Chapter 14: Oscillations
Oscillations are back-and forth motions. Sometimes the word
vibration is used in place of oscillation; for our purposes, we can
consider the two words to represent the same class of motions.
In the ideal case of no friction, free oscillations are a sub-class of
periodic motions; that is, in the absence of friction, all free
oscillations are periodic motions, but not all periodic motions
are oscillations. For example, uniform circular motion (which we
studied in PHYS 1P21) is periodic, but not considered an
oscillation.
Uniform circular motion can be modelled by sine or cosine
functions of time (think back to the unit circle in high-school
math when you were learning trigonometry). (Sine and cosine
functions are collectively known as sinusoidal functions, or
sinusoids for short.) If the restoring force that causes oscillation
is a linear function of displacement, then the resulting
oscillatory motion can also be modelled by a sinusoidal function
of time.
There is a close relationship between uniform circular motion
and oscillatory motion caused by a linear restoring force; we
won't explore this, but check the textbook if you're interested.
What is a restoring force? What is a linear restoring force?
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Notice that the formula for Hooke's law is represented
by the graph; the magnitude of the restoring force is
proportional to the magnitude of the displacement
from equilibrium, and in the opposite direction. The
constant of proportionality is the stiffness constant of
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the spring.
Q: What are the units of k? What are some typical
values for the stiffness constant for coil springs in your
experience (ones in your car's shock absorbers, in your
ball-point pen, attached to your aluminum door, etc.)?
Here is an example position-time diagram for an
oscillation:
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Notice that the position-time diagram for the
oscillation resembles the graph of a sinusoidal
function. You'll get a chance to see that this must be
so for an oscillator that is subject to a linear restoring
force (Hooke's law) later in the chapter. Now is a good
time to review sinusoidal functions, so let's do it:
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Exercises
1. Determine the amplitude, period, frequency, and angular
frequency for each function. In each case, time t is measured in
seconds and displacement x is measured in centimetres.
(a)
(b)
c.
(d)
2. Sketch a graph of each position function in parts (a), (b), and (c)
of Exercise 1.
3. Calculus lovers only! Determine a formula for the derivative of
the sine function, and a formula for the derivative of the cosine
function, valid for angles measured in degrees.
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Exercises
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Exercises
4. Consider an oscillation with position function
x = 20 cos (4t)
where x is measured in cm and t is measured in s.
(a) Determine the positions on the x-axis that are turning
points.
(b) Determine the times at which the oscillator is at the
turning points.
(c) Determine the times at which the oscillator is at the
equilibrium position.
(d) Determine the times at which the speed of the
oscillator is at (i) a maximum, and (ii) a minimum.
(e) Determine the times at which the acceleration of the
oscillator is at (i) a maximum, and (ii) a minimum.
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Note that the period (and therefore also both the frequency
and angular frequency) does not depend on the amplitude of
the oscillation. This is interesting. Does this match with your
experience?
Exercises
5. What would the position-time graphs look like for an oscillator
that is released from several different starting amplitudes?
6. Consider an oscillator of mass 4 kg attached to a spring with
stiffness constant 200 N/m. The mass is pulled to an initial
amplitude of 5 cm and then released.
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amplitude of 5 cm and then released.
(a) Determine the angular frequency, frequency, and period of
the oscillation.
(b) Write a position-time function for the oscillator.
7. A block of mass 3.2 kg is attached to a spring. The resulting
position-time function of this oscillator is x = 23.7 sin(4.3t),
where t is measured in seconds. Determine the stiffness of the
spring.
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Exercises
8. Consider a block of mass 5.1 kg attached to a spring. The
position-time function of this oscillator is x = 8.2 sin(2.7t),
where x is measured in cm and t is measured in seconds.
(a) Determine the total mechanical energy of the oscillator.
(b) Determine the stiffness constant of the spring.
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(c) Determine the maximum potential energy.
(d) Determine the maximum kinetic energy.
(e) Determine the positions at which the kinetic energy
and the potential energy of the oscillator are equal.
Pendulum motion
Recall from earlier in these lecture notes that for a block on
the end of a spring, applying Newton's law to the block
results in
This is an example of a differential equation, and to solve a
differential equation means to determine a position function
x(t) that satisfies the equation. You'll learn how to do this in
second-year physics (PHYS 2P20), and second-year math
(MATH 2P08), but for now you can verify that position
functions of the form
and
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and
both satisfy the differential equation above, provided that
satisfies a certain condition (the same condition that was
observed earlier in the notes).
From a different perspective, we can also infer that if a
physical phenomenon is modelled by a differential equation of
the form given above, the phenomenon is an example of
simple harmonic motion.
Let's consider a simple pendulum. First draw a free-body
diagram:
In the radial direction, applying Newton's second law gives:
In the tangential direction, applying Newton's second law
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gives:
This looks very similar to the differential equation
written earlier that represents simple harmonic motion.
But not exactly; if the sine theta were replaced by just
theta, then the equation would have the form of the
SHM differential equation.
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SHM differential equation.
Note that for small angles, sine theta is very similar to
theta, provided that theta is measured in radians:
Construct a table of values using a calculator and
you'll see for yourself that sine theta is
approximately equal to theta for small angles. The
approximation is better for smaller angles.
Also note that
as long as theta is measured in radians; this is an
example of a power series, which you'll learn about
later in MATH 1P06 or MATH 1P02, if you are
taking either of these two courses.
Thus, for a pendulum with a small amplitude, the
motion is approximately SHM, described by the
differential equation
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Substituting the model
into the differential equation leads to expressions
for the period and frequency:
Substituting this expression into the differential
equation gives:
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Thus
and therefore
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Swinging your arms while running or walking; note
how the period of the swing is modified by
changing the effective length of your arms:
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Damped oscillations
Cars have shock absorbers to make the ride smoother.
A shock absorber consists of a stiff spring together with
a damping tube. (The damping tube consists of a piston
in an enclosed cylinder that is filled with a thick (i.e.,
viscous) oil.)
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Without the damping tube, a car would oscillate for a
long time after going over a bump in a road; the
damping tube helps to limit both the amplitude and
duration of the oscillations.
When the damping tube doesn't work anymore, the
car tends to oscillate for a long time after going over a
bump, which is annoying. The same thing happens with
a screen door when its damping tube malfunctions.
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(There is a spring attached to the door, but it is not
shown in the photograph.)
The position function for a damped oscillator is
modified as follows:
You can think of this position function as representing
a sort of sinusoid, but one with a variable amplitude;
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the amplitude is the constant A times the exponential
factor, which steadily decreases as time passes.
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The damping tube in a screen door is adjustable. If
the resistance is too great, then the door will take a
long time to shut after it is opened. If the resistance
is too little, then the door will swing back and forth
many times before shutting. There is an ideal
medium amount of resistance ("critical damping")
which works best; you'll learn more about this, and
see how these three cases (overdamping, critical
damping, and underdamping) follow naturally from
different classes of solutions to the appropriate
differential equation, in PHYS 2P20 and MATH 2P08.
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Resonance
Examples of driven oscillations:
•
•
child sloshing around in a bathtub
parent pushing a child on a playground swing
Some systems have a natural oscillating frequency; if you drive
the system at its natural oscillating frequency, its amplitude
can increase dramatically. This phenomenon is called
resonance.
In many situations, one tries to avoid resonant oscillations. For
example, that annoying vibration in your dashboard when you
are driving on the highway at a certain speed … is very
annoying. More seriously, soldiers are trained to break ranks
when they march across a bridge, because if their collective
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march is at the same frequency as the natural frequency of
the bridge, then there is danger that they could collapse the
bridge. (This is an ancient custom, from an age when many
bridges were made of wood.)
Engineers must be careful to design bridges and tall buildings
so that they don't have natural vibration frequencies;
otherwise an unlucky wind could cause dangerous largeamplitude vibrations.
http://en.wikipedia.org/wiki/Tacoma_Narrows_Bridge_(1940)
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Another good example of resonance is tuned
electrical circuits, such as the ones used in radio or
television reception.
Radio waves of many different frequencies are
incident on a radio receiver in your home; each tries
to "drive" electrical oscillations in an electrical
circuit. The natural frequency of the electrical circuit
can be adjusted so that it will resonate with only a
certain frequency of radio wave; this is how you
"tune in" to a certain radio station. The oscillations
due to the resonant frequency persist, while all the
other frequencies are rapidly damped.
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