009C Sample Final 3, Problem 9 - Revision history

Kayla Murray at 23:35, 3 December 2017

2017-12-03T23:35:18Z

Kayla Murray at 01:54, 3 December 2017

2017-12-03T01:54:42Z

Grad at 18:29, 19 March 2017

2017-03-19T18:29:14Z

Kayla Murray at 01:23, 5 March 2017

2017-03-05T01:23:54Z

Kayla Murray at 01:00, 5 March 2017

2017-03-05T01:00:52Z

Kayla Murray at 19:05, 18 February 2017

2017-02-18T19:05:26Z

Kayla Murray: Created page with "Compute ::a) $\lim_{n\rightarrow \infty} \frac{3-2n^2}{5n^2+n+1}$ ::b)
2017-02-18T02:52:18Z

Created page with "Compute ::a) <math style="vertical-align: -12px">\lim_{n\rightarrow \infty} \frac{3-2n^2}{5n^2+n+1}</math> ::b) <mat..."

New page
Compute

::a) <math style="vertical-align: -12px">\lim_{n\rightarrow \infty} \frac{3-2n^2}{5n^2+n+1}</math>

::b) <math style="vertical-align: -12px">\lim_{n\rightarrow \infty} \frac{\ln n}{\ln 3n}</math>

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'''Solution:'''

'''(a)'''

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'''(b)'''

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[[009C_Sample_Final_3|'''Return to Sample Exam''']]

← Older revision		Revision as of 23:35, 3 December 2017
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−	<span class="exam">A wheel of radius 1 rolls along a straight line, say the  <math>x</math>-axis. A point  <math style="vertical-align: 0px">P</math>  is located halfway between the center of the wheel and the rim. As the wheel rolls,  <math style="vertical-align: 0px">P</math>  traces a curve. Find parametric equations for the curve.	+	<span class="exam">A wheel of radius 1 rolls along a straight line, say the  <math>x</math>-axis. A point  <math style="vertical-align: 0px">P</math>  is located halfway between the center of the wheel and the rim; assume  <math style="vertical-align: 0px">P</math>  starts at the point  <math style="vertical-align: -15px">\bigg(0,\frac{1}{2}\bigg).</math>  As the wheel rolls,  <math style="vertical-align: 0px">P</math>  traces a curve. Find parametric equations for the curve.

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 <span class="exam">A wheel of radius 1 rolls along a straight line, say the &nbsp;<math>x</math>-axis. A point &nbsp;<math style="vertical-align: 0px">P</math>&nbsp; is located halfway between the center of the wheel and the rim. As the wheel rolls, &nbsp;<math style="vertical-align: 0px">P</math>&nbsp; traces a curve. Find parametric equations for the curve.
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
+<hr>
-!Foundations: &nbsp;
+[[009C Sample Final 3, Problem 9 Solution|'''<u>Solution</u>''']]
-Essentially, equations of motion can often be broken into parts, and
+[[009C Sample Final 3, Problem 9 Detailed Solution|'''<u>Detailed Solution</u>''']]
 [[009C_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]

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 !Foundations: &nbsp;
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+|Many concepts in physics involve the notion of a relative frame. For example, if I'm in a box dropped from an airplane, I won't be moving relative to the box. However, I'm still heading towards the ground with acceleration <math style="vertical-align: -5px">10\,\textrm{m/sec}^{2}. </math> &thinsp;Say it drops for 5 seconds, so the box is going <math style="vertical-align: -5px">50\,\textrm{m/sec} </math> when it hits the ground. Even if I jump with all my might and pull off something like <math style="vertical-align: -5px">5\,\textrm{m/sec} </math> of upward velocity, I'll still feel the impact of hitting the ground at
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+|If a wheel of radius one is resting at the origin, its axis will be at the point <math style="vertical-align: -5px">(1,0). </math> For this solution, we will assume the point <math style="vertical-align: 0px">P </math> is below the axle, although the problem does not state the position of <math style="vertical-align: 0px">P </math>. When the wheel rotates clockwise, it will move to the right. Since the length of the arc defined by an angle <math style="vertical-align: 0px">\theta </math> on a circle of radius <math style="vertical-align: 0px">R </math> is <math style="vertical-align: -4px">L=R\cdot\theta=1\cdot\theta=\theta, </math> &thinsp;the wheel will roll forward the length of the arc, which is just <math style="vertical-align: 0px">\theta. </math>
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+Moreover, the axle's <math style="vertical-align: 0px">x </math> position will change in the same manner,
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+while the height of the axle will always be fixed at <math style="vertical-align: -4px">y=1 </math>. This
+means we can describe the position of the axle as a function of <math style="vertical-align: -4px">\theta, </math>
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+|Since the wheel is rotating, we also know that our point <math style="vertical-align: 0px">P </math> will rotate around the axle. As described in the problem, it is halfway
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+|We therefore have a frame (the axle) that is moving, and a point <math style="vertical-align: 0px">P </math> that is moving relative to the frame. To get the movement relative to the stationary "world", we simply add them up to find
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 [[009C_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]

← Older revision		Revision as of 01:23, 5 March 2017
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−	<span class="exam">A wheel of radius 1 rolls along a straight line, say the <math>x</math>-axis. A point <math>P</math> is located halfway between the center of the wheel and the rim. As the wheel rolls, <math>P</math> traces a curve. Find parametric equations for the curve.	+	<span class="exam">A wheel of radius 1 rolls along a straight line, say the  <math>x</math>-axis. A point  <math style="vertical-align: 0px">P</math>  is located halfway between the center of the wheel and the rim. As the wheel rolls,  <math style="vertical-align: 0px">P</math>  traces a curve. Find parametric equations for the curve.

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-|&nbsp;&nbsp; '''(a)'''
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 [[009C_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]

← Older revision		Revision as of 19:05, 18 February 2017
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−	<span class="exam">~~Compute~~	+	<span class="exam">A wheel of radius 1 rolls along a straight line, say the <math>x</math>-axis. A point <math>P</math> is located halfway between the center of the wheel and the rim. As the wheel rolls, <math>P</math> traces a curve. Find parametric equations for the curve.
−
−	::<~~span class="exam"~~>a) <math ~~style="vertical-align: -12px"~~>~~\lim_{n\rightarrow \infty} \frac{3~~-~~2n^2}{5n^2+n+1}~~</math>
−
−	::<~~span class="exam"~~>b) <math ~~style="vertical-align: -12px"~~>~~\lim_{n\rightarrow \infty} \frac{\ln n}{\ln 3n}~~</math>

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