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?

Chapter 14: Oscillations

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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

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.

1.

(a)

(b)

c.

(d)

Sketch a graph of each position function in parts (a), (b), and (c) of Exercise 1.

2.

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.

3.

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Exercises

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Exercises

Consider an oscillation with position function4.

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

What would the position-time graphs look like for an oscillator that is released from several different starting amplitudes?

5.

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.

6.

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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.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.

7.

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Exercises

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.

8.

(a) Determine the total mechanical energy of the oscillator.(b) Determine the stiffness constant of the spring.

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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

(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.

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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.

Thus, for a pendulum with a small amplitude, the motion is approximately SHM, described by the differential equation

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.

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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

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

Examples of driven oscillations:

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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 large-amplitude 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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