qu.1.topic=Questions@

qu.1.1.mode=Multiple Choice@
qu.1.1.name=What is the natural frequency of a simple pendulum?@
qu.1.1.comment=<p>
	The equation of motion for the pendulum is&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle fontfamily="Times New Roman" mathsize="12pt" mathcolor="#000000" veryverythinmathspace="0.0555556em" verythinmathspace="0.111111em" thinmathspace="0.166667em" mediummathspace="0.222222em" thickmathspace="0.277778em" verythickmathspace="0.333333em" veryverythickmathspace="0.388889em" scriptlevel="0" scriptsizemultiplier="0.71" scriptminsize="8.0pt"><mrow><mover><mrow><mi>θ</mi></mrow><mi>¨</mi></mover><mfenced open="(" close=")" separators=","><mrow><mi>t</mi></mrow></mfenced><mo lspace="0.2222222em" rspace="0.2222222em">+</mo><mfenced open="(" close=")" separators=","><mrow><mfrac><mi>g</mi><mrow><mi>l</mi></mrow></mfrac></mrow></mfenced><mi>θ</mi><mfenced open="(" close=")" separators=","><mrow><mi>t</mi></mrow></mfenced><mo mathvariant="italic" lspace="0.2777778em" rspace="0.2777778em">=</mo><mn mathvariant="italic">0</mn><mo mathvariant="italic" lspace="0.0em" rspace="0.0em"> </mo></mrow></mstyle></math>​.</p>@
qu.1.1.editing=useHTML@
qu.1.1.solution=<p>
	<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msqrt><mrow><mfrac><mi>l</mi><mrow><mi>g</mi></mrow></mfrac></mrow></msqrt></mrow></math>​</p>@
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qu.1.1.question=<p>
	What is the natural frequency of a simple pendulum of length l and mass m?&nbsp;</p>
<p>
	<img alt="" src="__BASE_URI__spring-mass-analog_pendulum.PNG" style="width: 389px; height: 329px; float: left;"></p>
<p>
	<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi></mi></mrow></math></p>
@
qu.1.1.answer=5@
qu.1.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msqrt><mrow><mi>g</mi></mrow></msqrt></mrow></mstyle></math>@
qu.1.1.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msqrt><mrow><mi>lgm</mi></mrow></msqrt></mrow></mstyle></math>@
qu.1.1.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msqrt><mrow><mfrac><mi>l</mi><mrow><mi>g</mi></mrow></mfrac></mrow></msqrt></mrow></mstyle></math>@
qu.1.1.choice.4=Cannot be determined from the information provided.@
qu.1.1.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msqrt><mrow><mfrac><mi>g</mi><mrow><mi>l</mi></mrow></mfrac></mrow></msqrt></mrow></mstyle></math>@
qu.1.1.fixed=@

qu.1.2.mode=Multiple Choice@
qu.1.2.name=Phase difference@
qu.1.2.comment=<p>
	Since the displacement is a sinusoidal function of time, its derivative with respect to time will be a sinusoidal function with a phase difference of Pi/2 radians.&nbsp;</p>@
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qu.1.2.description=For a friction-less spring-mass oscillator, what is the phase difference (in radians) between the displacement and the rate of change of the displacement?@
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qu.1.2.question=<p>
	For a friction-less spring-mass oscillator, what is the phase difference (in radians) between the displacement and the rate of change of the displacement?&nbsp;</p>
<p>
	<img alt="" src="__BASE_URI__spring-mass.PNG" style="width: 400px; height: 248px;"></p>
@
qu.1.2.answer=3@
qu.1.2.choice.1=0@
qu.1.2.choice.2=Pi/4 @
qu.1.2.choice.3=Pi/2 @
qu.1.2.choice.4=Pi@
qu.1.2.fixed=@

qu.1.3.mode=Multiple Choice@
qu.1.3.name=Number of complete oscillations@
qu.1.3.comment=<p>
	The number of complete oscillations is given by&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle fontfamily="Times New Roman" mathsize="12pt" mathcolor="#000000" veryverythinmathspace="0.0555556em" verythinmathspace="0.111111em" thinmathspace="0.166667em" mediummathspace="0.222222em" thickmathspace="0.277778em" verythickmathspace="0.333333em" veryverythickmathspace="0.388889em" scriptlevel="0" scriptsizemultiplier="0.71" scriptminsize="8.0pt"><mrow><mfrac><msub><mrow><mi>ω</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mn>2</mn><mrow><mi>π</mi></mrow></mrow></mfrac><mo lspace="0.0em" rspace="0.0em">⋅</mo><mi>t</mi></mrow></mstyle></math>​.</p>@
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qu.1.3.uid=6231ace7-4047-4d51-9424-807b72fba81c@
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qu.1.3.description=@
qu.1.3.difficulty=0.0@
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qu.1.3.attributeAuthor=true@
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qu.1.3.question=<p>
	If the equation of motion of a system is given by</p>
<p>
	&nbsp;</p>
<p>
	<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>2</mn><mo>⋅</mo><mi mathvariant="italic">x</mi><mo>'</mo><mo>'</mo><mfenced open="(" close=")" separators=","><mrow><mi mathvariant="italic">t</mi></mrow></mfenced><mo>+</mo><mn>18</mn><mo>⋅</mo><mi mathvariant="italic">x</mi><mo>=</mo><mn>0</mn></mrow></math> ​</p>
<p>
	&nbsp;</p>
<p>
	where&nbsp;<em>x</em> is the displacement, how many oscillations does it complete in 4 seconds?&nbsp;</p>
@
qu.1.3.answer=1@
qu.1.3.choice.1=6/Pi@
qu.1.3.choice.2=3/Pi@
qu.1.3.choice.3=9@
qu.1.3.choice.4=4@
qu.1.3.choice.5=Pi@
qu.1.3.fixed=@

qu.1.4.mode=Multiple Choice@
qu.1.4.name=U-tube manometer natural frequency@
qu.1.4.comment=<p>
	The equation of motion of the fluid is given by</p>
<p>
	<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle fontfamily="Times New Roman" mathsize="12pt" mathcolor="#000000" veryverythinmathspace="0.0555556em" verythinmathspace="0.111111em" thinmathspace="0.166667em" mediummathspace="0.222222em" thickmathspace="0.277778em" verythickmathspace="0.333333em" veryverythickmathspace="0.388889em" scriptlevel="0" scriptsizemultiplier="0.71" scriptminsize="8.0pt"><mrow><mfenced open="[" close="]" separators=","><mrow><mi>ρ</mi><msub><mi>A</mi><mrow><mi>c</mi></mrow></msub><mi>l</mi></mrow></mfenced><mover><mrow><mi>x</mi></mrow><mi>¨</mi></mover><mfenced open="(" close=")" separators=","><mrow><mi>t</mi></mrow></mfenced><mo lspace="0.2222222em" rspace="0.2222222em">+</mo><mfenced open="[" close="]" separators=","><mrow><mi>ρg2x</mi><mfenced open="(" close=")" separators=","><mrow><mi>t</mi></mrow></mfenced></mrow></mfenced><msub><mi>A</mi><mrow><mi>c</mi></mrow></msub><mo lspace="0.2777778em" rspace="0.2777778em">=</mo><mn>0</mn></mrow></mstyle></math>​</p>
<p>
	&nbsp;</p>@
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qu.1.4.algorithm=@
qu.1.4.uid=452d3640-a4e6-4290-b901-24dc703ab8cf@
qu.1.4.privacy=10@
qu.1.4.allowRepublish=false@
qu.1.4.description=@
qu.1.4.difficulty=1.0@
qu.1.4.modifiedBy=23b262f5-37a1-4326-89fa-49a0d9525aa2@
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qu.1.4.question=<p>
	What is the natural frequency of oscillation of the fluid in the following U-tube manometer?</p>
<p>
	<img alt="" src="__BASE_URI__u-tube.PNG" style="width: 500px; height: 224px;"></p>
@
qu.1.4.answer=1@
qu.1.4.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msqrt><mrow><mfrac><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>g</mi></mrow><mrow><mi>l</mi></mrow></mfrac></mrow></msqrt></mrow></mstyle></math>@
qu.1.4.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msqrt><mrow><mfrac><mi>g</mi><mrow><mi>l</mi></mrow></mfrac></mrow></msqrt></mrow></mstyle></math>@
qu.1.4.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msqrt><mrow><mfrac><mi>g</mi><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>l</mi></mrow></mfrac></mrow></msqrt></mrow></mstyle></math>@
qu.1.4.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msqrt><mrow><mfrac><mrow><mn>4</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>g</mi></mrow><mrow><mi>l</mi></mrow></mfrac></mrow></msqrt></mrow></mstyle></math>@
qu.1.4.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msqrt><mrow><mfrac><mi>g</mi><mrow><mn>4</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>l</mi></mrow></mfrac></mrow></msqrt></mrow></mstyle></math>@
qu.1.4.choice.6=Cannot be determined from the information provided.@
qu.1.4.fixed=@

qu.1.5.mode=Multiple Choice@
qu.1.5.name=Underdamped motion@
qu.1.5.comment=<p>
	Since the object crosses the equilibrium position more than once, the motion is underdamped.&nbsp;</p>@
qu.1.5.editing=useHTML@
qu.1.5.solution=@
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qu.1.5.question=<p>
	The following plot shows the displacement of an object with respect to time. What kind of motion is this?</p>
<p>
	<img alt="" src="__BASE_URI__under-damped(1).PNG" style="width: 600px; height: 220px;"></p>
@
qu.1.5.answer=1@
qu.1.5.choice.1=Underdamped motion@
qu.1.5.choice.2=Overdamped motion@
qu.1.5.choice.3=Critically damped motion@
qu.1.5.fixed=@

qu.1.6.question=<p>
	If the equation of motion of a system is given by</p>
<p>
	<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12pt' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>3</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>y</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>14</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>y</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>k</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>y</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>0</mn></mrow></mstyle></math> </span> ​</p>
<p>
	find the value of&nbsp;<em>k</em> for which the system will be critically damped. (Answer using 3 significant figures)&nbsp;</p>@
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qu.1.6.numStyle=thousands scientific dollars arithmetic@
qu.1.6.mode=Numeric@
qu.1.6.name=Find k for critically damped motion@
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qu.1.7.question=<p>
	The natural frequency of a spring-mass oscillator with damping is $wn rad/s, the damping ratio is</p>
<p>
	$zeta, the initial displacement is $x0 m and the initial velocity is $v0 m/s. If the response is in the form</p>
<p>
	&nbsp;</p>
<p>
	<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">x</mi><mfenced open="(" close=")" separators=","><mrow><mi mathvariant="italic">t</mi></mrow></mfenced><mo>=</mo><msup><mi mathvariant="italic">e</mi><mrow><mo>−</mo><mrow><mi mathvariant="italic">ζ</mi><msub><mrow><mi mathvariant="italic">ω</mi></mrow><mrow><mi mathvariant="italic">n</mi></mrow></msub><mi mathvariant="italic">t</mi></mrow></mrow></msup><mn>sin</mn><mfenced open="(" close=")" separators=","><mrow><msub><mrow><mi mathvariant="italic">ω</mi></mrow><mrow><mi mathvariant="italic">d</mi></mrow></msub><mi mathvariant="italic">t</mi><mo>+</mo><mi mathvariant="italic">φ</mi></mrow></mfenced></mrow></math></p>
<p>
	&nbsp;</p>
<p>
	​what is the phase?</p>
<p>
	&nbsp;</p>
<p>
	Provide the answer in radians rounded to two significant figures.&nbsp;</p>@
qu.1.7.answer.num=$ans@
qu.1.7.answer.units=@
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qu.1.7.grading=exact_value@
qu.1.7.negStyle=minus@
qu.1.7.numStyle=thousands scientific dollars arithmetic@
qu.1.7.mode=Numeric@
qu.1.7.name=Phase@
qu.1.7.comment=<p>
	The phase is given by</p>
<p>
	<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle fontfamily="Times New Roman" mathsize="12pt" mathcolor="#000000" veryverythinmathspace="0.0555556em" verythinmathspace="0.111111em" thinmathspace="0.166667em" mediummathspace="0.222222em" thickmathspace="0.277778em" verythickmathspace="0.333333em" veryverythickmathspace="0.388889em" scriptlevel="0" scriptsizemultiplier="0.71" scriptminsize="8.0pt"><mrow><mi>φ</mi><mo mathvariant="italic" lspace="0.2777778em" rspace="0.2777778em">=</mo><msup><mi>tan</mi><mrow><mo lspace="0.2222222em" rspace="0.2222222em">−</mo><mn>1</mn></mrow></msup><mfenced open="(" close=")" separators=","><mrow><mfrac><mrow><msub><mi>x</mi><mrow><mn>0</mn></mrow></msub><mo lspace="0.0em" rspace="0.0em">⋅</mo><mrow><msub><mi>ω</mi><mrow><mi>d</mi></mrow></msub></mrow></mrow><mrow><msub><mi>v</mi><mrow><mn>0</mn></mrow></msub><mo lspace="0.2222222em" rspace="0.2222222em">+</mo><mrow><mi>ζ</mi><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow><mrow><msub><mi>x</mi><mrow><mn>0</mn></mrow></msub></mrow></mrow></mrow></mfrac></mrow></mfenced></mrow></mstyle></math>​</p>
<p>
	In this case, since the numerator of the argument of the inverse tan function is negative and the denominator is positive, the angle will lie in the fourth quadrant.&nbsp;</p>@
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qu.1.7.solution=$ans@
qu.1.7.algorithm=$wn=decimal(1,range(1,1.5,0.1));
$zeta=decimal(1,range(0.1,0.4,0.1));
$x0=range(-5,-1,1);
$v0=range(10,15,1);
$ans=arctan(($x0*sqrt(1-$zeta^2)*$wn)/($v0+$zeta*$wn*$x0));@
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qu.1.8.mode=True False@
qu.1.8.name=Is kinetic friction viscous damping?@
qu.1.8.comment=<p>
	Both static and kinetic friction are not directly proportional to an object's velocity. Therefore, the friction that opposes the motion of an object sliding on a surface is not an example of viscous damping.&nbsp;</p>@
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qu.1.8.question=<p>
	True or False: The friction that resists the motion of an object sliding on a surface is an example of viscous damping.</p>@
qu.1.8.answer=2@
qu.1.8.choice.1=True@
qu.1.8.choice.2=False@
qu.1.8.fixed=@

qu.1.9.mode=Blanks@
qu.1.9.name=If the displacement is +ve acc. is -ve.@
qu.1.9.comment=<p>
	The following equations show the forms of the displacement, velocity and acceleration of a spring-mass oscillator.&nbsp;</p>
<p>
	<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle fontfamily="Times New Roman" mathsize="12pt" mathcolor="#000000" veryverythinmathspace="0.0555556em" verythinmathspace="0.111111em" thinmathspace="0.166667em" mediummathspace="0.222222em" thickmathspace="0.277778em" verythickmathspace="0.333333em" veryverythickmathspace="0.388889em" scriptlevel="0" scriptsizemultiplier="0.71" scriptminsize="8.0pt"><mrow><mi>x</mi><mfenced open="(" close=")" separators=","><mrow><mi>t</mi></mrow></mfenced><mo mathvariant="italic" lspace="0.2777778em" rspace="0.2777778em">=</mo><mi>A</mi><mo mathvariant="italic" lspace="0.0em" rspace="0.0em"> </mo><mi>sin</mi><mfenced open="(" close=")" separators=","><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>n</mi></mrow></msub><mi>t</mi><mo lspace="0.2222222em" rspace="0.2222222em">+</mo><mi>φ</mi></mrow></mfenced><mspace height="0.0ex" width="0.0em" depth="0.0ex" linebreak="newline"></mspace><mover><mrow><mi>x</mi></mrow><mi>˙</mi></mover><mfenced open="(" close=")" separators=","><mrow><mi>t</mi></mrow></mfenced><mo mathvariant="italic" lspace="0.2777778em" rspace="0.2777778em">=</mo><mi>A</mi><mo mathvariant="italic" lspace="0.0em" rspace="0.0em"> </mo><msub><mi>ω</mi><mrow><mi>n</mi></mrow></msub><mi mathvariant="normal">cos</mi><mfenced open="(" close=")" separators=","><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>n</mi></mrow></msub><mi>t</mi><mo lspace="0.2222222em" rspace="0.2222222em">+</mo><mi>φ</mi></mrow></mfenced><mo lspace="0.0em" rspace="0.0em"> </mo><mspace height="0.0ex" width="0.0em" depth="0.0ex" linebreak="newline"></mspace><mover><mrow><mi>x</mi></mrow><mi>¨</mi></mover><mfenced open="(" close=")" separators=","><mrow><mi>t</mi></mrow></mfenced><mo mathvariant="italic" lspace="0.2777778em" rspace="0.2777778em">=</mo><mo mathvariant="italic" lspace="0.2222222em" rspace="0.2222222em">−</mo><mi>A</mi><mo mathvariant="italic" lspace="0.0em" rspace="0.0em"> </mo><msup><msub><mi>ω</mi><mrow><mi>n</mi></mrow></msub><mrow><mn>2</mn></mrow></msup><mi mathvariant="normal">sin</mi><mfenced open="(" close=")" separators=","><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>n</mi></mrow></msub><mi>t</mi><mo lspace="0.2222222em" rspace="0.2222222em">+</mo><mi>φ</mi></mrow></mfenced><mspace height="0.0ex" width="0.0em" depth="0.0ex" linebreak="auto"></mspace><mspace height="0.0ex" width="0.0em" depth="0.0ex" linebreak="auto"></mspace></mrow></mstyle></math>​</p>
<p>
	&nbsp;</p>@
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qu.1.9.question=For an oscillating spring-mass system: a) if the displacement is positive then the acceleration has to be  <1> . b) if the displacement is zero then the acceleration has to be  <2> . c) if the velocity is negative then the displacement has to be  <3> . @
qu.1.9.blank.1=negative@
qu.1.9.blank.2=zero@
qu.1.9.blank.3=decreasing@
qu.1.9.extra=positive,increasing@

qu.1.10.mode=Multiple Choice@
qu.1.10.name=Harmonic 1. Beats@
qu.1.10.comment=<p>
	This rapid oscillation of increasing and decreasing amplitude corresponds to the phenomena of beats.&nbsp;</p>@
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qu.1.10.question=<p>
	<img alt="" src="__BASE_URI__Beats.PNG" style="width: 700px; height: 206px;"></p>
<p>
	&nbsp;</p>
<p>
	Which phenomena does this response correspond to?&nbsp;</p>
@
qu.1.10.answer=1@
qu.1.10.choice.1=Beats@
qu.1.10.choice.2=Resonance@
qu.1.10.choice.3=Simple Harmonic Motion@
qu.1.10.fixed=@

qu.1.11.mode=Multiple Choice@
qu.1.11.name=Harmonic 1. Resonance@
qu.1.11.comment=<p>
	Since the amplitude of the displacement keeps on increasing, this response is an example of resonance.&nbsp;</p>@
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qu.1.11.question=<p>
	<img alt="" src="__BASE_URI__Resonance.PNG" style="width: 700px; height: 306px;"></p>
<p>
	&nbsp;</p>
<p>
	Which phenomena does this response correspond to?</p>
@
qu.1.11.answer=2@
qu.1.11.choice.1=Beats@
qu.1.11.choice.2=Resonance@
qu.1.11.choice.3=Simple Harmonic Motion@
qu.1.11.fixed=@

qu.1.12.mode=Formula@
qu.1.12.name=Harmonic 1. Resonance slope@
qu.1.12.comment=<p>
	The undmaped response at resonance for a cosine forcing function of magnitude <em>F</em>&nbsp;is given by</p>
<p>
	<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>x</mi><mfenced open="(" close=")" separators=","><mrow><mi>t</mi></mrow></mfenced><mo lspace="0.2777778em" rspace="0.2777778em">=</mo><mfrac><msub><mi>v</mi><mrow><mn>0</mn></mrow></msub><mrow><mi>ω</mi></mrow></mfrac><mo lspace="0.0em" rspace="0.0em">⋅</mo><mn>sin</mn><mfenced open="(" close=")" separators=","><mrow><mi>ω</mi><mo mathvariant="italic" lspace="0.0em" rspace="0.0em">⋅</mo><mi>t</mi></mrow></mfenced><mo lspace="0.2222222em" rspace="0.2222222em">+</mo><msub><mi>x</mi><mrow><mn>0</mn></mrow></msub><mo lspace="0.0em" rspace="0.0em">⋅</mo><mn>cos</mn><mfenced open="(" close=")" separators=","><mrow><mi>ω</mi><mo mathvariant="italic" lspace="0.0em" rspace="0.0em">⋅</mo><mi>t</mi></mrow></mfenced><mo lspace="0.2222222em" rspace="0.2222222em">+</mo><mfrac><msub><mi>f</mi><mrow><mn>0</mn></mrow></msub><mrow><mn>2</mn><mo lspace="0.0em" rspace="0.0em">⋅</mo><mrow><mi>ω</mi></mrow></mrow></mfrac><mo lspace="0.0em" rspace="0.0em">⋅</mo><mi>t</mi><mo lspace="0.0em" rspace="0.0em">⋅</mo><mn>sin</mn><mfenced open="(" close=")" separators=","><mrow><mi>ω</mi><mo mathvariant="italic" lspace="0.0em" rspace="0.0em">⋅</mo><mi>t</mi></mrow></mfenced></mrow></math> where&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>f</mi><mrow><mn>0</mn></mrow></msub><mo lspace="0.2777778em" rspace="0.2777778em">=</mo><mrow><mfrac><mi>F</mi><mrow><mi>m</mi></mrow></mfrac></mrow></mrow></math> ​.</p>
<p>
	&nbsp;</p>
<p>
	Therefore, the slope of the dashed line shown in the plot is&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><msub><mi>f</mi><mrow><mn>0</mn></mrow></msub><mrow><mn>2</mn><mo lspace="0.0em" rspace="0.0em">⋅</mo><mrow><mi>ω</mi></mrow></mrow></mfrac></mrow></math> ​or, in terms of the given parameters, <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mi>F</mi><mrow><mn>2</mn><mo lspace="0.0em" rspace="0.0em">⋅</mo><mrow><msqrt><mrow><mi>k</mi><mo lspace="0.0em" rspace="0.0em">⋅</mo><mi>m</mi></mrow></msqrt></mrow></mrow></mfrac></mrow></math> ​ .</p>@
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qu.1.12.question=<p>
	A spring-mass system of mass&nbsp;<em>m</em> and spring constant&nbsp;<em>k</em>&nbsp;with zero initial displacement and zero initial velocity is excited by a forcing function of the form&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>F</mi><mo mathvariant="italic" lspace="0.0em" rspace="0.0em">⋅</mo><mi>cos</mi><mfenced open="(" close=")" separators=","><mrow><msqrt><mrow><mfrac><mi>k</mi><mrow><mi>m</mi></mrow></mfrac></mrow></msqrt><mo lspace="0.0em" rspace="0.0em">⋅</mo><mi>t</mi></mrow></mfenced></mrow></math> ​. What is the slope of the dashed lines in the plot in terms of the above parameters?&nbsp;</p>
<p>
	<img alt="" src="__BASE_URI__Resonance-slope.PNG" style="width: 700px; height: 311px;"></p>@
qu.1.12.answer=F/2/sqrt(m*k)@

qu.1.13.question=<p>
	A system modelled as a spring-mass-damper has the following parameters:</p>
<p>
	&nbsp;</p>
<p>
	Spring constant,&nbsp;<em>k</em>= $k N/m</p>
<p>
	Damping coefficient, <em>c</em>= $c N·s/m</p>
<p>
	Mass, <i>m</i>= $m kg</p>
<p>
	&nbsp;</p>
<p>
	The system is excited by a sinusoidal forcing function of amplitude $F N with an angular frequency <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">ω</mi><mo>=</mo></mrow></math> ​$omega rad/s.&nbsp;</p>
<p>
	&nbsp;</p>
<p>
	By how many degrees does the steady state response lag the forcing function? (Round the answer to the first decimal place.)</p>
<p>
	&nbsp;</p>@
qu.1.13.answer.num=$theta@
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qu.1.13.numStyle=thousands scientific dollars arithmetic@
qu.1.13.mode=Numeric@
qu.1.13.name=Harmonic 2. Phase lag@
qu.1.13.comment=<p>
	In terms of the parameters given in the question, the phase lag is given by</p>
<p>
	<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12pt' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi mathvariant='normal'>tan</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi>c</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mi>&omega;</mi></mrow></mrow><mrow><mi>k</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>m</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><msup><mi>&omega;</mi><mrow><mn>2</mn></mrow></msup></mrow></mrow></mfrac></mrow></mfenced></mrow></mstyle></math> </span>​</p>@
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$c=range(5,10,1);
$m=range(20,25,1);
$F=range(20,30,1);
$omega=range(2,4,1);
$zeta=$c/2/sqrt($k*$m);
$omegan=sqrt($k/$m);
$theta=180/Pi*(if(lt($omega/$omegan,1),atan(2*$zeta*$omegan*$omega/($omegan^2-$omega^2)),atan(2*$zeta*$omegan*$omega/($omegan^2-$omega^2))+Pi));@
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qu.1.14.question=<p>
	A system modelled as a spring-mass-damper has the following parameters:</p>
<p>
	&nbsp;</p>
<p>
	Spring constant,&nbsp;<em>k</em>= $k N/m</p>
<p>
	Damping coefficient, <em>c</em>= $c N·s/m</p>
<p>
	Mass, <i>m</i>= $m kg</p>
<p>
	&nbsp;</p>
<p>
	If the system is excited by a sinusoidal forcing function of amplitude F= $F N , what should the angular frequency (in rad/s) of the forcing function be to get vibrations of the greatest amplitude?</p>
<p>
	&nbsp;</p>
<p>
	(Provide the answer using two significant digits. Answer 0 if the amplitude does not have a peak with respect to the frequency.)</p>
<p>
	&nbsp;</p>@
qu.1.14.answer.num=$ans@
qu.1.14.answer.units=@
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qu.1.14.numStyle=thousands scientific dollars arithmetic@
qu.1.14.mode=Numeric@
qu.1.14.name=Harmonic 2. Peak frequency@
qu.1.14.comment=<p>
	In terms of the parameters given in the question, the peak 
frequency is given by</p>
<p>
	<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle 
fontfamily='Times New Roman' mathsize='12pt' mathcolor='#000000'  
veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 
thinmathspace='0.166667em' mediummathspace='0.222222em' 
thickmathspace='0.277778em' verythickmathspace='0.333333em' 
veryverythickmathspace='0.388889em' scriptlevel='0' 
scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msqrt><mrow>
<mfrac><mi>k</mi><mrow><mi>m</mi></mrow></mfrac><mo lspace='0.0em' 
rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow>
<mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo>
<mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' 
close=')' separators=','><mrow><mfrac><msup><mi>c</mi><mrow><mn>2</mn>
</mrow></msup><mrow><mn>4</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo>
<mi>k</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow>
</mfrac></mrow></mfenced></mrow></mfenced></mrow></msqrt></mrow></mstyle>
</math> </span> ​</p>
<p>
	This is only valid when the term under the square root sign is 
greater than zero. If it is less than zero, then the amplitude of the 
vibrations decreases with increasing frequency and is greatest just above 0 
rad/s.&nbsp;</p>
<p>
	&nbsp;</p>@
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qu.1.14.algorithm=$k=range(10,20,1);
$c=range(20,30,1);
$m=range(20,25,1);
$F=range(20,25,1);
$zeta=$c/2/sqrt($k*$m);
$omegan=sqrt($k/$m);
$ans=if(lt($zeta,1/sqrt(2)),$omegan*sqrt(1-2*$zeta^2),0);@
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qu.1.15.question=<p>
	A system modelled as a spring-mass-damper has the following parameters:</p>
<p>
	&nbsp;</p>
<p>
	Spring constant,&nbsp;<em>k</em>= $k N/m</p>
<p>
	Damping coefficient, <em>c</em>= $c N·s/m</p>
<p>
	Mass, <i>m</i>= $m kg</p>
<p>
	&nbsp;</p>
<p>
	The system is excited by a sinusoidal forcing function of amplitude <em>F</em>= $F N with an angular frequency <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">ω</mi><mo>=</mo></mrow></math> ​$omega rad/s.&nbsp;</p>
<p>
	&nbsp;</p>
<p>
	What is the percentage change in the amplitude of the steady-state vibrations if the excitation frequency is increased by $p %? (Round the answer to three significant figures)</p>
<p>
	&nbsp;</p>@
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qu.1.15.name=Harmonic 2. Percent change amplitude@
qu.1.15.comment=<p>
	The amplitude of the steady-state vibrations is given by&nbsp;</p>
<p>
	<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12pt' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>X</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mi>F</mi></mrow><mrow><mi>m</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><msqrt><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msubsup><mi>&omega;</mi><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><msup><mi>&omega;</mi><mrow><mn>2</mn></mrow></msup></mrow></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&zeta;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><msub><mi>&omega;</mi><mrow><mi>n</mi></mrow></msub><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&omega;</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mrow></mfrac></mrow></mrow></mstyle></math> </span> ​</p>
<p>
	where</p>
<p>
	<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12pt' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msub><mi>&omega;</mi><mrow><mi>n</mi></mrow></msub><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msqrt><mrow><mfrac><mi>k</mi><mrow><mi>m</mi></mrow></mfrac></mrow></msqrt></mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math> </span> ​</p>
<p>
	and</p>
<p>
	<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12pt' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&zeta;</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>c</mi><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><msqrt><mrow><mi>k</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow></msqrt></mrow></mrow></mfrac></mrow></mrow></mstyle></math> </span>​</p>
<p>
	&nbsp;</p>@
qu.1.15.editing=useHTML@
qu.1.15.solution=@
qu.1.15.algorithm=$p=range(2,5,1);
$k=range(12,16,1);
$c=range(1,3,1);
$m=range(20,25,1);
$F=range(20,30,1);
$omega=range(0.7,0.8,0.05);
$zeta=$c/2/sqrt($k*$m);
$omegan=sqrt($k/$m);
$X1=$F/$m/sqrt(($omegan^2-$omega^2)^2+(2*$zeta*$omegan*$omega)^2);
$X2=$F/$m/sqrt(($omegan^2-((1+$p/100)*$omega)^2)^2+(2*$zeta*$omegan*((1+$p/100)*$omega))^2);
$ans=100*($X2-$X1)/$X1;@
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