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<h2 class="hd hd-2 unit-title">Pset 2 checker</h2>
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<p><a href="/assets/courseware/v1/3180836d21871bb2b5d928cda57403d8/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/mites_1803_s24_ps2.pdf" target="[object Object]">Problem set 2 (PDF)</a></p>
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<h3 class="hd hd-3 problem-header" id="ps2-1-check-problem-title" aria-describedby="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps2-1-check-problem-progress" tabindex="-1">
Problem 1
</h3>
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<p>
To check your answers put them in the appropriate box and click the &#8216;Check' button. The checker can do arithmetic. So, 3^2, [mathjaxinline]5/3[/mathjaxinline] and [mathjaxinline]3*4[/mathjaxinline] are okay. </p>
<p>
Unless otherwise stated the checker expects numerical answers to 3 significant digits. </p>
<p>
For formulas you should use Matlab notation. Remember to use a * for multiplication, i.e. [mathjaxinline]3*x[/mathjaxinline] <b class="bf">NOT </b> [mathjaxinline]3x[/mathjaxinline]. </p>
<p>
The checker knows many of the standard functions. Use exp(x) and ln(x) for the exponential and natural log functions. </p>
<p>
The formula checker expects a specific letter for the variables, so be sure to use the one specified. </p>
<p><b class="bf">1a(i)</b> Give the 3 roots of [mathjaxinline]-1[/mathjaxinline] in the form [mathjaxinline]a + b\, i[/mathjaxinline] </p>
<p>
Enter them as comma separated list. Enter complex numbers as, e.g. 2.123 - 3.432*i. You can write your answer using the square root function, e.g. <tt class="tt">5 +sqrt(7)[mathjaxinline]*[/mathjaxinline]i</tt>. If you use decimals, give the answer correct to 3 decimal places. </p>
<p>
<p style="display:inline">Roots of -1: </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps2-1-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps2-1-check_2_1" id="input_ps2-1-check_2_1" aria-describedby="status_ps2-1-check_2_1" value="" class="math" size="70"/>
<span class="trailing_text" id="trailing_text_ps2-1-check_2_1"/>
<span class="status unanswered" id="status_ps2-1-check_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps2-1-check_2_1" class="answer"/>
<div id="display_ps2-1-check_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps2-1-check_2_1_dynamath" name="input_ps2-1-check_2_1_dynamath"/>
</div>
</div></div>
</p>
<p><b class="bf">1a(ii)</b> Give the 3 roots of [mathjaxinline]1+\sqrt{3}\, i[/mathjaxinline] in the form [mathjaxinline]r\, e^{i\theta }[/mathjaxinline] </p>
<p>
<p style="display:inline">Roots of [mathjaxinline]1+\sqrt{3}\, i[/mathjaxinline]: </p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_ps2-1-check_3_1" class="text-input-dynamath capa_inputtype inline textline">
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<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps2-1-check_3_1" id="input_ps2-1-check_3_1" aria-describedby="status_ps2-1-check_3_1" value="" class="math" size="70"/>
<span class="trailing_text" id="trailing_text_ps2-1-check_3_1"/>
<span class="status unanswered" id="status_ps2-1-check_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps2-1-check_3_1" class="answer"/>
<div id="display_ps2-1-check_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps2-1-check_3_1_dynamath" name="input_ps2-1-check_3_1_dynamath"/>
</div>
</div></div>
</p>
<p><b class="bf">1(b)</b><em>No answer checker given.</em>But, here's a hint: look at the spacing between the roots. </p>
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<h3 class="hd hd-3 problem-header" id="ps2-2-check-problem-title" aria-describedby="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps2-2-check-problem-progress" tabindex="-1">
Problem 2
</h3>
<div class="problem-progress" id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps2-2-check-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bf">2</b>. <em>No answer checker given.</em>But here's a hint. </p>
<p>
Give the trigonometric identity: [mathjaxinline]\cos (u-v) =[/mathjaxinline] <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps2-2-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps2-2-check_2_1" id="input_ps2-2-check_2_1" aria-describedby="status_ps2-2-check_2_1" value="" class="math" size="50"/>
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<span class="status unanswered" id="status_ps2-2-check_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps2-2-check_2_1" class="answer"/>
<div id="display_ps2-2-check_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps2-2-check_2_1_dynamath" name="input_ps2-2-check_2_1_dynamath"/>
</div>
</div></div> </p>
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<div id="problem_ps2-3-check" class="problems-wrapper" role="group"
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data-problem-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps2-3-check" data-url="/courses/course-v1:MITx+ES.1803+2023_Fall/xblock/block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps2-3-check/handler/xmodule_handler"
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<h3 class="hd hd-3 problem-header" id="ps2-3-check-problem-title" aria-describedby="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps2-3-check-problem-progress" tabindex="-1">
Problem 3
</h3>
<div class="problem-progress" id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps2-3-check-problem-progress"></div>
<div class="problem">
<div>
<p>
The checker essentially gives you the answer. So, on your pset, you need to explain clearly how you arrived at your answer. </p>
<p>
<b class="bf">3(a)</b>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_ps2-3-check_2_1">
<fieldset aria-describedby="status_ps2-3-check_2_1">
<div class="field">
<input type="radio" name="input_ps2-3-check_2_1" id="input_ps2-3-check_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="ps2-3-check_2_1-choice_1-label" for="input_ps2-3-check_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_ps2-3-check_2_1">
<text> Stable</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_ps2-3-check_2_1" id="input_ps2-3-check_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="ps2-3-check_2_1-choice_2-label" for="input_ps2-3-check_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_ps2-3-check_2_1">
<text> Unstable</text>
</label>
</div>
<span id="answer_ps2-3-check_2_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_ps2-3-check_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>
<b class="bf">3(b)</b>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_ps2-3-check_3_1">
<fieldset aria-describedby="status_ps2-3-check_3_1">
<div class="field">
<input type="radio" name="input_ps2-3-check_3_1" id="input_ps2-3-check_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="ps2-3-check_3_1-choice_1-label" for="input_ps2-3-check_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_ps2-3-check_3_1">
<text> Stable</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_ps2-3-check_3_1" id="input_ps2-3-check_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="ps2-3-check_3_1-choice_2-label" for="input_ps2-3-check_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_ps2-3-check_3_1">
<text> Unstable</text>
</label>
</div>
<span id="answer_ps2-3-check_3_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_ps2-3-check_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
<p>
<b class="bf">3(c)</b>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div class="choicegroup capa_inputtype" id="inputtype_ps2-3-check_4_1">
<fieldset aria-describedby="status_ps2-3-check_4_1">
<div class="field">
<input type="radio" name="input_ps2-3-check_4_1" id="input_ps2-3-check_4_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="ps2-3-check_4_1-choice_1-label" for="input_ps2-3-check_4_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_ps2-3-check_4_1">
<text> Stable</text>
</label>
</div>
<div class="field">
<input type="radio" name="input_ps2-3-check_4_1" id="input_ps2-3-check_4_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="ps2-3-check_4_1-choice_2-label" for="input_ps2-3-check_4_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_ps2-3-check_4_1">
<text> Unstable</text>
</label>
</div>
<span id="answer_ps2-3-check_4_1"/>
</fieldset>
<div class="indicator-container">
<span class="status unanswered" id="status_ps2-3-check_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
</div></div>
</p>
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Problem 4a
</h3>
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<div class="problem">
<div>
<p><b class="bf">4(a)</b> The checker is not yet sophisticated enough to check general solutions. Give the solution with initial conditions </p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto">
<tr>
<td class="equation" style="width:80%; border:none">[mathjax]x(0) = 5, \quad x'(0) = 2, \quad x' '(0) = 1, \quad x' ' '(0) = 0.[/mathjax]</td>
<td class="eqnnum" style="width:20%; border:none">&#160;</td>
</tr>
</table>
<p>
(Hint, in the order I wrote my general solution, the coefficients for this are [mathjaxinline]c_1=2[/mathjaxinline], [mathjaxinline]c_2 = 1[/mathjaxinline], [mathjaxinline]c_3=2[/mathjaxinline], [mathjaxinline]c_4=1[/mathjaxinline].) </p>
<p>
<p style="display:inline">[mathjaxinline]x(t) =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps2-4a-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
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<span class="status unanswered" id="status_ps2-4a-check_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps2-4a-check_2_1" class="answer"/>
<div id="display_ps2-4a-check_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps2-4a-check_2_1_dynamath" name="input_ps2-4a-check_2_1_dynamath"/>
</div>
</div></div>
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<h3 class="hd hd-3 problem-header" id="ps2-4b-check-problem-title" aria-describedby="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps2-4b-check-problem-progress" tabindex="-1">
Problem 4b
</h3>
<div class="problem-progress" id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps2-4b-check-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bf">4(b)</b> The checker is not yet sophisticated enough to check general solutions. Give the periodic solution with initial conditions [mathjaxinline]x(0) = 4[/mathjaxinline], [mathjaxinline]x'(0)=1[/mathjaxinline]. </p>
<p>
(Hint, in the order I wrote my general solution, the coefficients for this are 4 and 1.) </p>
<p>
<p style="display:inline">[mathjaxinline]x(t) =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps2-4b-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
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<span class="trailing_text" id="trailing_text_ps2-4b-check_2_1"/>
<span class="status unanswered" id="status_ps2-4b-check_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps2-4b-check_2_1" class="answer"/>
<div id="display_ps2-4b-check_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps2-4b-check_2_1_dynamath" name="input_ps2-4b-check_2_1_dynamath"/>
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Problem 4cde
</h3>
<div class="problem-progress" id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps2-4cde-check-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bf">4(c)</b> The checker is not yet sophisticated enough to check general solutions. Give the solution that goes to 0 with initial condition [mathjaxinline]x(0) = 4[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]x(t) =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps2-4cde-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps2-4cde-check_2_1" id="input_ps2-4cde-check_2_1" aria-describedby="status_ps2-4cde-check_2_1" value="" class="math" size="70"/>
<span class="trailing_text" id="trailing_text_ps2-4cde-check_2_1"/>
<span class="status unanswered" id="status_ps2-4cde-check_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps2-4cde-check_2_1" class="answer"/>
<div id="display_ps2-4cde-check_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps2-4cde-check_2_1_dynamath" name="input_ps2-4cde-check_2_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<b class="bf">4(d)</b>
<em>No answer checker given.</em>
</p>
<p>
<b class="bf">4(e)</b>
<em>No answer checker given.</em>
</p>
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Problem 5ab
</h3>
<div class="problem-progress" id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps2-5ab-check-problem-progress"></div>
<div class="problem">
<div>
<p>
<b class="bf">5(a)</b>
<em>No answer checker given.</em>
</p>
<p><b class="bf">5(b)</b> Ranges are given in the usual notation, e.g. [2,5) is the range [mathjaxinline]2\le \zeta &lt; 5[/mathjaxinline]. For infinity, you can enter: infty (all lower case). </p>
<p>
Give the range of [mathjaxinline]\zeta[/mathjaxinline] where the system is overdamped: <br/><div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps2-5ab-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_ps2-5ab-check_2_1" id="input_ps2-5ab-check_2_1" aria-describedby="status_ps2-5ab-check_2_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps2-5ab-check_2_1"/>
<span class="status unanswered" id="status_ps2-5ab-check_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps2-5ab-check_2_1" class="answer"/>
<div id="display_ps2-5ab-check_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps2-5ab-check_2_1_dynamath" name="input_ps2-5ab-check_2_1_dynamath"/>
</div>
</div></div> </p>
<p>
Give the range of [mathjaxinline]\zeta[/mathjaxinline] where the system is underdamped:<br/><div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_ps2-5ab-check_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="unanswered inline">
<input type="text" name="input_ps2-5ab-check_3_1" id="input_ps2-5ab-check_3_1" aria-describedby="status_ps2-5ab-check_3_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps2-5ab-check_3_1"/>
<span class="status unanswered" id="status_ps2-5ab-check_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps2-5ab-check_3_1" class="answer"/>
<div id="display_ps2-5ab-check_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps2-5ab-check_3_1_dynamath" name="input_ps2-5ab-check_3_1_dynamath"/>
</div>
</div></div> </p>
<p>
Give the value of [mathjaxinline]\zeta[/mathjaxinline] where the system is critically damped:<br/><div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_ps2-5ab-check_4_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps2-5ab-check_4_1" id="input_ps2-5ab-check_4_1" aria-describedby="status_ps2-5ab-check_4_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps2-5ab-check_4_1"/>
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<div id="display_ps2-5ab-check_4_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps2-5ab-check_4_1_dynamath" name="input_ps2-5ab-check_4_1_dynamath"/>
</div>
</div></div> </p>
<p>
Give the value of [mathjaxinline]\zeta[/mathjaxinline] where the system is undamped:<br/><div class="inline" tabindex="-1" aria-label="Question 4" role="group"><div id="inputtype_ps2-5ab-check_5_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps2-5ab-check_5_1" id="input_ps2-5ab-check_5_1" aria-describedby="status_ps2-5ab-check_5_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps2-5ab-check_5_1"/>
<span class="status unanswered" id="status_ps2-5ab-check_5_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps2-5ab-check_5_1" class="answer"/>
<div id="display_ps2-5ab-check_5_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps2-5ab-check_5_1_dynamath" name="input_ps2-5ab-check_5_1_dynamath"/>
</div>
</div></div> </p>
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data-problem-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps2-5c-check" data-url="/courses/course-v1:MITx+ES.1803+2023_Fall/xblock/block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps2-5c-check/handler/xmodule_handler"
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<h3 class="hd hd-3 problem-header" id="ps2-5c-check-problem-title" aria-describedby="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps2-5c-check-problem-progress" tabindex="-1">
Problem 5c
</h3>
<div class="problem-progress" id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps2-5c-check-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bf">5(c)</b> Give the time when each case reaches practical equilibrium. Give your answer accurate to one decimal place. </p>
<p>
<p style="display:inline">[mathjaxinline]\zeta _1=0.8[/mathjaxinline]: &#8201; [mathjaxinline]t=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps2-5c-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
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<input type="text" name="input_ps2-5c-check_2_1" id="input_ps2-5c-check_2_1" aria-describedby="status_ps2-5c-check_2_1" value="" class="math" size="25"/>
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<span class="status unanswered" id="status_ps2-5c-check_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps2-5c-check_2_1" class="answer"/>
<div id="display_ps2-5c-check_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps2-5c-check_2_1_dynamath" name="input_ps2-5c-check_2_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<p style="display:inline">[mathjaxinline]\zeta _2=1.0[/mathjaxinline]: &#8201; [mathjaxinline]t=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_ps2-5c-check_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps2-5c-check_3_1" id="input_ps2-5c-check_3_1" aria-describedby="status_ps2-5c-check_3_1" value="" class="math" size="25"/>
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<span class="status unanswered" id="status_ps2-5c-check_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps2-5c-check_3_1" class="answer"/>
<div id="display_ps2-5c-check_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps2-5c-check_3_1_dynamath" name="input_ps2-5c-check_3_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<p style="display:inline">[mathjaxinline]\zeta _3=1.1[/mathjaxinline]: &#8201; [mathjaxinline]t=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="inputtype_ps2-5c-check_4_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
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<input type="text" name="input_ps2-5c-check_4_1" id="input_ps2-5c-check_4_1" aria-describedby="status_ps2-5c-check_4_1" value="" class="math" size="25"/>
<span class="trailing_text" id="trailing_text_ps2-5c-check_4_1"/>
<span class="status unanswered" id="status_ps2-5c-check_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps2-5c-check_4_1" class="answer"/>
<div id="display_ps2-5c-check_4_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps2-5c-check_4_1_dynamath" name="input_ps2-5c-check_4_1_dynamath"/>
</div>
</div></div>
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<h3 class="hd hd-3 problem-header" id="ps2-5de-check-problem-title" aria-describedby="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps2-5de-check-problem-progress" tabindex="-1">
Problem 5de
</h3>
<div class="problem-progress" id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps2-5de-check-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bf">5(d) (for fun)</b> Give the damping ratio that causes [mathjaxinline]x[/mathjaxinline] to reach practical equilibrium the fastest. Give your answer to 2 decimal places. </p>
<p>
<p style="display:inline">[mathjaxinline]\zeta =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps2-5de-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
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<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<div id="display_ps2-5de-check_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps2-5de-check_2_1_dynamath" name="input_ps2-5de-check_2_1_dynamath"/>
</div>
</div></div>
</p>
<p>
<b class="bf">5(e) (for fun)</b>
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Problem 6
</h3>
<div class="problem-progress" id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps2-6-check-problem-progress"></div>
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<p>
<b class="bf">6.</b>
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