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November 28, 2006 Version A

PHYSICS 207: Mid-Term Exam 3

Print your name and section clearly on all pages. (If you do not know your section number, write your TA's name.) Show all work in the space immediately below each problem. Your final answer must be placed in the boxes provided. Problems will be graded on reasoning and intermediate steps as well as on the final answer. Be sure to include units wherever necessary, and the direction of vectors. Each problem is worth between 4 and 16 points. In doing the problems, try to be neat. Check your answers to see that they have the correct dimensions (units) and are the right order of magnitude. You are allowed one 8.5 $\times$ 11" sheet of notes and no other references. The exam lasts 90 minutes.

NOTE: for the purposes of this exam you should assume $g=10.$ m/s$^2$.

Possibly useful moments of inertia (about high symmetry axes):
Solid Sphere $= \frac{2}{5} MR^2$,                    Solid Cylinder $=\frac{1}{2} MR^2$
Thin Hollow Cylinder $=MR^2$
Long thin rod (axis through center) $= \frac{1}{12} ML^2$
Thin rectangular Plate (axis through center) $= \frac{1}{12} M(a^2+b^2)$

Do NOT write below this point.

SCORE

PROBLEM 1:

PROBLEM 2:

PROBLEM 3:

PROBLEM 4:

PROBLEM 5:

PROBLEM 6:

PROBLEM 7:

PROBLEM 8:

PROBLEM 9:

PROBLEM 10:

PROBLEM 11:

PROBLEM 12:

PROBLEM 13:

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PROBLEM 1: 4 pts.
Three observers, A, B and C are listening to a moving source of sound. The diagram at right shows the locations of the wave-crests of the moving source with respect to the three stationary observers. Which of the following is true:

Circle the correct letter:

(A)

The wavefronts move faster at $A$ than at $B$ and $C$.

(B)

The wavefronts move faster at $C$ than at $A$ and $B$.

(C)

The frequency of the sound is highest at $A$.

(D)

The frequency of the sound is highest at $B$.

(E)

The frequency of the sound is highest at $C$.

PROBLEM 2: 6 pts. Two metal wires, each 5.0 meters long, have cross-sectional areas of 0.20 m$^2$ and 1.0 m$^2$ respectively are mounted as shown. Young's modulus is $5.0 \times 10^9$ N/m$^2$ and the bulk modulus is $2.0 \times 10^6$ N/m$^2$. A mass of 5.0$\times 10^4$ kg is hung as shown. The pulley is massless and frictionless. What is the fractional change in length of the 10 m section of wire?

fractional change$=$  

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PROBLEM 3: (4 pts.)
A lead weight is fastened to the top of a large solid cylindrical piece of foam that floats in a container of water. Because of the weight of the lead, the water line is flush with the top surface of the foam. If the piece of foam is turned upside down, so that the weight is now suspended underneath it, the water level in the container

Circle the correct letter.

(A)

rises.

(B)

drops.

(C)

remains the same.

(D)

is impossible to deduce without more information.

PROBLEM 4: (4 pts.)
A mass attached to a spring oscillates back and forth as indicated in the position vs. time plot below. At point P, the mass has
$\parbox{0.55\linewidth}{ \begin{enumerate} \item [(A)] positive velo... ...item [(G)] zero velocity and zero acceleration. \end{enumerate} \hspace*{.3in}}$$\parbox{0.30\linewidth}{~~ \vspace*{-0.5in} \begin{center} \includegraphics[width=3in]{mt3_f06_f4.eps} \end{center}}$

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PROBLEM 5: (2,3,3,3,3 pts.) A traveling wave on string is given as $y(x,t)= 3 \sin(2 x - 5 t)$ where the displacement is in units of meters (all units are SI).



















(a) Is it harmonic?              
(b) What is the period?       
(c) What is the wavelength? 
(d) At what speed does the energy in the wave travel?                
(e) What is the maximum velocity of a mass element on the string? 

PROBLEM 6:(6 pts)
A simple pendulum of mass 10 kg and length 3.0 m is placed in an elevator on the Earth's surface (g = 10. m/s$^2$). The elevator is accelerating upwards at 2.0 m/s$^2$. What is the period of the pendulum while the elevator is accelerating.















$T=$  

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PROBLEM 7: (16 pts., 4,6,6 )

A single large square shaped pipe, 5.0 cm on a side, ends and opens into two smaller square pipes each 2.0 cm on a side. The pipes are horizontally mounted. The pressure in the large pipe is 100. atm (where 1 atm is $1.0 \times 10^5$ Pascals and 1 Pa = 1 N/m$^2$) and water ($\rho=1.00$ gm/cm$^3$=$10^3$ kg/m$^3$) is flowing at 100. liters (1 liter is 1000 cm$^3$) per second (in the large pipe) into both smaller pipes. The bulk modulus of water is 2.2 $\times 10^9$ N/m$^2$

(a) What is speed of the water in the large pipe?
(b) What is the pressure in one of the smaller pipes?
(c) Assuming the velocity of the water in the large and small pipes were 50 m/s and 100 m/s respectively then how long would it take a 1.0 kHz sound wave to travel 200 meters along the pipes starting in the large pipe and traveling 100 meters in both?

  

$v=$  












$P_2=$  












$t=$  

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PROBLEM 8: (6 pts.)
A 3.0 cm radius cylinder contains a liquid which fills the cylinder to a height of 10. cm. Into the cylinder you drop a hollow plastic spherical ball of radius 3.0 cm. It floats such that exactly half of the sphere is submerged. By how much, $h$, does the level of the fluid rise? (The volume of a sphere is $\frac{4}{3} \pi r^3$.)

$h=$  

PROBLEM 9: (6,6 pts.)
A 10 kg mass is attached to two Hooke's Law springs of spring constants 1000 N/m and 3000 N/m per meter as shown in the figure below.

(a) What is the resonant frequency in radians per second?
(b) Initially the mass is displaced 2.00 cm to the right of its equilibrium position and, after one full cycle, the mass returns to 1.95 cm to the right of its equilibrium position. What is the drag coefficient $b$?









$\omega=$  









$b=$  

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PROBLEM 10: (6 pts.)

A 4.0 m long massless rod hangs vertically and can pivot about a point 1.0 m from the end. Attached to each end of the rod are small masses, m, of 2.0 kg each. What is the angular frequency of the pendulum in the limit of small oscillations?

$\omega=$  

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PROBLEM 11: (6,6 pts.)
A pied-piper, initially at rest, starts playing a flute at 400 Hz as he falls off the edge of an 80 m high cliff ($g=10$m/s$^2$). He continues to play the flute as he falls. The flute's sound travels at 320 m/s. Do not consider the piper's physical height.
(a) After he falls just one meter he can first hear the reflection of the sound off the ground. If he is able to emit 0.40 W of power, how loud is the reflected sound (in dBs) assuming the ground is perfectly reflective.
(b) At the instant immediately before he hits the ground what is the frequency of the reflected sound he hears?










$dB=$  







$f=$  

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PROBLEM 12:(4 pts)
A wave pulse is moving, as illustrated, with uniform speed $v$ along a rope.Which of the graphs 1 thru 4 below correctly shows the relationship between the displacement $y$ at point P as a function of time $t$? (Look carefully at the axes.)
Circle the correct letter.

PROBLEM 13:(6 pts)
A very elastic string of length $L$ and linear density of $\mu$ is under a tension T. If the tension is quadrupled then the string stretches by a factor of two.
(a) By what factor will the power transmitted change for harmonic waves of identical frequency and amplitude?

$\cal{P}_{\mbox{\tiny 2L}} / \cal{P}_{\mbox{\tiny L}}=$