Let’s go back to thinking about masses and springs. Imagine a mass held between two springs as shown in the animation below. If the mass is moved away from the equilibrium position the restoring forces provided by the springs will make the mass oscillate back and forth in a similar manner to the playground swing example shown on the previous page.

What happens if we change the size of the mass?

An increase in mass increases the *inertia* (reluctance to change velocity i.e. reluctance to accelerate and decelerate) of the object. (BTW In this example we have switched off gravity. Note – massive objects have inertia *even in outer space where they have no weight!*). An increased inertia means that the springs will not be able to make the mass change direction as quickly. This increases the time period of the oscillation. The greater the inertia of an oscillating object the greater the time period. This therefore lowers the frequency of its oscillations.

The oscillating objects we’ve looked at up to now all vibrate with a rather special pattern or *waveform*. Let’s think some more about that.

# Sine Waves

The animation below shows how an oscillating pattern (*waveform*) can be modelled by placing a marker on the edge of a rotating disc. If we just think about the VERTICAL location of this marker (i.e. view the disc edge-on) then we can see that it moves up and down with **simple harmonic motion**. It generates the sine-wave (*sinusoidal*) waveform we have been using for mass-spring systems and pendulums. Objects which oscillate sinusoidally are described as oscillating with simple harmonic motion.

This ‘disc’ idea is very useful for thinking about *phase* of oscillation – we’ll come back to this later.

## Simple Harmonic Motion and Resonant Frequencies

Here’s the maths which underlies what we’ve been talking about. An oscillation follows simple harmonic motion if it fulfills the following two rules:

- Acceleration is always in the opposite direction to the displacement from the equilibrium position
- Acceleration is proportional to the displacement from the equilibrium position

The acceleration and displacement are linked by the following equation:

Acceleration = – ω^{2} * x

Here ω (Greek character omega) is called the angular frequency of oscillation, and is given by 2π /T or 2πf

T is the period of oscillation (s), f=1/T = frequency of oscillation (Hz) and x is the displacement (m).

Using Newton’s 2nd Law (F=ma) we can show that the Force on the object due to inertia will be:

F=-m\omega^2xUsing Hooke’s Law for springs (F=-kx), we know that the force on the oscillating mass due to the springs is simply

F=-kxThese two forces are always in balance, so

m\omega^2x-kx=0From this we can find the resonant frequency:

\omega^2=\frac{kx}{mx}=\frac{k}{m} \omega=\sqrt{\frac{k}{m}} f=\frac{1}{2}\sqrt{\frac{k}{m}}This might look a little tricky, but it is a massively useful equation.

# Damping

When we were talking about playground swings, we mentioned *damping – *a loss of energy from, in that example, movement (kinetic) energy to heat. There other examples of damping we could think about – here’s one:

Imagine hitting a cymbal. This causes the cymbal to oscillate. These oscillations cause the air around the cymbal to vibrate, and these vibrations travel to your ear and you hear this as sound. The sound of the cymbal will eventually die away, as the air resistance and internal losses as the metal flexes reduce the restoring forces, causing the oscillations to get smaller and smaller. Placing your hand on the cymbal after hitting it greatly speeds up the damping process, as your fingers are soft and absorb the kinetic energy very effectively.

Normally it is hard to see a cymbal vibrating because it is moving too fast. Here we’ve slowed it down to 1/ 80th of normal speed. You can see that the oscillations take ages to die away, as damping is small. For many oscillations (including this one) the damping forces are roughly proportional to velocity, and this leads to an exponential decay of amplitude over time.

Some drums use extra damping pads to control the time the skin rings for, after striking. Many drummers put a pillow in the kick (bass) drum – the extra damping creates more of a thud than a boom. The world of acoustics and sound is full of examples where damping is intentionally used to control vibrations and sounds.

Some systems such as shock absorbers on cars, or vibration dampers on bridges, have a very high level of damping. The controlled mass (car body, bridge deck etc) will ideally return to equilibrium in the shortest possible time once vibrated, rather than bouncing around either side of it as we have seen up to now. Such systems are said to be **critically damped**.

Next: Free Oscillations, Forced Oscillations and Resonance