## Simple Harmonic Motion

You may have heard the one about the opera singer breaking a wineglass with their voice. (In other versions it’s a greenhouse(?!), or even the specs worn by members of the audience). Well, if the singer could pull it off, they would be exploiting oscillations. Later, we’ll show you how to break a wineglass with sound, but first we’ll try to work out some rules for things which vibrate, or oscillate.

There are many forms of oscillation in the real world – oscillations determine the sound of a musical instrument, the colour of a rainbow, the ticking of a clock and even the temperature of a cup of tea. A mechanical oscillation is a repeating movement – an electrical oscillation is a repeating change in voltage and current.

Mechanical oscillations are probably easiest to think about, so we’ll start there. Let’s think about the movement of an object back and forth over a fixed range of positions, such as the movement of a swing in a playground, or the bouncing up and down of a weight on the end of a spring (this last one is one of those examples which teachers / scientists / engineers love, but which doesn’t seem to have anything to do with the real world. As we’ll see, it does!!).

The pendulum and the mass-spring system are both oscillating. How can we describe their oscillation?

## Displacement

The animation shows how displacement describes the distance (and also direction as displacement can be negative as well as positive) of the object from its equilibrium position.

The displacement amplitude tells us how ‘big’ the oscillations are. We can use the peak value (the maximum positive displacement from the equilibrium position) or the peak-to-peak value (the distance between biggest negative and biggest positive displacement).

The time period of the oscillation is the time taken for the object to travel through one complete cycle. We can measure from the equilibrium position, as in the animation, or from any other point on the cycle, as long as we measure the time taken to return to the same point with the same direction of travel.

Can you see why the first time the mass returns to the equilibrium position, we’ve only reached half a cycle?

Displacement on its own will sometimes do. But often it is also useful to understand the oscillation in terms of velocity or acceleration. These three things are very similar.

# Velocity and Acceleration

When our oscillating object reaches maximum displacement (when it is as far from equilibrium as it can get) it changes direction. This must mean that there is an instant in time when it is not moving. So, at the point of maximum displacement, its speed is zero. Then it speeds up in the opposite direction, and travels fast through the equilibrium position before starting to slow again in preparation for the next change in direction.

Velocity is just another word for speed, with the extra feature that it has direction and therefore can be negative or positive. (If you walk backwards at 5 km/h. – difficult without falling over – then your velocity is minus 5 km/h.)

So – when velocity is maximum, displacement is zero, and when displacement is maximum, velocity is zero. The period of oscillation is the same, but the plots for displacement and velocity are shifted compared to the other. This shift along in time is called a phase angle, which is an idea we’ll come back to later.

At maximum displacement the object has to come to rest, and then start moving in the opposite direction. Imagine braking in a car (moving forwards) before immediately moving off fast in reverse – you’d feel rapid deceleration during braking, followed acceleration in reverse gear. Like displacement and velocity, acceleration can be negative or positive depending on direction. In this example, braking would cause negative acceleration, as would moving off in reverse. It sounds weird, but the point in time where the acceleration would be (negative) maximum is the moment when the car stops moving – just as it changes direction.

Thinking about our animated mass-spring system, we can see this effect. Conversely as the object moves through its equilibrium position, the velocity is only changing very slowly and the acceleration actually becomes zero. This is shown in the acceleration vs time graph below:

Note again – the period is the same, but things don’t line up in time. There’s some phase change in there too.

# Equilibrium and Restoring Forces

In our animations, the mass on a spring bounces up and down with a regular pattern or waveform. Why does this happen? Let’s think about playground swings. What’s the first thing we do to get the swing moving?

The swing is first pulled back, and then pushed forward. It then oscillates back and forth ‘on its own’ until it slowly comes to rest at its equilibrium position – the middle position between the two extremes of displacement. This is usually the place where an object will naturally rest if no external forces are applied to it.

Once the swing has been pulled away from its equilibrium position, the force of gravity will act to bring it back. This force will always act in a direction towards the equilibrium position, and is known as a restoring force. If the swing is pulled higher into the air (larger displacement amplitude) then there will be a larger restoring force acting on it, and when the pusher lets go it will travel further. This shows that the restoring force is proportional to the distance of the swing from the equilibrium position.
Towards the end of the the animation the forces are shown.

So why doesn’t the swing just return directly to the equilibrium position, and stay there?

The restoring forces are large enough to make the swing overshoot. This is when a restoring force in the opposite direction takes over and pushes the swing back down. Again, the force is large enough for the swing to overshoot equilibrium and travel up the other side. The process keeps repeating and an oscillation is formed. If no other forces are applied to the system, the swing will keep swinging back and forth forever, with a constant period and constant amplitude.

Does this happen in real life? (Silly question – of course not). Then why not?

In all real-life oscillating systems, damping comes into play. Damping is all about the loss of energy, often due to resistance or friction. The air displaced by the swinger adds some resistance, and even if the swing were in a vacuum (and the swinger were, therefore, dead) the pivots at the end of the chain generate some friction with their supports and slowly, almost immeasurably, heat up. This uses energy, which has to come from somewhere. And so the amplitude of oscillation decreases.

If you have no pusher, and have to propel yourself on the swing, you have to put some effort into it to keep the swing moving. This effort is overcoming damping. If there were no damping (and also if you could switch off your heart, brain and other energy-consumptive body functions) you could swing forever without needing to eat!

Next: Inertia

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