Simple Harmonic Motion

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Simple Harmonic Motion

All waves are caused by a vibration source, including mechanical waves.

The conditions for simple harmonic motion (SHM) are as follows:

The force acting on the body always acts towards the equilibrium position

The force acting on the body is proportional to its displacement from the equilibrium position

The features of SHM can be summarised in the equations:

The minus sign signifies that the force and acceleration are in the opposite direction to the displacement.

When at rest, the object hangs in equilibrium position, A is the amplitude of the oscillation - greatest displacement from equilibrium.

Time period does not depend on the amplitude of the oscillations, provided the amplitude is small. If a spring or pendulum is overstretched, the motion ceases to be simple harmonic.

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Mathematical Description of SHM

The above graph shows how the displacement of mass varies with time, if it is released from rest with amplitude A.

Since the graph has a cosine function, it can be written as:

Once we have the equation that links displacement with time, we can produce the equation that links velocity with time, and acceleration with time:

Since the maximum value of a sine or cosine function is 1, we can write maximum values for v and a as follows:

The velocity of an oscillating particle at any known displacement:

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Time Period of Oscillations

Two prior equations can be combined to link the period (T) to the mass (m) and force per unit displacement (k).

However, for small angles this is simplified to:

x can be related to pendulum length:

And again for small angles, this is simplified to:

Combining equations gives:

When the pendulum is released, the restoring force acts in the opposite direction, so:

This is the defining equation of SHM, as the acceleration is proportional and opposite to the displacement:

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Energy in SHM

As a pendulum swings from 1 to 5, and back there is a continuing transfer of energy from one form to another.

At 1: Velocity is 0, kinetic energy is 0, but potential energy is at max.

3: Velocity is max, kinetic energy is max, potential energy is min.

5: Velocity is 0 again, kinetic energy is 0, potential energy is max.

Potential energy is the work done to take a mass to displacement x:

As amplitude is maximum displacement, maximum potential energy is as follows:

Kinetic energy of the oscillator at velocity v is:

Assuming no other energy transfers take place, total energy remains constant across the system.

These three energies can be linked:

Since

When velocity is substituted into the prior equation, it gives:

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Damped Oscillations

In theory, free oscillations carry on indefinitely since there are no external forces to stop them.

In reality, all pendulums stop swinging eventually because frictional forces transfer its energy to heat and sound.

All mechanical oscillations are damped oscillations, as energy is transferred to the surroundings. With light damping, energy is transferred slowly, with heavy damping, energy is transferred quickly.

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Forced Oscillations

When you let a pendulum swing freely, it swings at its natural frequency - determined by its length.The pendulum gradually transfers its energy to the surroundings and slows down. It is also possible to force a pendulum to oscillate at a different frequency by pushing it at regular intervals.

The resonant frequency of an oscillator is the same as its natural frequency. When an oscillator is driven at its natural frequency, it resonates and the amplitude of the oscillations grow larger.

Barton's pendulums demonstrate resonance. X is the driving pendulum, when it is swung, the string PQ swings, and begins to drive the other pendulums. Most of them swing with low amplitude, but the pendulums of similar length swing with large amplitude. This is because the natural frequency is similar to the driving frequency, resonance occurs.

Examples of resonance:

On a guitar, when a string is plucked it vibrates at its natural frequency

The frequency of microwaves are matched to the natural frequency of water molecules, so the water absorbs the energy and begin to vibrate, dissipating energy as heat

Bridges have a natural frequency of oscillation, if the wind matches that frequency, the bridge can oscillate which is dangerous. To avoid this, structures are designed so oscillations are damped, or the natural frequency is extremely high.