Simple harmonic motion

What is simple harmonic motion?

Narrated video. Transcript:

A particle moves to and fro along the x-axis, between x equals plus A and minus A, about the origin. Watch it with a strobe light: dots far apart in the middle, crowded together near the ends.

So it moves fastest through the centre and slows down, stops and turns round at each end. Now plot its displacement x against time.

The graph is a smooth cosine curve. When the displacement varies with time like this, x of t equals A cos of ωt plus φ, the motion is called simple harmonic motion, or SHM.

A is the amplitude: the largest displacement on either side of the mean position. It is always positive, and the particle always stays between minus A and plus A.

The whole angle, ωt plus φ, is the phase. It tells us exactly where the particle is, and which way it is moving, at time t.

φ is the phase constant: the phase at t equals zero. Watch two particles with the same amplitude and period but different φ. Their graphs are the same shape, just shifted along the time axis.

ω is the angular frequency. The motion repeats after one period T, so ωT must equal 2π. That gives ω equals 2π over T, which is 2πν, measured in radians per second.

Worked example: x equals 6 cos of πt plus π by 3, in centimetres. The amplitude is 6 cm. ω is π radians per second, so T is 2π over π, which is 2 seconds. At t equals zero, x is 6 cos 60°, which is 3 cm.

Be careful: SHM is not just any periodic motion. A puck bouncing between two walls repeats, but its graph is a zig-zag. Only a sine or cosine curve of time is simple harmonic.

SHM and uniform circular motion

Narrated video. Transcript:

Here is a beautiful link. Draw a circle of radius A, centred at the origin. A point P moves round it anticlockwise at a constant angular speed ω. We call this the reference circle.

At t equals zero, the radius OP makes an angle φ with the x-axis. After a time t, it has turned through ωt more, so the angle is ωt plus φ.

Now drop a perpendicular from P onto the x-axis. Its foot is P′. The x-coordinate of P′ is A cos of ωt plus φ. That is exactly the equation of SHM.

So as P goes round steadily, its shadow P′ slides to and fro along the diameter in simple harmonic motion. Shine a light from above and watch the shadow on a screen below.

Match the parts. The radius of the circle is the amplitude. The angular speed of P is the angular frequency of the SHM. And the angle OP makes with the x-axis is the phase.

One full turn of P takes 2π over ω. In that time P′ goes from plus A to minus A and back again: one complete oscillation. So the period of the SHM is also T equals 2π over ω.

Worked example: P goes round a circle of radius 3 cm once every 2 seconds, starting at 45° to the x-axis. Then ω is 2π over 2, or π radians per second, and φ is π by 4. So x equals 3 cos of πt plus π by 4, in centimetres.

The projection of P on the y-axis, A sin of ωt plus φ, is also simple harmonic. Remember, though: P itself moves in a circle, which is not SHM. Only its projection on a diameter is.

Try it: the reference circle

Interactive simulation. Things to try:

• Press Play (or Space). P goes round the circle; its foot P′ moves to and fro on the x-axis.
• Change A: the circle and the graph grow together. The period does not change.
• Change φ: the starting point on the circle moves, and the whole x–t graph shifts sideways.
• Halve T: P goes round twice as fast and the graph squeezes up.

Where does it start?

A particle moves with x = 4 cos(πt + π/2) cm. Where is it at t = 0, and which way is it moving?

1. At x = +4 cm, momentarily at rest
2. At x = 0, moving towards +x
3. At x = 0, moving towards −x
4. At x = −4 cm, momentarily at rest

Read the equation

A particle in SHM has x = 5 cos(4πt + π/3) cm, with t in seconds. What are its period and its displacement at t = 0?

1. T = 0.5 s, x = 2.5 cm
2. T = 2 s, x = 2.5 cm
3. T = 0.5 s, x = 4.3 cm
4. T = 4π s, x = 5 cm

Summary

• Displacement in SHM: x(t) = A cos(ωt + φ)
• Angular frequency: ω = 2π/T = 2πν
• Phase: ωt + φ
• Reference circle: radius = A, angular speed = ω

• In simple harmonic motion, the displacement is a sine or cosine function of time.
• The amplitude A is the maximum displacement. The phase ωt + φ fixes the state of motion; the phase constant φ is its value at t = 0.
• SHM is the projection of uniform circular motion on a diameter of the circle.
• Not every periodic motion is SHM: only one whose x–t graph is a single sinusoid.