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| − | <math> | + | <span class="texhtml">''s''''i''''n''<sup>2</sup>(''x'') + ''c''''o''''s''<sup>2</sup>(''x'') = 1</span> |
| − | % Christopher Allen
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| − | % callen@alum.dartmouth.org
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| − | | |
| − | \documentclass[letterpaper,10pt]{article}
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| − | | |
| − | \oddsidemargin=39pt
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| − | \evensidemargin=39pt
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| − | \marginparwidth=68pt
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| − | \textwidth=390pt
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| − | | |
| − | \usepackage{pstricks}
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| − | \usepackage{pst-coil}
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| − | \usepackage{pst-3d}
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| − | \usepackage{amstext}
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| − | \usepackage{amsmath}
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| − | \usepackage{amssymb}
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| − | | |
| − | \begin{document}
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| − | | |
| − | \center{\textbf{\LARGE{AP Physics B Problem Sheet: Spring \#1}}}\\
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| − | | |
| − | \begin{enumerate}
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| − | | |
| − | \item A long straight wire carries a constant current horizontally. A wire of diameter $d$ and resistivity $\rho$ is shaped into a rectangular loop with sides of lengths $l$ and $w$. A small battery of voltage $V$ is attached to the wire loop. The mass of
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| − | the loop and battery together is $m$. This loop is placed so that the long straight wire is directly above its top by a height $h$ and so that they lie in the same plane, as shown below. When released, the loop remains suspended in mid-air.
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| − | \begin{center}
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| − | \begin{pspicture}(0,.3)(12,2)
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| − | % Loop
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| − | \psline(6.1,.2)(7.5,.2)(7.5,1.2)(7.5,1.2)(4.5,1.2)(4.5,.2)(5.9,.2)
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| − | % Battery
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| − | \psline[linewidth=0.4mm](5.9,.1)(5.9,.3)
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| − | \psline[linewidth=0.4mm](6.1,-.1)(6.1,.5)
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| − | % Labels
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| − | \rput(5.8,.5){$V$}
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| − | \psline[linestyle=dashed,dash=2pt 2pt]{<->}(4.3,.2)(4.3,1.2)
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| − | \rput(4.1,.7){$w$}
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| − | \psline[linestyle=dashed,dash=2pt 2pt]{<->}(4.3,2.2)(4.3,1.2)
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| − | \rput(4.1,1.7){$h$}
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| − | \psline[linestyle=dashed,dash=2pt 2pt]{<->}(4.5,1.4)(7.5,1.4)
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| − | \rput(6,1.6){$l$}
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| − | % Long straight wire
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| − | \psline(0,2.2)(12,2.2)
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| − | \end{pspicture}
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| − | \end{center}
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| − | \begin{enumerate}
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| − | \item How much current flows through the loop? (4 pts)
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| − | \item What is the direction of the magnetic field from the long straight wire at the loop's location? (1 pt)
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| − | \item In what direction does current flow through the long straight wire? (1 pts)
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| − | \item What is the magnitude of the current in the long straight wire? (4 pts)
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| − | \end{enumerate}
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| − | | |
| − | \item Two capacitor plates are set up to launch a proton from rest into a uniform magnetic field, as shown below. The voltage between the plates is 100 V. The magnetic field has a strength of 0.05 T and is directed into the page.
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| − | \begin{center}
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| − | \begin{pspicture}(0,.3)(4.4,3.9)
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| − | % Capacitor plates
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| − | \psline(.1,0)(.1,1.95)
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| − | \psline(.1,2.05)(.1,4)
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| − | \psline(1.1,0)(1.1,1.95)
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| − | \psline(1.1,2.05)(1.1,4)
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| − | % Magnetic field
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| − | \multirput(1.4,.5)(0,1){4}{\multirput(0,0)(1,0){4}{$\times$}}
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| − | % Proton
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| − | \psdots(0,2)
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| − | % Possible paths
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| − | \psline{->}(0.2,2)(1.1,2)
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| − | \psarc{->}(1.1,3.5){1.5}{-85}{20}
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| − | \rput(2.75,3.8){$a$}
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| − | \psarcn{->}(1.1,.5){1.5}{85}{-20}
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| − | \rput(2.75,.2){$b$}
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| − | \end{pspicture}
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| − | \end{center}
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| − | \begin{enumerate}
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| − | \item At what speed does the proton enter the magnetic field? (3 pts)
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| − | \item Will the proton follow path $a$ or path $b$? (1 pt)
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| − | \item What will the radius of this path be? (3 pts)
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| − | \item How long after it enters the magnetic field will the proton hit the back of the capacitor plate? (3 pts)
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| − | \end{enumerate}
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| − | | |
| − | \item Two long straight wires of length $L$ and mass $M$ hang side-by-side from very light strings of length $l$, with $l\ll L$. These wires each carry a current of the same magnitude. Each wire's strings make an angle $\theta$ to the vertical.
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| − | \begin{center}
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| − | \begin{pspicture}(0,-.45)(4.8,2.8)
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| − | \psset{viewpoint=2 -2 1}
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| − | % Backs of wires
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| − | \ThreeDput[normal=0 -1 0](0,5,0){\pscustom[fillstyle=solid,fillcolor=white]{\pscircle(0,0){.2}}}
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| − | \ThreeDput[normal=0 -1 0](0,5,0){\pscustom[fillstyle=solid,fillcolor=white]{\pscircle(2,0){.2}}}
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| − | % Sides of wires
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| − | \ThreeDput[normal=1 0 1](1.911,0,.179){\pscustom[fillstyle=solid,fillcolor=white]{\psline(5,-.4)(0,-.4)(0,0)(5,0)}}
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| − | \ThreeDput[normal=1 0 1](-.089,0,.179){\pscustom[fillstyle=solid,fillcolor=white]{\psline(5,-.4)(0,-.4)(0,0)(5,0)}}
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| − | % Fronts of wires
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| − | \ThreeDput[normal=0 -1 0]{\pscustom[fillstyle=solid,fillcolor=white]{\pscircle(0,0){.2}}}
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| − | \ThreeDput[normal=0 -1 0]{\pscustom[fillstyle=solid,fillcolor=white]{\pscircle(2,0){.2}}}
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| − | % Strings
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| − | \ThreeDput[normal=0 -1 0](0,.5,0){\psline(.089,.179)(1,2)(1.911,.179)}
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| − | \ThreeDput[normal=0 -1 0](0,4.5,0){\psline(.089,.179)(1,2)(1.911,.179)}
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| − | % Support
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| − | \ThreeDput[normal=1 0 0](1,0,0){\psline[linewidth=.6mm](0,2)(5,2)}
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| − | % Angles
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| − | \ThreeDput[normal=0 -1 0](0,4.5,0){\psline[linestyle=dashed](1,0)(1,1.8)}
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| − | \ThreeDput[normal=0 -1 0](0,4.5,0){\psarc(1,2){.7}{243.4}{296.6}}
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| − | \rput(3.65,1.7){$\theta$}
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| − | \rput(4.05,1.7){$\theta$}
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| − | % Wire length label
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| − | \ThreeDput[normal=1 0 0](2.8,0,0){\psline[linestyle=dashed]{<->}(5,0)}
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| − | \rput(4.2,-.2){$L$}
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| − | % String length label
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| − | \ThreeDput[normal=0 -1 0](0,-.5,0){\psline[linestyle=dashed]{<->}(1,2)}
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| − | \rput(-.3,.7){$l$}
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| − | \end{pspicture}
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| − | \end{center}
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| − | \begin{enumerate}
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| − | \item Using an end-on view, draw a free body diagram for each wire. (3 pts)
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| − | \item Find the magnitude of the magnetic force between the wires. (2 pts)
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| − | \item Do the currents run parallel or anti-parallel? (1 pt)
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| − | \item Find the magnitude of the current in the wires. (4 pts)
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| − | \end{enumerate}
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| − | | |
| − | \item Two resistors and an inductor are attached to a battery, as shown below. The switch is originally open. The switch is then closed and remains closed.
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| − | \begin{center}
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| − | \begin{pspicture}(0,.2)(8,1.8)
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| − | % Wires and switch
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| − | \psline(1,1)(1,1.8)(2.2,1.8)
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| − | \psline(1,.8)(1,0)(2.2,0)
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| − | \psline(2.8,1.8)(7,1.8)(7,1.2)
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| − | \psline(4,0)(4,.6)
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| − | \psline(4,1.2)(4,1.8)
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| − | \psline(2.29,.3)(2.8,0)(7,0)(7,.6)
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| − | \psdots(2.2,0)(2.8,0)(4,0)(4,1.8)
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| − | % Upper resistor
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| − | \pszigzag[coilarm=.01,coilwidth=.18,linewidth=0.4mm](2.2,1.8)(2.8,1.8)
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| − | \rput(2.5,1.5){4000 $\Omega$}
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| − | % Inductor
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| − | \pscoil[coilarm=.01,coilwidth=.18](7,.6)(7,1.2)
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| − | \rput(7.7,.9){3 mH}
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| − | % Middle resistor
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| − | \pszigzag[coilarm=.01,coilwidth=.18,linewidth=0.4mm](4,.6)(4,1.2)
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| − | \rput(4.8,.9){2000 $\Omega$}
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| − | % Battery
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| − | \psline[linewidth=0.4mm](.7,1)(1.3,1)
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| − | \psline[linewidth=0.4mm](.9,.8)(1.1,.8)
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| − | \rput(.2,.9){12 V}
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| − | \end{pspicture}
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| − | \end{center}
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| − | \begin{enumerate}
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| − | \item How much current flows through each resistor and the inductor immediately after the switch is closed? \mbox{(4 pts)}
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| − | \item How much current flows through each resistor and the inductor a long time later? \mbox{(4 pts)}
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| − | \item How much energy is stored in the inductor a long time later? \mbox{(2 pts)}
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| − | \end{enumerate}
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| − | | |
| − | \item An inclined plane is set up with two conductive rails running along its sides. The rails are electrically connected at the top of the incline. A bar, of mass $m$ and length $l$, slides down the frictionless rails, making electrical contact with them
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| − | . The net resistance of the circuit is $R$. The plane is inclined by an angle $\theta$ to the horizontal. There is a uniform magnetic field, of magnitude $B$, directed straight downward.
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| − | \begin{center}
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| − | \begin{pspicture}(0,-.1)(5,3.6)
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| − | \psset{viewpoint=2 -4 1}
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| − | % Inside base line of back
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| − | \ThreeDput[normal=1 0 0](.24,0,0){\psline(.3,0)(2.7,0)}
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| − | % Third triangle
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| − | \ThreeDput[normal=0 -1 0](0,2.7,0){\pscustom[fillstyle=solid,fillcolor=white]{\psline(.24,0)(.24,2.82)(4,0)(.24,0)}}
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| − | % Second triangle
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| − | \ThreeDput[normal=0 -1 0](0,.3,0){\pscustom[fillstyle=solid,fillcolor=white]{\psline(.24,0)(.24,2.82)(4,0)(.24,0)}}
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| − | % Front triangle
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| − | \ThreeDput[normal=0 -1 0]{\pscustom[fillstyle=solid,fillcolor=white]{\psline(0,0)(0,3)(4,0)(0,0)}}
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| − | % Top
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| − | \ThreeDput[normal=3 0 4](4,0,0){\pscustom[fillstyle=solid,fillcolor=white]{\psline(.3,0)(.3,4.7)(2.7,4.7)(2.7,0)(3,0)(3,5)(0,5)(0,0)(.3,0)}}
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| − | % Angle and its label
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| − | \ThreeDput[normal=0 -1 0]{\psarc(4,0){.75}{143}{180}}
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| − | \rput(2.7,0){$\theta$}
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| − | % Bar's front end
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| − | \ThreeDput[normal=0 -1 0]{\pscustom[fillstyle=solid,fillcolor=white]{\psline(2,1.5)(2.2,1.35)(2.35,1.55)(2.15,1.7)(2,1.5)}}
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| − | %Bar's front
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| − | \ThreeDput[normal=4 0 -3](2.2,0,1.35){\pscustom[fillstyle=solid,fillcolor=white]{\psline(0,0)(3,0)(3,.25)(0,.25)(0,0)}}
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| − | % Bar's top
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| − | \ThreeDput[normal=3 0 4](2.35,0,1.55){\pscustom[fillstyle=solid,fillcolor=white]{\psline(0,0)(3,0)(3,.25)(0,.25)(0,0)}}
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| − | % Velocity and its label
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| − | \ThreeDput[normal=0 -1 0](0,1.5,0){\psline{->}(2.475,1.3)(3.075,.85)}
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| − | \rput(3,.9){$\mathbf{v}$}
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| − | % Length and its label
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| − | \ThreeDput[normal=1 0 0](-.2,0,.15){\psline[linestyle=dashed]{<->}(0,3)(3,3)}
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| − | \rput(.4,3.6){$l$}
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| − | %Magnetic field and its label
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| − | \ThreeDput[normal=0 -1 0](0,1.5,0){\psline{->}(4.5,3.5)(4.5,2.5)}
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| − | \rput(5,2.85){$\mathbf{B}$}
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| − | \end{pspicture}
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| − | \end{center}
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| − | \begin{enumerate}
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| − | \item Using an end-on view, draw a free body diagram for the bar. (2 pts)
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| − | \item Find the current through the bar when it reaches terminal velocity. (4 pts)
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| − | \item Determine the terminal velocity of the bar. (4 pts)
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| − | \end{enumerate}
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| − | | |
| − | \end{enumerate}
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| − | | |
| − | \end{document}
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| − | </math> | |