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%TCIDATA{Created=Tue Sep 23 17:59:11 1997}

%TCIDATA{LastRevised=Fri Sep 26 11:27:35 1997}



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\begin{document}



\SetTitle{Phy206 Exam I}

\SetAuthor{T. Curtright}

\Setdate{25 September 1997}

\TitlePage{This is a ``closed book'', multiple-choice exam. You are NOT allowed to use

any notes. Circle only one answer out of the five choices for each problem.

Do ALL problems. \\

You may ask the instructor questions, especially if you think a problem is

not clearly stated.\\

\bigskip \bigskip Good luck!\\

\bigskip \bigskip ``On my honor, I have neither given nor received any aid

on this examination.''\\

\bigskip

Name:\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\\

\bigskip ID \#:

\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}



NOTE: This is the exam key. The answer (a) is correct for each question.



\smallskip



\eject[1] A mass $m$ moves along an elliptical orbit in the $xy$-plane about

a fixed mass $M$ (i.e. $M\gg m$) which is located at one focus of the

ellipse, as shown in the accompanying Figure 1. Which of the following

expressions correctly gives the $z$-component of the angular momentum of $m$

about $M$?



(a) $m(x+c)v_{y}-myv_{x}$



(b) $m(x+c)v_{y}+myv_{x}$



(c) $m(x+c)v_{x}-myv_{y}$



(d) $m(x-c)v_{y}-myv_{x}$



(e) $xv_{y}/a^{2}+yv_{x}/b^{2}$



\noindent In these expressions, $v_{x}$ and $v_{y}$ are the $x$ and $y$

components of velocity, respectively, for $m$.



\bigskip



[2] A mass $m$ moves along an elliptical orbit in the $xy$-plane about a

fixed mass $M$ (i.e. $M\gg m$) which is located at one focus of the ellipse,

as shown in the accompanying Figure 1. Which of the following expressions

correctly gives a constraint satisfied at all times by the coordinates $%

(x,y) $ and velocity components $(v_{x},v_{y})$ of the mass $m$?



(a) $xv_{x}/a^{2}+yv_{y}/b^{2}=0$



(b) $xv_{x}/b^{2}+yv_{y}/a^{2}=0$



(c) $xv_{x}/a^{2}+yv_{y}/b^{2}=1$



(d) $xv_{y}/a^{2}+yv_{x}/b^{2}=0$



(e) $(x-c)v_{y}-yv_{x}=0$



\noindent (Hint: differentiate the equation for the ellipse.)



\bigskip



[3] A sinusoidal wave on a string, with mass/length $=\mu $, is given by $%

y(x,t)=A\sin (kx-\omega t).$ What is the total energy (kinetic plus

potential) contained in a length $\Delta x$ of the string which is one

wavelength long, $\Delta x=\lambda $ $=2\pi /k\,$?



(a) $\pi \,\mu \,A^{2}\omega ^{2}/k$



(b) $\pi \,\mu \,A\omega ^{2}/k$



(c) $2\pi \,\mu \,A^{2}\omega ^{2}/k$



(d) $\pi \,\mu \,A^{2}\omega /k$



(e) $2\pi \,\mu \,A^{2}\omega v$



\bigskip



[4] Moving along a streamline in a \emph{compressible,} nonviscous, flowing

fluid, which of the following is guaranteed by the conservation of both mass

and energy \emph{not }to change from point to point?



(a) $p/\rho +\frac{1}{2}\,v^{2}+gy$



(b) $p+\frac{1}{2}\,\rho v^{2}+\rho gy$



(c) $p+\frac{1}{2}\,\rho v^{2}$



(d) $p/\rho +\frac{1}{2}\,v^{2}$



(e) $\rho $



\noindent In these expressions, $v$ is the speed of the fluid, $\rho $ is

its mass density, $p$ is the pressure, $y$ is the height of the streamline

point above some point of reference, and $g=9.80$ m/sec$^{2}$ is the

acceleration of gravity (which we take to be constant throughout the

fluid).\smallskip



\bigskip



\eject[5] A stone drops into a pond, creating a perfect circular wavecrest

that moves outward from the point of impact of the stone. If the wave has

maximum amplitude $A=5.0\,cm$ when it reaches a distance of $1.0\,meter$

from the point of impact, what is the maximum amplitude of the wave when it

reaches a distance of $100.\,meters$ from the point of impact? (Neglect all

effects of viscosity and dispersion!)



(a) $0.50\,cm$



(b) $0.05\,cm$



(c) $5.0\,cm$



(d) $50\,cm$



(e) $1.0\,cm$



\bigskip



[6] Two harmonic waves on a string are given by $y_{1}(x,t)=0.04\sin

(31.4x-157t)$ and $y_{2}(x,t)=0.04\cos (31.4x+157t-\pi /2)$. What is the

superposition, $y_{1}+y_{2}$, of these waves?



(a) $0.08\sin (31.4x)\cos (157t)$



(b) $0.08\cos (31.4x)\sin (157t)$



(c) $0.04\sin (31.4x)\cos (157t)$



(d) $0.04\cos (31.4x)\sin (157t)$



(e) $0.08\sin (31.4x)\sin (157t)$



\bigskip



[7] For convenience in this problem, suppose the speed of sound in air is $%

v=340\,m/s$. A student is driving down Dixie Highway across from the

University with a speed of $v=34\,m/s$. Of course, she is driving faster

than the legal speed limit, so a perspicacious policeman pursues her. When

his speed matches hers, he turns on his siren which has a frequency of $%

f=440\,Hz$. What frequency does the student hear when the sound reaches her?



(a) $440\,Hz$



(b) $400\,Hz$



(c) $484\,Hz$



(d) $396\,Hz$



(e) $489\,Hz$



\bigskip



[8] Olive oil (density $\rho _{oil}$) floats on water (density $\rho

_{water} $). A bouillon \emph{cube} (density $\rho _{bouillon}$) of height $%

H $ floats at the oil-water interface with its four sides vertical and its

top and bottom horizontal. The cube is submerged to a depth $D$ into the

water. Which of the following gives the relation between $D$ and $H$ when

the cube is at rest?



(a) $\left( \rho _{water}-\rho _{oil}\right) D=\left( \rho _{bouillon}-\rho

_{oil}\right) H$



(b) $\left( \rho _{bouillon}-\rho _{oil}\right) D=\left( \rho _{water}-\rho

_{oil}\right) H$



(c) $\left( \rho _{water}-\rho _{bouillon}\right) D=\left( \rho

_{water}-\rho _{oil}\right) H$



(d) $\left( \rho _{water}-\rho _{oil}\right) D=\left( \rho _{water}-\rho

_{bouillon}\right) H$



(e) $\left( \rho _{water}-\rho _{oil}\right) D=\left( \rho _{oil}-\rho

_{bouillon}\right) H$



\bigskip



\eject[9] A tank of constant cross sectional area $A$ is filled with water

to a height $x$ above a small hole of area $a$. Water flows out of the hole.

Using Bernoulli's relation, and assuming that the water has speed $v=0$ at

the top surface, what is the rate of change of the water height in the tank?



(a) $\frac{dx}{dt}=-\frac{a}{A}\;\sqrt{2gx}$



(b) $\frac{dx}{dt}=-\frac{A}{a}\;\sqrt{2gx}$



(c) $\frac{dx}{dt}=-\frac{a}{A}\;\sqrt{gx}$



(d) $\frac{dx}{dt}=-\frac{A}{a}\;\sqrt{gx}$



(e) $\frac{dx}{dt}=-\frac{a}{A}\;gx$



\bigskip



[10] Two pulses traveling on the same string are described by $y_{1}=\frac{8%

}{(8x-8t)^{4}+4}$ and $y_{2}=\frac{-8}{(8x+8t-16)^{4}+4}$ . (All lengths are

in $\emph{meters}$ and all times are in \emph{seconds}.) At one particular

time, let's call it $t_{0}$, the pulses cancel everywhere. Also, at one

particular location, let's call it $x_{0}$, the pulses cancel for all times.

What are $t_{0}$ and $x_{0}$?



(a) $t_{0}=1.0$ sec, $x_{0}=1.0$ meters



(b) $t_{0}=0.75$ sec, $x_{0}=1.0$ meters



(c) $t_{0}=0.00$ sec, $x_{0}=0.00$ meters



(d) $t_{0}=0.75$ sec, $x_{0}=-1.0$ meters



(e) None of the other answers is correct.



\bigskip



[11] A sinusoidal wave is traveling along a rope. The oscillator that

generates the wave completes 100. vibrations in 10.0 seconds. Also, a given

maximum travels 500 cm along the rope in 10.0 seconds. What is the

wavelength of the wave?



(a) 5.00 cm



(b) 31.9 cm



(c) 0.50 cm



(d) 50.0 cm



(e) There is not enough information given to determine the wavelength.



\bigskip



[12] Which of the following is the linear wave equation that describes small

transverse displacement, as given by $y(x,t)$, waves on a string which are

moving in either the $+x$ or the $-x$ direction? (Note that $F$ is the

tension and $\mu $ is the mass/length for the string.)



(a) $\frac{\partial ^{2}y}{\partial x^{2}}=\frac{\mu }{F}\,\frac{\partial

^{2}y}{\partial t^{2}}$



(b) $\frac{\partial ^{2}y}{\partial x^{2}}=\frac{F}{\mu }\,\frac{\partial

^{2}y}{\partial t^{2}}$



(c) $\frac{\partial y}{\partial x}=\sqrt{\frac{\mu }{F}}\,\frac{\partial y}{%

\partial t}$



(d) $\frac{\partial y}{\partial x}=\sqrt{\frac{F}{\mu }}\,\frac{\partial y}{%

\partial t}$



(e) None of the other answers is correct.



\bigskip



\eject[13] A thin hemispherical shell of radius $R=1/\sqrt{\pi }=\allowbreak

.\,56419\,meters$ is placed in airtight contact with a flat surface, and

then air is pumped out until the pressure inside the hemisphere is $p=$ $%

0.5\times \,p_{atm}\,.$ What minimum magnitude force must be applied (normal

to the flat surface) to remove the hemisphere from the surface? (Recall that 

$p_{atm}=1.01\times 10^{5}N/m^{2}.$ Also note that you are not allowed to

consider altering the hemisphere, say by punching a hole in it, nor are you

allowed to damage the wall!)



(a) $5\times 10^{4}\,N$



(b) $1\times 10^{5}\,N$



(c) $2\times 10^{5}\,N$



(d) $4\times 10^{5}\,N$



(e) $2/3\times 10^{5}\,N$



\bigskip



[14] Old Faithful Geyser in Yellowstone Park erupts at approximately 1-hour

intervals, and produces a fountain of (mostly) hot water whose height

reaches $40$ meters. What is the pressure (above atmospheric) in the heated

underground chamber, from which the fountain issues, if its depth is $175$

meters?



(a) $2100\;kiloPascals$



(b) $390\;kiloPascals$



(c) $930\;kiloPascals$



(d) $1200\;kiloPascals$



(e) The answer to this problem given in the back of the text is different

than the answer given by the grader, so I have an excuse for not knowing

which is right!



\bigskip



[15] Two wires are welded together. The wires are of the same material, but

one is twice the diameter of the other one. They are subjected to a tension

of $9.2$ N. The thin wire has a length of $40$ cm and a linear mass density

of $2.0$ g/m. The combination is fixed at both ends and vibrated in such a

way that two antinodes are present with a node between them which is located

exactly at the weld. How long is the thick wire?



(a) $20$ cm



(b) $40$ cm



(c) $10$ cm



(d) $4$ cm



(e) $100$ cm



\bigskip



\eject[16] Two harmonic waves are described by 

\[

y_{1}=(9.0\,m)\sin \left[ 2\pi \left( 6.0x-1800t\right) \right] \;\;\text{%

and\ \ }y_{2}=(9.0\,m)\sin \left[ 2\pi \left( 6.0x-1800t-0.50\right) \right] 

\]

where all lengths are in \emph{meters} and $t$ is in \emph{seconds}. These

waves interfere. What is the frequency of the resultant wave?



(a) $f=1800\;Hz$



(b) $f=300\;Hz$



(c) $f=6.0\,\pi \;Hz$



(d) $f=10800\;Hz$



(e) $f=1800\,\pi \;Hz$



\smallskip 



NOTE: There was an error in the statement of this problem on the distributed

exam! The correct answer was not given considering the error. Therefore NO

answer was the correct one on the distributed exam. In any case, ALL answers

were accepted.



\bigskip



\eject[17] Which of the following ``trig identities'' is wrong?



(a) $\sin (\theta )+\sin (\phi )=\cos \left( \theta -\phi \right) \sin

\left( \theta +\phi \right) $



(b) $\sin \left( \theta +\phi \right) =\sin (\theta )\cos (\phi )+\cos

(\theta )\sin (\phi )$



(c) $\sin \left( \theta -\phi \right) =\sin (\theta )\cos (\phi )-\cos

(\theta )\sin (\phi )$



(d) $\cos \left( \theta +\phi \right) =\cos (\theta )\cos (\phi )-\sin

(\theta )\sin (\phi )$



(e) $\cos \left( \theta -\phi \right) =\cos (\theta )\cos (\phi )+\sin

(\theta )\sin (\phi )$



\bigskip



[18] A spherical neutron star has a radius of only $10$ km, yet its mass is

equal to that of the sun, $M_{sun}=2.0\times 10^{30}$ kg. What is the

(average) density of such a star?



(a)$\allowbreak $ $4.\,8\times 10^{17}\;kg/m^{3}$



(b) $1.\,6\times 10^{21}\;kg/m^{3}$



(c) $\allowbreak 2.0\times 10^{18}\;kg/m^{3}$



(d)$\allowbreak $ $4.\,8\times 10^{26}\;kg/m^{3}$



(e) $\allowbreak 1.6\times 10^{17}\;kg/m^{3}$



\bigskip



[19] A vertical, rectangular dam of height $h$ and width $w$ holds back

water in a lake. The lake is only partially full: water covers the face of

the dam up to a height $\frac{2}{3}\,h$. What is the magnitude of the total

force on that part of the face of the dam which is underwater? You may

assume that the water is incompressible with constant density $\rho _{water}$%

, that the acceleration of gravity $g$ is constant over the heights

involved, and that the air pressure at the surface of the lake is $p_{atm}$.



(a)$\left( p_{atm}+\frac{1}{3}\,\rho _{water}hg\right) \frac{2}{3}\,h\,w$



(b)$\left( p_{atm}+\frac{1}{2}\,\rho _{water}hg\right) \,h\,w$



(c)$\left( p_{atm}+\frac{1}{3}\,\rho _{water}hg\right) h\,w$



(d)$\left( p_{atm}+\frac{1}{2}\,\rho _{water}hg\right) \frac{2}{3}\,h\,w$



(e) None of the other answers is correct.



\bigskip



[20] A ``left-moving'' wave is described by the function $y(x,t)=\sin

(x+5t). $ Which one of the following derivatives is \emph{not} correct.



(a) $\frac{\partial ^{2}y}{\partial x^{2}}$ $=\sin (x+5t)$



(b) $\frac{\partial y}{\partial x}$ $=\cos (x+5t)$



(c) $\frac{\partial y}{\partial t}$ $=5\cos (x+5t)$



(d)$\frac{\partial ^{2}y}{\partial x\partial t}=-5\sin (x+5t)$



(e) $\frac{\partial ^{2}y}{\partial t^{2}}$ $=-25\sin (x+5t)$



\smallskip



\eject



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\end{document}