Physics 18 Spring 2011
Homework 11
Wednesday April 6, 2011
Make sure your name is on your homework, and please box your final answer. Because
we will be giving partial credit, be sure to attempt all the problems, even if you don’t finish
them. The homework is due at the beginning of class on Wednesday, April 13th. Because
the solutions will be posted immediately after class, no late homeworks can be accepted! You
are welcome to ask questions during the discussion session or during office hours.
1. An ultrasonic transducer, of the type used in medical ultrasound imaging, is a very thin
disk (m = 0.10 grams) driven back and forth in SHM at 1.0 MHz by an electromagnetic
coil.
(a) The maximum restoring force that can be applied to the disk without breaking it
is 40,000 N. What is the maximum oscillation amplitude that won’t rupture the
disk?
(b) What is the disk’s maximum speed at this amplitude?
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2. An object of mass m is suspended from a vertical spring of force constant 1800 N/m.
When the object is pulled down 2.50 cm from equilibrium and released from rest, the
object oscillates at 5.50 Hz.
(a) Find m.
(b) Find the amount the spring is stretched from its unstretched length when the
object is in equilibrium.
(c) Write expressions for the displacement x, the velocity vx , and the acceleration ax
as functions of time.
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3. You have designed a cat door that consists of a square piece of plywood that is 1.0 in
thick and 6.0 in on a side, and is hinged at its top. To make sure the cat has enough
time to get through it safely, the door should have a natural period of at least 1.0 s.
Will your design work? If not, explain qualitatively what you would need to do to
make it meet your requirements.
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4. In one of your chemistry labs, you determine that one of the vibrational modes of
the HCl molecule has a frequency of 8.969 × 1013 Hz. Using the result of Problem 8,
find the “effective spring constant” between the H atom and the Cl atom in the HCl
molecule.
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5. A straight tunnel is dug through Earth, as
shown in the figure. Assume that the walls
of the tunnel are frictionless.
(a) The gravitational force exerted by
Earth on a particle of mass m at
a distance r from the center of
Earth when r < RE is Fr =
3
− (GmME /RE
) r, where ME is the
mass of Earth and RE is its radius.
Show that the net force on a particle of mass m a distance x from
the middle of the tunnel is given by
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Fx = − (GmME /RE
) x and that the
motion of the particle is therefore
simple harmonic motion.
(b) Show that the period of the motion is
independent of the length ofpthe tunnel and is given by T = 2π RE /g.
(c) Find its numerical value in minutes.
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