Mechanical Oscillations

A mechanical oscillation is a periodic conversion of energy from P.E. to K.E to P.E. etc.

If a body is to oscillate it must be acted on by a force which is directed towards the equilibrium position. This force is called the restoring force.

The simplest type of oscillation is called simple harmonic motion (s.h.m.).

This definition can be expressed mathematically as

where x is the displacement of the body from the equilibrium position.

The magnitude of the constant is equal to

or

where m is the mass of the oscillating body and F is the restoring force.

We would therefore expect the value of the constant for a given oscillation to depend on

The amplitude of an oscillation is the maximum displacement from the equilibrium position.

The frequency of an oscillation is the number of oscillations per second.

If an object moving with constant speed in a circular path is observed from a distant point (in the plane of the motion), it will appear to be oscillating with simple harmonic motion. Similarly, the shadow of a pendulum bob moves with s.h.m. when the pendulum itself is either oscillating (through a small angle) or moving in a circle with constant speed, as shown in the diagram below.

This observation suggests that for any s.h.m. we can find a corresponding circular motion. When we say that a circular motion "corresponds to" a given s.h.m., we mean that

By using the comparison between circular motion and s.h.m., it can easily be shown that the constant of proportionality between acceleration and displacement for an object moving with s.h.m. is equal to the square of the angular velocity of the corresponding circular motion. For this reason the "s.h.m. equation" is usually written as

So, to find the value of the constant for a given oscillation, we
simply measure the time period and then use the relation

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© David Hoult 2008