Periodic and oscillatory motion

Motions that repeat

Put your fingers on your wrist and feel your pulse. It beats again and again, at nearly equal gaps of time. The second hand of a clock comes back to 12 every minute. The sea rises and falls at a beach about twice a day. The Earth turns once on its axis every day.

Each of these motions repeats itself after a fixed time. We call it a periodic motion. The smallest time after which the motion repeats is its period, $T$. The SI unit of the period is the second.

Oscillations

Now look at the needle of a sewing machine. It goes down, comes back up, and goes down again. It moves to and fro about a middle position. That middle position is the mean position, or the position of equilibrium. A periodic motion to and fro about a mean position is an oscillation. We also call it a vibration, mostly when it is fast and small.

Is every periodic motion an oscillation? No. The Earth going round the Sun repeats every year, but it does not move to and fro about a mean position. It goes round. So it is periodic, but it is not an oscillation. Every oscillation, on the other hand, is periodic.

Why does a body oscillate? At the mean position, the net force on it is zero. Pull it a little to one side, and a force pulls it back. It overshoots, and the force pulls it back again. In course 132 we study this force in detail.

Period and frequency

The frequency $f$ is the number of repeats in each second:

Its SI unit is the hertz (Hz). One hertz is one oscillation each second. A heart that beats $72$ times in $60\ \text{s}$ has a period of $\dfrac{60\ \text{s}}{72} = 0.833\ \text{s}$ and a frequency of $1.2\ \text{Hz}$.

Fast vibrations have large frequencies. A note on a guitar may be a few hundred hertz. The vibrations of atoms in a solid are near $10^{13}\ \text{Hz}$. Slow ones have small frequencies. The tides have a period of about half a day.

Displacement as a function of time

To describe an oscillation, we need a quantity that changes as the body moves. We call it the displacement. For the needle, it is the distance from the mean position, in metres. For a swinging door, it can be an angle. For a sound wave, it can be the change in pressure. The word is the same, and the idea is the same.

A periodic motion has a displacement that is a periodic function of time: $x(t + T) = x(t)$ for every $t$. The simplest periodic functions are the sine and the cosine:

Add $T$ to $t$. The angle grows by $2\pi$, and the sine and cosine come back to the same values.

What if we add two sines? If one has the period $T$ and the other $T/2$, the sum still repeats every $T$. Graph (b) shows such a sum. A useful result: any periodic function can be built from sines and cosines of the periods $T$, $T/2$, $T/3$ and so on.

Graph (c) is different. The curve $x = e^{-t/\tau}$ falls for ever. It never comes back to an earlier value, so it is not periodic.

Practise with new numbers

Try it in Python

The program finds the period and the frequency of three motions from a count, as you would with a stopwatch.

Common mistakes

• Mixing up period and frequency. The period is a time, in seconds. The frequency is a count each second, in hertz.
• Calling every periodic motion an oscillation. A body going round a circle is periodic, but it does not go to and fro.
• Taking half a trip as one oscillation. One oscillation goes from a point, out and back, and on to the same point moving the same way.
• Thinking the displacement is always a distance. It can be an angle, a pressure or a voltage too.

Remember these

• A periodic motion repeats after a fixed time, the period $T$.
• An oscillation is a periodic motion to and fro about a mean position.
• $f = \dfrac{1}{T}$, in hertz. One hertz is one oscillation each second.
• A periodic displacement satisfies $x(t + T) = x(t)$. The sine and cosine are the simplest examples.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A child on a swing goes forward and back $15$ times in one minute. Find the period and the frequency of the swing.

2. Is the motion of the hands of a clock periodic? Is it an oscillation?

3. A string on a musical instrument vibrates with a frequency of $250\ \text{Hz}$. What is its period, in milliseconds?

4. A displacement is $x = 2\sin(\pi t) + \sin(2\pi t)$, with $t$ in seconds. Is the motion periodic? If so, what is its period?