "/courses/course-v1:MITx+8.01.4x+1T2019/xblock/block-v1:MITx+8.01.4x+1T2019+type@video+block@L42v04/handler/publish_completion", "completionEnabled": false, "autoplay": false, "completionPercentage": 0.95, "captionDataDir": null, "duration": 339.9, "saveStateUrl": "/courses/course-v1:MITx+8.01.4x+1T2019/xblock/block-v1:MITx+8.01.4x+1T2019+type@video+block@L42v04/handler/xmodule_handler/save_user_state", "ytApiUrl": "https://www.youtube.com/iframe_api", "start": 0.0, "transcriptLanguage": "en", "lmsRootURL": "https://openlearninglibrary.mit.edu", "prioritizeHls": false, "autoAdvance": false, "transcriptLanguages": {"en": "English"}, "autohideHtml5": false, "transcriptTranslationUrl": "/courses/course-v1:MITx+8.01.4x+1T2019/xblock/block-v1:MITx+8.01.4x+1T2019+type@video+block@L42v04/handler/transcript/translation/__lang__", "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.01.4x+1T2019/xblock/block-v1:MITx+8.01.4x+1T2019+type@video+block@L42v04/handler/transcript/available_translations", "ytMetadataEndpoint": "", "recordedYoutubeIsAvailable": true, "sources": ["https://d2f1egay8yehza.cloudfront.net/MITx8.01.4x-V002000_DTH.mp4", "https://d2f1egay8yehza.cloudfront.net/MITx8.01.4x-V002000/MITx8.01.4x-V002000.m3u8"], "showCaptions": "true", "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="L42v04"></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_L42v04">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L42v04">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+8.01.4x+1T2019/xblock/block-v1:MITx+8.01.4x+1T2019+type@video+block@L42v04/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+8.01.4x+1T2019/xblock/block-v1:MITx+8.01.4x+1T2019+type@video+block@L42v04/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-init="VerticalStudentView" data-has-score="False" data-runtime-version="1" data-graded="True" data-usage-id="block-v1:MITx+8.01.4x+1T2019+type@vertical+block@vert-L40q08" data-runtime-class="LmsRuntime" data-block-type="vertical" data-request-token="0a2d133e038811efb62b02329aca76dd" data-course-id="course-v1:MITx+8.01.4x+1T2019">
<h2 class="hd hd-2 unit-title">L42Q2: Pendulum</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@SHM-008">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-init="XBlockToXModuleShim" data-has-score="True" data-runtime-version="1" data-graded="True" data-usage-id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@SHM-008" data-runtime-class="LmsRuntime" data-block-type="problem" data-request-token="0a2d133e038811efb62b02329aca76dd" data-course-id="course-v1:MITx+8.01.4x+1T2019">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_SHM-008" class="problems-wrapper" role="group"
aria-labelledby="SHM-008-problem-title"
data-problem-id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@SHM-008" data-url="/courses/course-v1:MITx+8.01.4x+1T2019/xblock/block-v1:MITx+8.01.4x+1T2019+type@problem+block@SHM-008/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="SHM-008-problem-title" aria-describedby="block-v1:MITx+8.01.4x+1T2019+type@problem+block@SHM-008-problem-progress" tabindex="-1">
Pendulum
</h3>
<div class="problem-progress" id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@SHM-008-problem-progress"></div>
<div class="problem">
<div>
<p>Consider an ideal pendulum consisting of a "bob" of mass <i> m</i> hanging from a light (massless) string of length <i>L</i>.
The pendulum swings back and forth in simple harmonic motion (SHM). You may assume that the oscillations are small,
so that the motion is "ideal" SHM. </p>
<p>If the mass of the pendulum bob is changed to 4<i>m</i>, the frequency will: </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_SHM-008_2_1">
<fieldset aria-describedby="status_SHM-008_2_1">
<div class="field">
<input type="radio" name="input_SHM-008_2_1" id="input_SHM-008_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="SHM-008_2_1-choice_1-label" for="input_SHM-008_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_SHM-008_2_1"> increase by a factor of 4.
</label>
</div>
<div class="field">
<input type="radio" name="input_SHM-008_2_1" id="input_SHM-008_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="SHM-008_2_1-choice_2-label" for="input_SHM-008_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_SHM-008_2_1"> increase by a factor of 2.
</label>
</div>
<div class="field">
<input type="radio" name="input_SHM-008_2_1" id="input_SHM-008_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="SHM-008_2_1-choice_3-label" for="input_SHM-008_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_SHM-008_2_1"> decrease by a factor of 4.
</label>
</div>
<div class="field">
<input type="radio" name="input_SHM-008_2_1" id="input_SHM-008_2_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="SHM-008_2_1-choice_4-label" for="input_SHM-008_2_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_SHM-008_2_1"> decrease by a factor of 2.
</label>
</div>
<div class="field">
<input type="radio" name="input_SHM-008_2_1" id="input_SHM-008_2_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="SHM-008_2_1-choice_5-label" for="input_SHM-008_2_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_SHM-008_2_1"> none of the above.
</label>
</div>
<span id="answer_SHM-008_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_SHM-008_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>If the string length is changed to 4<i>L</i>, the frequency will: </p>
<p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_SHM-008_3_1">
<fieldset aria-describedby="status_SHM-008_3_1">
<div class="field">
<input type="radio" name="input_SHM-008_3_1" id="input_SHM-008_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="SHM-008_3_1-choice_1-label" for="input_SHM-008_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_SHM-008_3_1"> increase by a factor of 4.
</label>
</div>
<div class="field">
<input type="radio" name="input_SHM-008_3_1" id="input_SHM-008_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="SHM-008_3_1-choice_2-label" for="input_SHM-008_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_SHM-008_3_1"> increase by a factor of 2.
</label>
</div>
<div class="field">
<input type="radio" name="input_SHM-008_3_1" id="input_SHM-008_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="SHM-008_3_1-choice_3-label" for="input_SHM-008_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_SHM-008_3_1"> decrease by a factor of 4.
</label>
</div>
<div class="field">
<input type="radio" name="input_SHM-008_3_1" id="input_SHM-008_3_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="SHM-008_3_1-choice_4-label" for="input_SHM-008_3_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_SHM-008_3_1"> decrease by a factor of 2.
</label>
</div>
<div class="field">
<input type="radio" name="input_SHM-008_3_1" id="input_SHM-008_3_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="SHM-008_3_1-choice_5-label" for="input_SHM-008_3_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_SHM-008_3_1"> none of the above.
</label>
</div>
<span id="answer_SHM-008_3_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_SHM-008_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<div class="solution-span">
<span id="solution_SHM-008_solution_1"/>
</div></div>
<div class="action">
<input type="hidden" name="problem_id" value="Pendulum" />
<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_SHM-008" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_SHM-008">
<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="SHM-008-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="SHM-008-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="SHM-008-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-init="VerticalStudentView" data-has-score="False" data-runtime-version="1" data-graded="True" data-usage-id="block-v1:MITx+8.01.4x+1T2019+type@vertical+block@vert-L42v05" data-runtime-class="LmsRuntime" data-block-type="vertical" data-request-token="0a2d133e038811efb62b02329aca76dd" data-course-id="course-v1:MITx+8.01.4x+1T2019">
<h2 class="hd hd-2 unit-title">L42v5: Physical Pendulum</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.01.4x+1T2019+type@video+block@L42v05">
<div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-init="XBlockToXModuleShim" data-has-score="False" data-runtime-version="1" data-graded="True" data-usage-id="block-v1:MITx+8.01.4x+1T2019+type@video+block@L42v05" data-runtime-class="LmsRuntime" data-block-type="video" data-request-token="0a2d133e038811efb62b02329aca76dd" data-course-id="course-v1:MITx+8.01.4x+1T2019">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Video"}
</script>
<h3 class="hd hd-2">L42v5: Physical Pendulum</h3>
<div
id="video_L42v05"
class="video closed"
data-metadata='{"speed": null, "saveStateEnabled": false, "streams": "1.00:PuZdy0pQnno", "savedVideoPosition": 0.0, "ytTestTimeout": 1500, "generalSpeed": 1.0, "poster": "https://mitx-edx-video-meta-storage.s3.amazonaws.com/media/video-images/96a67dc1c5e54b8ab68a0a7ee77ed067.png?Expires=1714107131&AWSAccessKeyId=ASIA4QDFFSURWFHTV653&Signature=lyw8xqtk4KqmuiGM7jJtIadyZe4%3D&x-amz-security-token=IQoJb3JpZ2luX2VjEG0aCXVzLWVhc3QtMSJGMEQCIG9RZAb1J8GI9%2BhQpxrjXiWznrDyfsktvRlNrjme6oztAiA6rhY391LbQm4h2YeXw2MGjMGw3SRXQ%2BfWht7FPe6vzSrDBQi2%2F%2F%2F%2F%2F%2F%2F%2F%2F%2F8BEAMaDDg1OTIwNTYzNzQxMSIMCZiZiNi687vp3dm7KpcF4n9eJc4CVj5q9cGJLyBUWfvRcSEMZLzdCBZPkBa5CAarcNGMD6WS%2BT2JBmGcjukpPKN%2FJjv6bLrn755vjh%2BbFof1%2FRmiJ8Fs8nhOvjvTOznSOIYHPKu42Ea8IUgfwntjQaxSrIasVJQNiPhA6KPOrV67l1HUmFKF4KyidUbuBdDGvD5X7D%2F%2F7aIwOUVjcCBlDlMmuZokl%2Be499k13KrwrkBpY%2FMVJ4uthX7I8quRA2JsqktY86TWlbEa9G9m%2FdFfJbvtZkO%2F%2Fd9q70%2BKU7PZxFgV4hcKjQOulzckD1O5UTRks1xRoOKC%2BZr9WiH5pfe3wa%2FM74%2BCS4pD4MTpOgBGX5xJ4G39nMvjyUguSDGykc8XES4lhUD2%2Fj3sV83nSeNejU6gFo0jxrLRjMuuO86ELZiuIqlxQ%2FXlk%2BEhbBvgGG0DD4Bm5BDJPyKGDHp927NlxsiDsy32eQ%2FR%2F7xORLcB7l2JxmP8ibbDjkpO5ITqExAnlhioh5qQJ4%2BjYHQP6T9X0JfzLiMC2REVn1hVuxIbr3LuJDNHRMMDUx8FRwIro0XnN%2BhE%2BqoyIOSstBKZnHKit4n5%2B0W6JCxKjLA9mKM1rIk06SNqI47U6lwpncwpZCBOErDNKkyMrJ1xxe%2FyCoXs3N5OB2iGB3GZnlyEeNUmn8ZCyTMLRSF5elwA7o84Oe%2FZOmUH8Jwa8Z2yl%2FFcNkkF5TLYwRKLUIqOwbdkGbnZbWANcQEtSEIw%2Fk1MUy%2F5iHOVg%2F3rbIkWW6O325nVXqhq5A87RXkiLqvhtiC0SqyHtNvNgGbGreQjHbHRiyuQRjzstADFOQ3Gn96LxiP%2FNoKK6619WLCFexqdKz%2BJ%2B0ZwUQDm0LNKenqgwIw5DHrlUuKULWjtSjwZMNHYrLEGOrIBiysEq4lkIE3m0NJ3dXmeUq2n4b5Hi6YpOUW3igSQBKSv0S9grSeQ74KDG5EdVMboPuSligPLbgMoA9QGzL02viywFMCvE0JmwoZoXzWExCu7w%2FhTQIsC2sY586BexbxKlAkN06GYNciSJyV0csRMru5JssXZYoWRW6X7JJdqNyEsUS1FUFJD%2BNmKIyNMIkoKVfhxN7%2FzxpRibnviqnHkMTGteKqMzY4c8CfefgsGvQ8vzw%3D%3D", "publishCompletionUrl": "/courses/course-v1:MITx+8.01.4x+1T2019/xblock/block-v1:MITx+8.01.4x+1T2019+type@video+block@L42v05/handler/publish_completion", "completionEnabled": false, "autoplay": false, "completionPercentage": 0.95, "captionDataDir": null, "duration": 223.85, "saveStateUrl": "/courses/course-v1:MITx+8.01.4x+1T2019/xblock/block-v1:MITx+8.01.4x+1T2019+type@video+block@L42v05/handler/xmodule_handler/save_user_state", "ytApiUrl": "https://www.youtube.com/iframe_api", "start": 0.0, "transcriptLanguage": "en", "lmsRootURL": "https://openlearninglibrary.mit.edu", "prioritizeHls": false, "autoAdvance": false, "transcriptLanguages": {"en": "English"}, "autohideHtml5": false, "transcriptTranslationUrl": "/courses/course-v1:MITx+8.01.4x+1T2019/xblock/block-v1:MITx+8.01.4x+1T2019+type@video+block@L42v05/handler/transcript/translation/__lang__", "transcriptAvailableTranslationsUrl": "/courses/course-v1:MITx+8.01.4x+1T2019/xblock/block-v1:MITx+8.01.4x+1T2019+type@video+block@L42v05/handler/transcript/available_translations", "ytMetadataEndpoint": "", "recordedYoutubeIsAvailable": true, "sources": ["https://d2f1egay8yehza.cloudfront.net/MITx8.01.4x-V001800_DTH.mp4", "https://d2f1egay8yehza.cloudfront.net/MITx8.01.4x-V001800/MITx8.01.4x-V001800.m3u8"], "showCaptions": "true", "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="L42v05"></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_L42v05">Downloads and transcripts</h3>
<div class="wrapper-downloads" role="region" aria-labelledby="video-download-transcripts_L42v05">
<div class="wrapper-download-transcripts">
<h4 class="hd hd-5">Transcripts</h4>
<ul class="list-download-transcripts">
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+8.01.4x+1T2019/xblock/block-v1:MITx+8.01.4x+1T2019+type@video+block@L42v05/handler/transcript/download" data-value="srt">Download SubRip (.srt) file</a>
</li>
<li class="transcript-option">
<a class="btn btn-link" href="/courses/course-v1:MITx+8.01.4x+1T2019/xblock/block-v1:MITx+8.01.4x+1T2019+type@video+block@L42v05/handler/transcript/download" data-value="txt">Download Text (.txt) file</a>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="xblock xblock-public_view xblock-public_view-vertical" data-init="VerticalStudentView" data-has-score="False" data-runtime-version="1" data-graded="True" data-usage-id="block-v1:MITx+8.01.4x+1T2019+type@vertical+block@vert-ls_ls43_ls43_03" data-runtime-class="LmsRuntime" data-block-type="vertical" data-request-token="0a2d133e038811efb62b02329aca76dd" data-course-id="course-v1:MITx+8.01.4x+1T2019">
<h2 class="hd hd-2 unit-title">L42Q3: A Disk at the End of an Oscilating Rod</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_03">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-init="XBlockToXModuleShim" data-has-score="True" data-runtime-version="1" data-graded="True" data-usage-id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_03" data-runtime-class="LmsRuntime" data-block-type="problem" data-request-token="0a2d133e038811efb62b02329aca76dd" data-course-id="course-v1:MITx+8.01.4x+1T2019">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_ls_ls43_ls43_03" class="problems-wrapper" role="group"
aria-labelledby="ls_ls43_ls43_03-problem-title"
data-problem-id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_03" data-url="/courses/course-v1:MITx+8.01.4x+1T2019/xblock/block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_03/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="0.5"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="ls_ls43_ls43_03-problem-title" aria-describedby="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_03-problem-progress" tabindex="-1">
A Disk at the End of an Oscillating Rod, part 1.
</h3>
<div class="problem-progress" id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_03-problem-progress"></div>
<div class="problem">
<div>
<center>
<img src="/assets/courseware/v1/79dcc1abba016117bb18ff9010b007e8/asset-v1:MITx+8.01.4x+1T2019+type@asset+block/images_exam_final_cq_4.svg" width="330"/>
</center>
<p>
A rigid body is composed of a uniform rod of mass [mathjaxinline]m[/mathjaxinline] and length [mathjaxinline]D[/mathjaxinline], and a uniform disk of mass [mathjaxinline]m[/mathjaxinline] and radius [mathjaxinline]R[/mathjaxinline]. The disk is rigidly fixed to one end of the rod (see figure above). This body is pivoted about point [mathjaxinline]S[/mathjaxinline]. Assume the pivot is frictionless. The downward acceleration of gravity is [mathjaxinline]g[/mathjaxinline], the moment of inertia of the rod about its center of mass is [mathjaxinline]mD^2/12[/mathjaxinline], and the moment of inertia of the disk about its center of mass is [mathjaxinline]mR^2/2[/mathjaxinline]. Suppose the angle [mathjaxinline]\theta[/mathjaxinline] is small throughout the motion. </p>
<p><b class="bfseries">(Part a)</b> Calculate [mathjaxinline]I_ S[/mathjaxinline], the moment of inertia of the rod-disk system with respect to the axis passing through point [mathjaxinline]S[/mathjaxinline] and perpendicular to the plane of the figure. Express your answer in terms of [mathjaxinline]m[/mathjaxinline], [mathjaxinline]R[/mathjaxinline], and [mathjaxinline]D[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]I_ S=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ls_ls43_ls43_03_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_ls_ls43_ls43_03_2_1" id="input_ls_ls43_ls43_03_2_1" aria-describedby="status_ls_ls43_ls43_03_2_1" value="" class="math" size="30"/>
<span class="trailing_text" id="trailing_text_ls_ls43_ls43_03_2_1"/>
<span class="status unanswered" id="status_ls_ls43_ls43_03_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ls_ls43_ls43_03_2_1" class="answer"/>
<div id="display_ls_ls43_ls43_03_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ls_ls43_ls43_03_2_1_dynamath" name="input_ls_ls43_ls43_03_2_1_dynamath"/>
</div>
</div></div>
<div class="solution-span">
<span id="solution_ls_ls43_ls43_03_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="A Disk at the End of an Oscillating Rod, part 1." />
<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_ls_ls43_ls43_03" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_ls_ls43_ls43_03">
<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="ls_ls43_ls43_03-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="ls_ls43_ls43_03-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="ls_ls43_ls43_03-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+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_03b">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-init="XBlockToXModuleShim" data-has-score="True" data-runtime-version="1" data-graded="True" data-usage-id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_03b" data-runtime-class="LmsRuntime" data-block-type="problem" data-request-token="0a2d133e038811efb62b02329aca76dd" data-course-id="course-v1:MITx+8.01.4x+1T2019">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_ls_ls43_ls43_03b" class="problems-wrapper" role="group"
aria-labelledby="ls_ls43_ls43_03b-problem-title"
data-problem-id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_03b" data-url="/courses/course-v1:MITx+8.01.4x+1T2019/xblock/block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_03b/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="0.5"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="ls_ls43_ls43_03b-problem-title" aria-describedby="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_03b-problem-progress" tabindex="-1">
A Disk at the End of an Oscillating Rod, part 2.
</h3>
<div class="problem-progress" id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_03b-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bfseries">(Part b)</b> At the instant shown in the figure, the object is at an angle [mathjaxinline]\theta[/mathjaxinline] with respect to the vertical. At that instant, calculate [mathjaxinline]\Sigma \vec{\mathbf{\tau }}_ S[/mathjaxinline], the total torque about point S exerted by the external forces on the rod-disk system. Express your answer in terms of [mathjaxinline]m[/mathjaxinline], [mathjaxinline]R[/mathjaxinline], [mathjaxinline]D[/mathjaxinline], [mathjaxinline]g[/mathjaxinline], theta for [mathjaxinline]\theta[/mathjaxinline], hati for [mathjaxinline]\hat{\mathbf{i}}[/mathjaxinline], hatj for [mathjaxinline]\hat{\mathbf{j}}[/mathjaxinline] and hatk for [mathjaxinline]\hat{\mathbf{k}}[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]\Sigma \vec{\mathbf{\tau }}_ S=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ls_ls43_ls43_03b_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_ls_ls43_ls43_03b_2_1" id="input_ls_ls43_ls43_03b_2_1" aria-describedby="status_ls_ls43_ls43_03b_2_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_ls_ls43_ls43_03b_2_1"/>
<span class="status unanswered" id="status_ls_ls43_ls43_03b_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ls_ls43_ls43_03b_2_1" class="answer"/>
<div id="display_ls_ls43_ls43_03b_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ls_ls43_ls43_03b_2_1_dynamath" name="input_ls_ls43_ls43_03b_2_1_dynamath"/>
</div>
</div></div>
<div class="solution-span">
<span id="solution_ls_ls43_ls43_03b_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="A Disk at the End of an Oscillating Rod, part 2." />
<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_ls_ls43_ls43_03b" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_ls_ls43_ls43_03b">
<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="ls_ls43_ls43_03b-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="ls_ls43_ls43_03b-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="ls_ls43_ls43_03b-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+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_03c">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-init="XBlockToXModuleShim" data-has-score="True" data-runtime-version="1" data-graded="True" data-usage-id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_03c" data-runtime-class="LmsRuntime" data-block-type="problem" data-request-token="0a2d133e038811efb62b02329aca76dd" data-course-id="course-v1:MITx+8.01.4x+1T2019">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_ls_ls43_ls43_03c" class="problems-wrapper" role="group"
aria-labelledby="ls_ls43_ls43_03c-problem-title"
data-problem-id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_03c" data-url="/courses/course-v1:MITx+8.01.4x+1T2019/xblock/block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_03c/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="0.5"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="ls_ls43_ls43_03c-problem-title" aria-describedby="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_03c-problem-progress" tabindex="-1">
A Disk at the End of an Oscillating Rod, part 3.
</h3>
<div class="problem-progress" id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_03c-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bfseries">(Part c)</b> Apply [mathjaxinline]\Sigma \vec{\mathbf{\tau }}_ S = I_ S\vec{\mathbf{\alpha }}[/mathjaxinline] to the rod-disk system to obtain an expression for [mathjaxinline]\dfrac {d^2\theta }{dt^2}[/mathjaxinline], the angular acceleration of the object. Express your answer in terms of [mathjaxinline]m[/mathjaxinline], [mathjaxinline]R[/mathjaxinline], [mathjaxinline]D[/mathjaxinline], [mathjaxinline]g[/mathjaxinline] and theta for [mathjaxinline]\theta[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]\displaystyle \frac{d^2\theta }{dt^2}=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ls_ls43_ls43_03c_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_ls_ls43_ls43_03c_2_1" id="input_ls_ls43_ls43_03c_2_1" aria-describedby="status_ls_ls43_ls43_03c_2_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_ls_ls43_ls43_03c_2_1"/>
<span class="status unanswered" id="status_ls_ls43_ls43_03c_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ls_ls43_ls43_03c_2_1" class="answer"/>
<div id="display_ls_ls43_ls43_03c_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ls_ls43_ls43_03c_2_1_dynamath" name="input_ls_ls43_ls43_03c_2_1_dynamath"/>
</div>
</div></div>
<div class="solution-span">
<span id="solution_ls_ls43_ls43_03c_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="A Disk at the End of an Oscillating Rod, part 3." />
<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_ls_ls43_ls43_03c" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_ls_ls43_ls43_03c">
<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="ls_ls43_ls43_03c-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="ls_ls43_ls43_03c-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="ls_ls43_ls43_03c-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+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_03d">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-init="XBlockToXModuleShim" data-has-score="True" data-runtime-version="1" data-graded="True" data-usage-id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_03d" data-runtime-class="LmsRuntime" data-block-type="problem" data-request-token="0a2d133e038811efb62b02329aca76dd" data-course-id="course-v1:MITx+8.01.4x+1T2019">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_ls_ls43_ls43_03d" class="problems-wrapper" role="group"
aria-labelledby="ls_ls43_ls43_03d-problem-title"
data-problem-id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_03d" data-url="/courses/course-v1:MITx+8.01.4x+1T2019/xblock/block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_03d/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="0.5"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="ls_ls43_ls43_03d-problem-title" aria-describedby="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_03d-problem-progress" tabindex="-1">
A Disk at the End of an Oscillating Rod, part 4.
</h3>
<div class="problem-progress" id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_03d-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bfseries">(Part d)</b> Assume that the amplitude of the oscillations is small, calculate [mathjaxinline]\omega _0[/mathjaxinline], the angular frequency of the oscillations. Express your answer in terms of [mathjaxinline]m[/mathjaxinline], [mathjaxinline]R[/mathjaxinline], [mathjaxinline]D[/mathjaxinline], and [mathjaxinline]g[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]\omega _0=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ls_ls43_ls43_03d_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_ls_ls43_ls43_03d_2_1" id="input_ls_ls43_ls43_03d_2_1" aria-describedby="status_ls_ls43_ls43_03d_2_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_ls_ls43_ls43_03d_2_1"/>
<span class="status unanswered" id="status_ls_ls43_ls43_03d_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ls_ls43_ls43_03d_2_1" class="answer"/>
<div id="display_ls_ls43_ls43_03d_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ls_ls43_ls43_03d_2_1_dynamath" name="input_ls_ls43_ls43_03d_2_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_ls_ls43_ls43_03d_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="A Disk at the End of an Oscillating Rod, part 4." />
<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_ls_ls43_ls43_03d" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_ls_ls43_ls43_03d">
<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="ls_ls43_ls43_03d-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="ls_ls43_ls43_03d-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="ls_ls43_ls43_03d-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-init="VerticalStudentView" data-has-score="False" data-runtime-version="1" data-graded="True" data-usage-id="block-v1:MITx+8.01.4x+1T2019+type@vertical+block@vert-ls_ls43_ls43_04" data-runtime-class="LmsRuntime" data-block-type="vertical" data-request-token="0a2d133e038811efb62b02329aca76dd" data-course-id="course-v1:MITx+8.01.4x+1T2019">
<h2 class="hd hd-2 unit-title">L42Q4: A Plate at the End of an Oscilating Rod</h2>
<div class="vert-mod">
<div class="vert vert-0" data-id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_04">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-init="XBlockToXModuleShim" data-has-score="True" data-runtime-version="1" data-graded="True" data-usage-id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_04" data-runtime-class="LmsRuntime" data-block-type="problem" data-request-token="0a2d133e038811efb62b02329aca76dd" data-course-id="course-v1:MITx+8.01.4x+1T2019">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_ls_ls43_ls43_04" class="problems-wrapper" role="group"
aria-labelledby="ls_ls43_ls43_04-problem-title"
data-problem-id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_04" data-url="/courses/course-v1:MITx+8.01.4x+1T2019/xblock/block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_04/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="0.5"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="ls_ls43_ls43_04-problem-title" aria-describedby="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_04-problem-progress" tabindex="-1">
A Square at the End of an Oscilating Rod, part1
</h3>
<div class="problem-progress" id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_04-problem-progress"></div>
<div class="problem">
<div>
<center>
<img src="/assets/courseware/v1/9f0c1374271ec031a4937153a525c054/asset-v1:MITx+8.01.4x+1T2019+type@asset+block/images_ls43_04.svg" width="440"/>
</center>
<p>
A rigid body is composed of a uniform rod of mass [mathjaxinline]m[/mathjaxinline] and length [mathjaxinline]D[/mathjaxinline], and a uniform square plate of mass [mathjaxinline]m[/mathjaxinline] and side [mathjaxinline]a[/mathjaxinline]. The center of mass of the plate is rigidly fixed to one end of the rod (see figure above). This body is pivoted about point [mathjaxinline]S[/mathjaxinline]. Assume the pivot is frictionless. The downward acceleration of gravity is [mathjaxinline]g[/mathjaxinline], the moment of inertia of the rod about its center of mass is [mathjaxinline]mD^2/12[/mathjaxinline], and the moment of inertia of the plate about its center of mass is [mathjaxinline]ma^2/6[/mathjaxinline]. Suppose the angle [mathjaxinline]\theta[/mathjaxinline] is small throughout the motion. </p>
<p><b class="bfseries">(Part a)</b> At the instant shown in the figure, the object is at an angle [mathjaxinline]\theta (t)[/mathjaxinline]. Set the zero of gravitational potential energy of the rod-plate system at center of mas of the plate when it is at the lowest point of its trajectory (the rod is vertical and [mathjaxinline]\theta = 0[/mathjaxinline]). Write an expression of [mathjaxinline]U[/mathjaxinline], the gravitational potential energy of the rod-plate system at the instant shown in the figure. Express your answer in terms of [mathjaxinline]m[/mathjaxinline], [mathjaxinline]a[/mathjaxinline], [mathjaxinline]D[/mathjaxinline], [mathjaxinline]g[/mathjaxinline], and theta for [mathjaxinline]\theta[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]U=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ls_ls43_ls43_04_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_ls_ls43_ls43_04_2_1" id="input_ls_ls43_ls43_04_2_1" aria-describedby="status_ls_ls43_ls43_04_2_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_ls_ls43_ls43_04_2_1"/>
<span class="status unanswered" id="status_ls_ls43_ls43_04_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ls_ls43_ls43_04_2_1" class="answer"/>
<div id="display_ls_ls43_ls43_04_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ls_ls43_ls43_04_2_1_dynamath" name="input_ls_ls43_ls43_04_2_1_dynamath"/>
</div>
</div></div>
<div class="solution-span">
<span id="solution_ls_ls43_ls43_04_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="A Square at the End of an Oscilating Rod, part1" />
<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_ls_ls43_ls43_04" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_ls_ls43_ls43_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="ls_ls43_ls43_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="ls_ls43_ls43_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="ls_ls43_ls43_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 class="vert vert-1" data-id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_04b">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-init="XBlockToXModuleShim" data-has-score="True" data-runtime-version="1" data-graded="True" data-usage-id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_04b" data-runtime-class="LmsRuntime" data-block-type="problem" data-request-token="0a2d133e038811efb62b02329aca76dd" data-course-id="course-v1:MITx+8.01.4x+1T2019">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_ls_ls43_ls43_04b" class="problems-wrapper" role="group"
aria-labelledby="ls_ls43_ls43_04b-problem-title"
data-problem-id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_04b" data-url="/courses/course-v1:MITx+8.01.4x+1T2019/xblock/block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_04b/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="0.5"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="ls_ls43_ls43_04b-problem-title" aria-describedby="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_04b-problem-progress" tabindex="-1">
A Square at the End of an Oscilating Rod, part 2
</h3>
<div class="problem-progress" id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_04b-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bfseries">(Part b)</b> Use the expressions of [mathjaxinline]U[/mathjaxinline] from part (a) to calculate [mathjaxinline]\dfrac {dU}{dt}[/mathjaxinline], the derivative of [mathjaxinline]U[/mathjaxinline] with respect to time, Express your answer in terms of [mathjaxinline]m[/mathjaxinline], [mathjaxinline]a[/mathjaxinline], [mathjaxinline]D[/mathjaxinline], [mathjaxinline]g[/mathjaxinline], theta for [mathjaxinline]\theta[/mathjaxinline], and omega_z for [mathjaxinline]\omega _ z=\dfrac {d\theta }{dt}[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]\dfrac {dU}{dt}=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ls_ls43_ls43_04b_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_ls_ls43_ls43_04b_2_1" id="input_ls_ls43_ls43_04b_2_1" aria-describedby="status_ls_ls43_ls43_04b_2_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_ls_ls43_ls43_04b_2_1"/>
<span class="status unanswered" id="status_ls_ls43_ls43_04b_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ls_ls43_ls43_04b_2_1" class="answer"/>
<div id="display_ls_ls43_ls43_04b_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ls_ls43_ls43_04b_2_1_dynamath" name="input_ls_ls43_ls43_04b_2_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_ls_ls43_ls43_04b_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="A Square at the End of an Oscilating Rod, part 2" />
<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_ls_ls43_ls43_04b" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_ls_ls43_ls43_04b">
<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="ls_ls43_ls43_04b-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="ls_ls43_ls43_04b-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="ls_ls43_ls43_04b-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+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_04c">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-init="XBlockToXModuleShim" data-has-score="True" data-runtime-version="1" data-graded="True" data-usage-id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_04c" data-runtime-class="LmsRuntime" data-block-type="problem" data-request-token="0a2d133e038811efb62b02329aca76dd" data-course-id="course-v1:MITx+8.01.4x+1T2019">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_ls_ls43_ls43_04c" class="problems-wrapper" role="group"
aria-labelledby="ls_ls43_ls43_04c-problem-title"
data-problem-id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_04c" data-url="/courses/course-v1:MITx+8.01.4x+1T2019/xblock/block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_04c/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="0.5"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="ls_ls43_ls43_04c-problem-title" aria-describedby="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_04c-problem-progress" tabindex="-1">
A Square at the End of an Oscilating Rod, part 3
</h3>
<div class="problem-progress" id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_04c-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bfseries">(Part c)</b> Calculate [mathjaxinline]I_ S[/mathjaxinline], the moment of inertia of the rod-plate system with respect to the axis passing through point [mathjaxinline]S[/mathjaxinline] and perpendicular to the (x,y) - plane. Express your answer in terms of [mathjaxinline]m[/mathjaxinline], [mathjaxinline]a[/mathjaxinline], [mathjaxinline]D[/mathjaxinline], [mathjaxinline]g[/mathjaxinline], and theta for [mathjaxinline]\theta[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]I_ S=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ls_ls43_ls43_04c_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_ls_ls43_ls43_04c_2_1" id="input_ls_ls43_ls43_04c_2_1" aria-describedby="status_ls_ls43_ls43_04c_2_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_ls_ls43_ls43_04c_2_1"/>
<span class="status unanswered" id="status_ls_ls43_ls43_04c_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ls_ls43_ls43_04c_2_1" class="answer"/>
<div id="display_ls_ls43_ls43_04c_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ls_ls43_ls43_04c_2_1_dynamath" name="input_ls_ls43_ls43_04c_2_1_dynamath"/>
</div>
</div></div>
<div class="solution-span">
<span id="solution_ls_ls43_ls43_04c_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="A Square at the End of an Oscilating Rod, part 3" />
<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_ls_ls43_ls43_04c" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_ls_ls43_ls43_04c">
<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="ls_ls43_ls43_04c-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="ls_ls43_ls43_04c-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="ls_ls43_ls43_04c-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+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_04d">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-init="XBlockToXModuleShim" data-has-score="True" data-runtime-version="1" data-graded="True" data-usage-id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_04d" data-runtime-class="LmsRuntime" data-block-type="problem" data-request-token="0a2d133e038811efb62b02329aca76dd" data-course-id="course-v1:MITx+8.01.4x+1T2019">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_ls_ls43_ls43_04d" class="problems-wrapper" role="group"
aria-labelledby="ls_ls43_ls43_04d-problem-title"
data-problem-id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_04d" data-url="/courses/course-v1:MITx+8.01.4x+1T2019/xblock/block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_04d/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="0.5"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="ls_ls43_ls43_04d-problem-title" aria-describedby="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_04d-problem-progress" tabindex="-1">
A Square at the End of an Oscilating Rod, part 4
</h3>
<div class="problem-progress" id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_04d-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bfseries">(Part d)</b> At the instant shown in the figure, the angular velocity of the rod-plate system is given by [mathjaxinline]\omega _ z = \dfrac {d\theta }{dt}[/mathjaxinline] . Calculate [mathjaxinline]K[/mathjaxinline], the kinetic energy of the rod-plate system. Express your answer in terms of [mathjaxinline]m[/mathjaxinline], [mathjaxinline]a[/mathjaxinline], [mathjaxinline]D[/mathjaxinline], [mathjaxinline]g[/mathjaxinline], theta for [mathjaxinline]\theta[/mathjaxinline], and omega_z for [mathjaxinline]\omega _ z = \dfrac {d\theta }{dt}[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]K=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ls_ls43_ls43_04d_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_ls_ls43_ls43_04d_2_1" id="input_ls_ls43_ls43_04d_2_1" aria-describedby="status_ls_ls43_ls43_04d_2_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_ls_ls43_ls43_04d_2_1"/>
<span class="status unanswered" id="status_ls_ls43_ls43_04d_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ls_ls43_ls43_04d_2_1" class="answer"/>
<div id="display_ls_ls43_ls43_04d_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ls_ls43_ls43_04d_2_1_dynamath" name="input_ls_ls43_ls43_04d_2_1_dynamath"/>
</div>
</div></div>
<div class="solution-span">
<span id="solution_ls_ls43_ls43_04d_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="A Square at the End of an Oscilating Rod, part 4" />
<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_ls_ls43_ls43_04d" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_ls_ls43_ls43_04d">
<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="ls_ls43_ls43_04d-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="ls_ls43_ls43_04d-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="ls_ls43_ls43_04d-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-4" data-id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_04e">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-init="XBlockToXModuleShim" data-has-score="True" data-runtime-version="1" data-graded="True" data-usage-id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_04e" data-runtime-class="LmsRuntime" data-block-type="problem" data-request-token="0a2d133e038811efb62b02329aca76dd" data-course-id="course-v1:MITx+8.01.4x+1T2019">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_ls_ls43_ls43_04e" class="problems-wrapper" role="group"
aria-labelledby="ls_ls43_ls43_04e-problem-title"
data-problem-id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_04e" data-url="/courses/course-v1:MITx+8.01.4x+1T2019/xblock/block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_04e/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="0.5"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="ls_ls43_ls43_04e-problem-title" aria-describedby="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_04e-problem-progress" tabindex="-1">
A Square at the End of an Oscilating Rod, part 5
</h3>
<div class="problem-progress" id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_04e-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bfseries">(Part e)</b> Calculate [mathjaxinline]\dfrac {dK}{dt}[/mathjaxinline], the derivative of the kinetic energy with respect to time, Express your answer in terms of [mathjaxinline]m[/mathjaxinline], [mathjaxinline]a[/mathjaxinline], [mathjaxinline]D[/mathjaxinline], [mathjaxinline]g[/mathjaxinline], theta for [mathjaxinline]\theta[/mathjaxinline], omega_z for [mathjaxinline]\omega _ z=\dfrac {d\theta }{dt}[/mathjaxinline], and alpha for [mathjaxinline]\alpha = \dfrac {d^2\theta }{dt^2}[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]\dfrac {dK}{dt}=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ls_ls43_ls43_04e_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_ls_ls43_ls43_04e_2_1" id="input_ls_ls43_ls43_04e_2_1" aria-describedby="status_ls_ls43_ls43_04e_2_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_ls_ls43_ls43_04e_2_1"/>
<span class="status unanswered" id="status_ls_ls43_ls43_04e_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ls_ls43_ls43_04e_2_1" class="answer"/>
<div id="display_ls_ls43_ls43_04e_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ls_ls43_ls43_04e_2_1_dynamath" name="input_ls_ls43_ls43_04e_2_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_ls_ls43_ls43_04e_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="A Square at the End of an Oscilating Rod, part 5" />
<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_ls_ls43_ls43_04e" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_ls_ls43_ls43_04e">
<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="ls_ls43_ls43_04e-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="ls_ls43_ls43_04e-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="ls_ls43_ls43_04e-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-5" data-id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_04f">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-init="XBlockToXModuleShim" data-has-score="True" data-runtime-version="1" data-graded="True" data-usage-id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_04f" data-runtime-class="LmsRuntime" data-block-type="problem" data-request-token="0a2d133e038811efb62b02329aca76dd" data-course-id="course-v1:MITx+8.01.4x+1T2019">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_ls_ls43_ls43_04f" class="problems-wrapper" role="group"
aria-labelledby="ls_ls43_ls43_04f-problem-title"
data-problem-id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_04f" data-url="/courses/course-v1:MITx+8.01.4x+1T2019/xblock/block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_04f/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="0.5"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="ls_ls43_ls43_04f-problem-title" aria-describedby="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_04f-problem-progress" tabindex="-1">
A Square at the End of an Oscilating Rod, part 6
</h3>
<div class="problem-progress" id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_04f-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bfseries">(Part f)</b> Because there is no friction at the pivot, the mechanical energy of the rod-plate system is constant. Starting from [mathjaxinline]\dfrac {dE}{dt}=0[/mathjaxinline], calculate an expression for [mathjaxinline]\dfrac {d^2\theta }{dt^2}[/mathjaxinline], the angular acceleration of the object. Express your answer in terms of [mathjaxinline]m[/mathjaxinline], [mathjaxinline]g[/mathjaxinline], [mathjaxinline]a[/mathjaxinline], [mathjaxinline]D[/mathjaxinline], and theta for [mathjaxinline]\theta[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]\dfrac {d^2\theta }{dt^2}=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ls_ls43_ls43_04f_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_ls_ls43_ls43_04f_2_1" id="input_ls_ls43_ls43_04f_2_1" aria-describedby="status_ls_ls43_ls43_04f_2_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_ls_ls43_ls43_04f_2_1"/>
<span class="status unanswered" id="status_ls_ls43_ls43_04f_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ls_ls43_ls43_04f_2_1" class="answer"/>
<div id="display_ls_ls43_ls43_04f_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ls_ls43_ls43_04f_2_1_dynamath" name="input_ls_ls43_ls43_04f_2_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_ls_ls43_ls43_04f_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="A Square at the End of an Oscilating Rod, part 6" />
<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_ls_ls43_ls43_04f" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_ls_ls43_ls43_04f">
<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="ls_ls43_ls43_04f-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="ls_ls43_ls43_04f-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="ls_ls43_ls43_04f-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-6" data-id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_04g">
<div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-init="XBlockToXModuleShim" data-has-score="True" data-runtime-version="1" data-graded="True" data-usage-id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_04g" data-runtime-class="LmsRuntime" data-block-type="problem" data-request-token="0a2d133e038811efb62b02329aca76dd" data-course-id="course-v1:MITx+8.01.4x+1T2019">
<script type="json/xblock-args" class="xblock-json-init-args">
{"xmodule-type": "Problem"}
</script>
<div id="problem_ls_ls43_ls43_04g" class="problems-wrapper" role="group"
aria-labelledby="ls_ls43_ls43_04g-problem-title"
data-problem-id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_04g" data-url="/courses/course-v1:MITx+8.01.4x+1T2019/xblock/block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_04g/handler/xmodule_handler"
data-problem-score="0.0"
data-problem-total-possible="0.5"
data-attempts-used="0"
data-content="
<h3 class="hd hd-3 problem-header" id="ls_ls43_ls43_04g-problem-title" aria-describedby="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_04g-problem-progress" tabindex="-1">
A Square at the End of an Oscilating Rod, part 7
</h3>
<div class="problem-progress" id="block-v1:MITx+8.01.4x+1T2019+type@problem+block@ls_ls43_ls43_04g-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bfseries">(Part g)</b> Assuming that the amplitude of the oscillation is small, calculate [mathjaxinline]\omega _0[/mathjaxinline], the angular frequency of oscillation. Express your answer in terms of [mathjaxinline]m[/mathjaxinline], [mathjaxinline]a[/mathjaxinline], [mathjaxinline]D[/mathjaxinline], [mathjaxinline]g[/mathjaxinline], and theta for [mathjaxinline]\theta[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]\omega _0=[/mathjaxinline] </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ls_ls43_ls43_04g_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_ls_ls43_ls43_04g_2_1" id="input_ls_ls43_ls43_04g_2_1" aria-describedby="status_ls_ls43_ls43_04g_2_1" value="" class="math" size="40"/>
<span class="trailing_text" id="trailing_text_ls_ls43_ls43_04g_2_1"/>
<span class="status unanswered" id="status_ls_ls43_ls43_04g_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ls_ls43_ls43_04g_2_1" class="answer"/>
<div id="display_ls_ls43_ls43_04g_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ls_ls43_ls43_04g_2_1_dynamath" name="input_ls_ls43_ls43_04g_2_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<div class="solution-span">
<span id="solution_ls_ls43_ls43_04g_solution_1"/>
</div></p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="A Square at the End of an Oscilating Rod, part 7" />
<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_ls_ls43_ls43_04g" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_ls_ls43_ls43_04g">
<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="ls_ls43_ls43_04g-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="ls_ls43_ls43_04g-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="ls_ls43_ls43_04g-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>
© All Rights Reserved