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Unformatted text preview: Physics53 Fall 06 Exam 3 A Name: M6 V Section: I certify that] have abitled by the rules and the spirit of the Duke Community Standard. “WW PLEASE READ CAREFULLY BEFORE YOU START “*i‘ Do not forget to write your name and section number above. You must do all multiple choice Questions and two of the longer problems. If you
attempt to do more than two of the longer problems, please, indicate below which two problems you want be counted. Remember that credit will only be given for solutions that show your work. Do not
just write down your result, but show how you got it. No work shown = no credit. ll‘ numbers are required, use g = 10 111/52. Record your MC answers (by letter A  E) here.
No credit will be given, ifyou do not record your answer here! mafia MC3_§Q MC4§ MCSQ Check here, which two of the long problems should be counted: MCI P1 P2 P3 P4 [To be ﬁlled in by graderz] Grading scores: MC P1 P2 P3 P4 Total: M ulti le Choice Problems: 6 Points Each 1. The staff of an excursion boat in distress on a small lake lowers the lifeboat together
with several frightened passengers into the water. They succeed without dropping a
passenger or letting any water into the lifeboat. Does the water level of the lake
change in the process? :jAgNo, the water level remains unchanged.
B) Yes, the water level rises.
C) Yes, the water level falls. Consider two identical oscillators x1(t) and x20). The maximum displacement of the
first oscillator, A1, is twice as large as that of the second oscillator, A2. The first
oscillator is released at rest from its position of maximum displacement, the second
oscillator is released at the same moment with positive velocity from the position
x220. If the period of both oscillators is T = 3.003, when do the two oscillators first
have the same displacement (x1=x2)? [Assume that all motion is frictionless] v? A l ,
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E) The two oscillators never have the same displacement at the same time.
A transverse wave propagates along a string, which has a gradually decreasing linear
mass density. As it propagates toward the thin end of the string, its
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B frequency decreases. f i’
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E) both its frequency and wavelength change. 1% Consider two vessels of equal volume. One is filled with air, the other with hydrogen
gas. Both gasses have the same pressure and temperature. Which vessel contains
more gas molecules? A) The one filled with hydrogen. B) The one filled with air. :5 W @Both contain the same number of molecules. 5. An organ pipe (Closed at one end), which was designed for a room with a temperature
T = 20°C, was produced outside of specifications. The organist determines that the
pipe plays with the correct pitch, if the room is heated to 30°C. The velocity of sound
in air is given by the expression v = (331 + 0.6T) m/s, where T is measured in degrees
Celsius. By how much was the pipe too long, if it was intended to play the note E
(330 Hz) at 20°C? [Neglect the thermal expansion of the pipe itself] A) 0.227 cm. _ z %%
’13” 0.303 cm. @0455 cm. % S if Mg 5 2 i
D) 0909 cm. g “T sf? E) The pipe was produced too short. The following problem is worth 4+6+5+5 = 20 points: P1. A pendulum is made of a uniform disk of mass M and radius R pivoted around a point at its edge, as shown. Consider small oscillations (small 0) of this disk about its stable
equilibrium position. a) Draw the free—body force diagram of the disk for the situation shown, in which
the center of mass (CM) is displaced by x. b) For a small displacement x of the disk, corresponding to a small rotation 0 about the pivot point, what is the torque of the disk about the pivot point in terms ofM,
R, and x? c) Use the result of (b) to derive an equation involving x and its second derivative,
dzx
dt2 d) What is the period of oscillation ifR : 0.500 m and M = 1.20 kg 7 which has the form + cozx = 0 . %
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a The following problem is worth 6+5+5+4 = 20 points: P2. Two identical cylindrical liquid containers of radius R and height h are connected by
a narrow pipe of radius r, as shown in the figure. The left container (A) is initially
filled with water; the container on the right (B) is initially empty. The pipe, which
leads from the bottom of container A to the middle of B, is controlled by a valve,
which is initially closed and then suddenly opened to allow water to flow from A to B
eventually equalizing the water levels in both containers. The pipe has a diameter of
11.00 cm. [Assume that water is an ideal fluid] 3 a) Find the speed with which water is ﬂowing through the pipe as a function of the
remaining water level in container A. b) Use the result obtained in (a) to obtain an equation for the amount (mass) of water
remaining above mid—level in container A as a function of the speed of ﬂow in the
connecting pipe. c) What is the rate (mass per unit time) of water flowing through the pipe, when the
container B is filled to one quarter of capacity? Use R 2 10cm, 11 = 50cm, and r =
0.5cm to obtain a number for the ﬂow rate. d) What is the total work done by gravity on the water when the water level has been
equalized and both containers are half full, for the numbers given in section c. The following problem is worth 5+5+5+5 = 20 points: 1’3. Consider two pipes of the same length L = 0.520m, each closed at one end. One of
them (pipe A) is equipped with a movable piston. which can be used to reduce the
length of the available air column in the pipe. The other pipe (B) is fixed. [Assume
that the air temperature is 20°C.] a) What is the fundamental frequency of pipe B, and what are its first two higher
harmonics? b) At which positions of the piston (measured relative to the open end of the pipe)
will the fundamental of pipe A have the same frequency as the first two higher
harmonics of pipe B? c) How long would a string need to be to have the same fundamental frequency as
pipe B, if the speed of wave propagation on it would be 265 m/s? d) What is the wavelength of sound emitted by the string described in (c), when it
vibrates in its second harmonic mode? B D
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and loudness (65 dB measured at 1.00 m distance from the speaker, when only one
speaker is turned on). Assume that the loudspeakers are in phase a) Find all the locations on the dashed line shown in the illustration (directly in front
of one of the speakers) where the two sound waves interfere most constructively. b) What is the loudness measured at a point 4.50 m in from of the center of the
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 Fall '07
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 Physics

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