Physics 53
Summer 2009
Exam III
Solutions
In questions or problems not requiring numerical answers, express the answers in terms of the symbols
given, and standard constants such as g. If numbers are required, use g = 10 m/s2.
Part A: Multiple choice questions. Check the best answer. Each question carries a value of 4 points.
1.
A hydraulic jack consists of a cylinder (1) with a small cross section area a,
connected by a pipe to a cylinder (2) with larger cross section A. Each cylinder is
fitted with a piston that can move vertically, and the system is filled with an
incompressible fluid. A heavy load to be raised rests
on the piston in 2; the total mass of the load and the
piston is M. A force F1 is exerted downward on the
2
1
piston in 1, resulting in an upward force F2 on the
piston in 2. Which of the following is NOT true?
The pressure in both cylinders is the same.
As the piston in 1 moves down distance d the piston in 2 moves up
distance d ! (a/ A) .
√
To lift the piston in 2 and the load at constant speed, we must have
F1 = Mg ! (A/ a) . [ F1 = Mg ! (a/ A) . The whole point of the device is that F1
is less than Mg.]
The work done by F1 is equal to the work done by F2 .
2.
Waves 1 and 2 exist simultaneously in a medium, described by the functions
y1 (x, t) = 0.01cos(10!x " 100! t)
y2 (x,t) = 0.01cos(9! x " 90!t)
Wave 1 travels faster than wave 2. [Same speed]
Superposition of these waves produces a standing wave.
√
The waves produce 5 beats/s. [Frequencies are 50 and 45 Hz]
None of the above is true.
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Physics 53
3.
Summer 2009
The frequencies of successive harmonics of an organ pipe are 250 and 350 Hz.
The fundamental frequency for this pipe is 250 Hz.
√
The fundamental frequency for this pipe is 50 Hz.
The pipe is open at both ends. [Closed at one end]
None of the above is true.
4.
Two sounds produce loudness levels of 90 db and 60 db for a listener. The ratio
of the intensities of these sounds is:
3/2.
30.
1030 .
√
103 . [ log(I 2 /I1 ) = 3 ]
Part B: True-false questions. Check the appropriate box.
5.
As water flowing in a horizontal pipe passes into a region where the pipe
diameter is smaller, the pressure increases.
True
√
6.
False [Decreases because the speed increases]
In a musical instrument like the flute, where standing wave resonance is used to
produce a well-defined frequency (pitch) of the note being played, the damping
of the system should be small.
√
True
False
7.
For two waves of the same amplitude, frequency and speed, but traveling in
opposite directions along a string, there is always complete destructive
interference at all points.
True
√
False [A standing wave is produced]
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Physics 53
Summer 2009
Part C: Problems. Work problems in the space provided, indicating your method clearly. Problems have
the point values shown
1a.
Archimedes put the king’s crown on a scale with an amount of pure gold that
had half the mass of the crown. The scale balanced as shown. He then lowered
the whole thing into a container of water until the hanging objects were
submerged, and watched to see if the scale still balanced.
Let the volumes of the crown and the gold be Vc and Vg , and let their average
densities be !c and ! g . Show that the
scale will balance with the objects
under water only if !c = ! g .
!
2!
[10 points]
The buoyant force on the crown is Fc = !wVc g , where Vc = Mc / !c , so
Fc = Mc g ! ( "w / "c ) . Similarly, the buoyant force on the gold is
Fg = M g g ! ( "w / " g ) . The scale will stay balanced if ! ! Fc = 2! ! Fg , or Fc = 2Fg . So
we require Mc / !c = 2Mg / ! g . Since Mc = 2Mg , this requires !c = ! g as claimed.
1b.
The siphon shown is used to move a liquid of density
ρ into a lower container. The dimensions are as
shown.
a.
Find the speed of flow of the fluid in the tube,
in terms of h, L, and g.
b.
What is the maximum length L for which the
siphon will function? Explain.
h
L
[10 points]
a.
Compare the top of the open tank with the bottom of the tube where the
fluid exits, using Bernoulli: P0 + ! g(L " h) = P0 + 12 ! v 2 . This gives
v 2 = 2g(L ! h) .
b.
The pressure in the top arm is obtained by comparing a point in that arm
with the bottom of the tube: P + 12 ! v 2 + ! gL = P0 + 12 ! v 2 . This gives
P = P0 ! " gL . Since pressure cannot be negative, we find L ! P0 / " g .
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Physics 53
2.
Summer 2009
Two blocks of mass m are attached at each end of an ideal spring of stiffness k.
The blocks are at rest on a horizontal frictionless table with the spring between
them at its equilibrium length. The blocks are pushed together, compressing the
spring a total amount x0 and released from rest. They oscillate in SHM.
a.
Describe the motion of the CM of the system.
b.
When one block is displaced distance x from its equilibrium position,
what is the force exerted on it by the spring? {What is the total
displacement of the spring?]
c.
Use this result to find the angular frequency ω of the oscillation.
d.
What is the maximum speed of each block in the oscillation, in terms of
the given quantities? [What is the total energy?]
[20 points]
a.
There is no net external force, so the CM remains at rest.
b.
The motion is symmetric, so when one block’s displacement is x the other
block’s displacement is –x. The total displacement of the spring is thus 2x,
and it exerts a force F = !2kx on both blocks (in opposite directions, of
course).
c.
The general formula for SHM is F = !m" 2 x , so we have m! 2 = 2k , or
! = 2k /m .
d.
The total energy is E = 12 kx0 2 , and this is also the maximum kinetic
energy, so we have 2 ! 12 mvmax 2 = 12 kx0 2 , or vmax = k / 2m ! x0 . This is the
same as vmax = ! " (x0 / 2) , where x0 / 2 is the amplitude of the oscillation
of each block.
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Physics 53
3a.
Summer 2009
A truck is moving directly toward a cliff, blowing its horn, which emits sound of
frequency 200 Hz. The truck’s speed is 1/11th of the speed of sound.
a.
What is the frequency that the driver of the truck hears in the sound
reflected from the cliff?
b.
What is the beat frequency when the reflected sound is combined with
sound directly from the horn?
[10 points]
3b.
1
11
= 200 " .
1 ! 1/11
10
This is reflected back to the driver, who hears frequency
11 12
fR = fc (1 + 1/11) = 200 ! ! = 240 Hz.
10 11
a.
The sound received by the cliff has frequency fc = f0
b.
The beat frequency is 240 – 200 = 40 Hz.
A piano tuner checks his work by sounding the strings for two notes and
listening for beats in the higher harmonics. He can count beats if the frequency is
less than 10 Hz. He sounds the strings for A (220 Hz) and C# (277 Hz). If the
strings are properly tuned, what beat frequency should he hear?
[Which harmonic of the A has nearly the same frequency as which harmonic of
the C#? Find the smallest pair.]
[10 points]
The 5th harmonic of A has frequency 1100 Hz, and the 4th harmonic of C# has
frequency 1108 Hz, so he should hear beat frequency 8 Hz.
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Physics 53
4.
Summer 2009
A loudspeaker is mounted at the same height as the ears of a listener, and aimed
at a wall as shown, sending a beam of sound waves toward the wall. The listener
starts from the wall and walks slowly toward the speaker. The speaker emits
sound of wavelength ! = 0.8d . At a certain time the listener is at distance x < d
from the wall.
a.
What is the phase difference (in radians) between the wave received by
the listener directly from the speaker and the wave received after it is
reflected from the wall? (There is no phase change in the waves on
reflection from the wall.) Give the answer in terms of x and d.
b.
For what values of x (≤ d) will the listener hear maximum intensity?
c.
As the listener approaches
close to the speaker, does the
sound intensity increase or
decrease? How do you
know?
x
[15 points]
d
a.
The reflected wave travels distance d + x while the direct wave travels
distance d ! x . The path difference is thus !x = 2x . The phase difference is
2#
x
! = k"x =
$ 2x = 5# .
0.8d
d
b.
Maximum intensity occurs when ! = 2m" , where m = 0,1, 2,... Since x = d
gives ! = 5" , the only possibilities for x ! d are ! = 0, 2" , 4" . These occur
at distances x = 0, 0.4d, 0.8d .
c.
As x ! d , ! " 5# , which gives destructive interference, so the intensity
will be decreasing.
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Physics 53
Summer 2009
Median = 74.0
SD = 11.4
Median = 65.7
SD = 12.7
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Physics 53
Summer 2009
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