Physics 102
Homework set 6
Solutions
L. Weinstein

Exercises:
----------
 
19.4: If the pendulum is shortened, its frequency increases and its
period T = 1/f decreases.  We showed this in class.

19.6: When you stand on the swing, you are shortening the length of
the pendulum.  Therefore the period  will decrease.

19.10: Use the relationship v = f lambda.  If v doubles and f is
unchanged then the wavelength lambda must double.

19.22: Consider the case where you shake one end of the thick rope up
and down at a frequency of 2 Hz (twice a second).  In that case, the
other end of the thick rope (where  it is attached to the  thin rope)
will also move up and down at a frequency of 2 Hz.  Therefore, the
end of the thin rope will also move up and down at 2 Hz.  Thus, the
frequency will be the same for both ropes.  Since v = f lambda, v and
lambda can both change.

19.28: The  picture below shows two full wavelengths of a standing
wave with the three nodes shown by asterisks.  If it is three
wavelengths long, then there will be 5 nodes.

| /\      /\     |
|/  \    /  \    |
----*---*---*-----
|    \  /    \  /|
|     \/      \/ |

19.32: If he plucks the middle of an open string, then he will excite
the lowest harmonic.  The nodes will be at the ends of the string.
The wavelength will be twice the length of the string.

19.36: a) If it moves towards you, then the frequency wil increase
(doppler effect).  b) The wavelength will decrease (since v = f
lambda). c) The speed will not change (since the speed of sound in air
does not depend on the speed of the source).

19.48: Yes, the  fish would create a sonic  boom in the water.  By
travelling faster than the speed of sound in water, the sound waves
would 'pile up' just like in figure  19.20.

20.2: If  the Moon blew up, there  would be  no air to carry the sound
to the Earth.  Sound does not propagate in vacuum.

20.6: v = f lambda.  If the  frequency increases then the wavelength
decreases (since the speed of sound does not depend on frequency).
Bats can therefore hear shorter wavelengths  than cats.

20.12: This is because  electricity travels much faster than sound.
The  runner furthest from the starting gun would be at disadvantage
since the sound would take longer to reach her.  If the extra distance
is 6 m (about 20 ft or the  width of the track), the time difference
would be about t = d/v = 6 m / 340 m/s = 0.02 s.  This is a big
difference in a short race (which can be decided by 0.001 s).


Problems:
--------------

19.4: The weight goes through one complete cycle (from down to up to
down) twice a second.  Therefore, the frequency is 2 Hz.  The period =
0.5 s.  If the distance travelled by the weight from bottom to top to
bottom is 20 cm, the distance from bottom to top is 10 cm, and the
amplitude is only 5 cm.

19.6: The frequency is 600 Hz.  The time between wingbeats is 
T = 1/f = 1.67*10^(-3) s.
Therefore, the distance traveled in this time is
d = vT = 340 m/s * 1.67*10^(-3) s = 0.57 m



Estimation:
-----------------

The radius of the Earth is 6400 km.  The circumference is therefore 
c = 2 pi r = 4*10^4 km = 4*10^7 m
The speed of sound v = 340 m/s.  Therefore the travel time is
t = c/v = (4*10^7 m) / (340 m/s) = 1.2*10^5 s
This is about 30 hours or a bit more than one day.