Oscillations and Waves Problems
- Oscillation: You have been asked to evaluate the
design for a simple device to measure the mass of small
rocks on the Moon. The rock is attached to the free end
of a lightweight spring which horizontal. The surface on
which the rock slides is almost frictionless. You are
worried that the kinetic energy of the rock may make this
device dangerous in some situations. The device
specifications state that a 150 gram rock will execute
harmonic motion, with a frequency of 0.32 Hz, described
by x(t) = A sin (bt - 35º) when the rock has an initial
speed of 1.2 cm/s.
- Oscillations: You and some friends are waiting in
line for "The Mixer", a new carnival ride. The
ride begins with the car and rider (150 kg combined) at
the top of a curved track. At the bottom of the track is
a 50 kg block of cushioned material which is attached to
a horizontal spring whose other end is fixed in concrete.
The car slides down the track ending up moving
horizontally when it crashes into the cushioned block,
sticks to it, and oscillates at 3 repetitions in about 10
seconds. Your friends estimate that the car starts from a
height of around 10 feet. You decide to use your physics
knowledge to see if they are right. After the collision,
you notice that the spring compresses about 15 ft from
equilibrium.
- Traveling Waves: You've been hired as a technical
consultant to the Minneapolis police department to design
a radar detector-proof device that measures the speed of
vehicles. (i.e. one that does not rely on sending out a
radar signal that the car can detect.) You decide to
employ the fact that a moving car emits a variety of
characteristic sounds. Your idea is to make a very small
and low device to be placed in the center of the road
that will pick out a specific frequency emitted by the
car as it approaches and then measure the change in that
frequency as the car moves off in the other direction.
The device will then send the initial and final
frequencies to its microprocessor, and then use this data
to compute the speed of the vehicle. You are currently in
the process of writing a program for the chip in your new
device. To complete the program, you need a formula that
determines the speed of the car using the data received
by the microprocessor. You may also include in your
formula any physical constants that you might need.
Because your reputation as a designer is on the line, you
realize that you'll need find ways to check the validity
of your formula, even though it contains no numbers.
- Traveling Waves: You have the perfect summer job
with a team of marine biologists studying dolphin
communication off the coast of Hawaii. Massive boulders
on the ocean floor can interrupt the reception of
underwater sound waves from the dolphins. To reduce these
disruptions, your team has decided to put several
"transceivers" (a device that receives a
signal, amplifies the signal, and then transmits it) at
strategic locations on the ocean floor. A transceiver
will receive sound waves from a dolphin and then
retransmit them to the researchers on the ship. The
ship's receiver is on a long cable so that it is at
approximately the same depth as the dolphins. Because of
your physics background, you worry that the frequency
received at the moving ship will be different than that
emitted by the dolphin. To determine the size of this
effect, you assume that the ship is moving at 35km/h away
from the stationary transceiver. Meanwhile, the dolphin
is moving at 60km/h towards the transceiver and at an
angle of 63º to the ship's path when it emits a sound
frequency of 660Hz.
- Wave Equation: A friend of yours, a guitarist,
knows you are taking physics this semester and asks for
assistance in solving a problem. Your friend explains
that he keeps breaking repeatedly the low E string (640
Hz) on his Gibson "Les Paul" when he tunes-up
before a gig. The cost of buying new strings is getting
out of hand, so your friend is desperate to resolve his
Delia. Your friend tells you that the E string he is now
using is made of copper and has a diameter of 0.063
inches. You do some quick calculations and, given the
length of the neck of your friends guitar, estimate that
the wave speed on the E string is 1900 ft/s. While
reading about stringed instruments in the library, you
discover that most musical instrument strings will break
if they are subjected to a strain greater than about 2%.
How do you suggest your friend solve his problem?
- Standing Waves: Your friend, an artist, has been
thinking about an interesting way to display a new wind
sculpture she has just created. In order to create an
aural as well as visual effect, she would like to use the
wires needed to hang the sculpture as a sort of a string
instrument. She decides that with three wires and some
luck, the strings will sound a C-major dyad (C - 262Hz, G
- 392 Hz) when the wind blows (note: A dyad is part of a
chord.). Her basic design involves attaching a piece of
wire from two eye-hooks on the ceiling that are
approximately a foot-and-a-half apart and then hanging
the 50 pound sculpture from another wire attached to the
first wire forming a "y-shaped" arrangement.
Your friend tells you that she has been successful in
hanging the sculpture but not in "tuning" the
sound. Desperate for success, she knows you are taking
physics and asks for your help. Before you tackle the
analysis, you use your knowledge of waves to gather some
more information. You take a sample of the wire back to
your lab and measure its linear mass density to be 5.0
g/m. You also determine that wire is some sort of iron or
steel from its color. What is your advice?
- Standing Waves: You have a summer job in a
biomedical engineering laboratory studying the technology
to enhance hearing. You have learned that the human ear
canal is essentially an air filled tube approximately 2.7
cm long which is open on one end and closed on the other.
You wonder if there is a connection between hearing
sensitivity and standing waves so you calculate the
lowest three frequencies of the standing waves that can
exist in the ear canal. From your trusty Physics
textbook, you find that the speed of sound in air is 343
m/s.
- Standing Waves: You have joined a team designing a
new skyway that is to link the Physics Building to the
Mechanical Engineering Building. To make sure it will be
stable in gusts of wind, you need to find the lowest
frequency that sets up a standing wave in the skyway
structure. Your group has decided to make a scale model
of the skyway and put it into a wind tunnel to determine
the frequency. Unfortunately the wind tunnel cannot be
pulsed at a very low frequency. While the model is in the
wind tunnel you pulse the wind until you find a frequency
which sets up a standing wave in the model. You then
slowly increase the frequency until you get the next
standing wave pattern. Using the two frequencies you have
measured together with the length of the skyway model you
then calculate the lowest frequency which will set up a
standing wave.
- Energy, Frequency: You have an exciting summer job
working on an oil tanker in the waters of Alaska. Your
Captain knows that the ship is near an underwater
outcropping of land and wishes to avoid running into it.
He estimates that it is about 6 km straight ahead of the
ship and asks you to use the sonar to check how fast the
ship is approaching it. The ship's instruments tell you
the ship is moving through still water at a speed of 31
km/hr but the captain cannot take any chances. A sonar
signal is sent out with a frequency of 980 Hz, bounces
off the underwater obstacle, and is detected on the ship.
If the ship's speed indicator is correct, what frequency
should you detect? You use your trusty Physics text to
find the speed of sound in seawater is 1522 m/s.
- Rotations: You are helping a friend build an
experiment to test behavior modification techniques on
rats. She needs to build an obstacle that swings across a
path every 1.0 second. To keep the experiment as
inexpensive as possible, she wants to use a meter stick
as the swinging obstacle. She asks you to determine where
to drill a hole in the meter stick so that, when it is
hung by a nail through that hole, it will do the job for
small swings.
- Rotations: Your friend is trying to construct a
clock for a craft show and asks you for some advice. She
has decided to construct the clock with a pendulum. The
pendulum will be a very thin, very light wooden bar with
a thin, but heavy, brass ring fastened to one end. The
length of the rod is 80 cm and the diameter of the ring
is 10 cm. She is planning to drill a hole in the bar to
place the axis of rotation 15 cm from one end. She wants
you to tell her the period of this pendulum.
- Rotations: The child of a friend has asked you to
help with a school project. She wants to build a clock
from common materials. She has found a meter stick which
has a mass of 300g and asks you to determine where to
drill a hole in it so that when it is hung by a nail
through that hole it will be a pendulum with a period of
2.0 seconds for small oscillations. A quick calculation
tells you that the moment of inertia of the meter stick
about its center of mass is 1/12 of its mass times the
square of its length.
- Rotations: You have a part time job at a software
company that is currently under contract to produce a
program simulating accidents in the modern commuter
railroad station being planned for downtown. Your task is
to determine the response of a safety system to prevent a
railroad car from crashing into the station. In the
simulation, a coupling fails causing a passenger car to
break away from a train and roll into the station.
Furthermore, the brakes on the passenger car have failed.
It cannot stop on its own so it keeps on rolling. The
safety system at the end of the track is a large
horizontal spring with a hook that will grab onto the car
when it hits preventing the car from crashing into the
station platform. After the car hits the spring, your
program must calculate the frequency and amplitude of the
car's oscillation based on the specifications of the
passenger car, the specifications of the spring, and the
speed of the passenger car. In your simulation, the
wheels of the car are disks with a significant mass and a
moment of inertia half that of a ring of the same mass
and radius. At this stage of your simulation, you ignore
any energy dissipation in the car's axle or in the
flexing of the spring, and the mass of the spring.
- Rotations: You have been asked to help design an
automated system for applying a resistive paint to
plastic sheeting in order to mass produce containers to
protect sensitive electronic components from static
electric charges. The object used to apply the paint is a
solid cylindrical roller. The roller is pushed back and
forth over the plastic sheeting by a horizontal spring
attached to a yoke, which in turn is attached to an axle
through the center of the roller. The other end of the
spring is attached to a fixed post. To apply the paint
evenly, the roller must roll without slipping over the
surface of the plastic. The machine simultaneously paints
two narrow strips of plastic that lay side by side
parallel to the axle of the roller. While the roller is
in contact with one strip, a feed mechanism pulls the
other strip forward to expose unpainted surface. In order
to determine how fast the process can proceed, you have
been assigned to calculate how the oscillation frequency
of the roller depends on its mass, radius and the
stiffness of the spring. You know that the moment of
inertia of a solid cylinder with respect to an axis
through its center is 1/2 that of a ring.