HARMONIC MOTION

In this presentation, you will learn that: The time period T of a harmonic oscillator is INDEPENDENT of its amplitude. This property is ideal for time-keeping. Although nothing is constant about this type of motion, we can relate it directly to CIRCULAR MOTION, which has a constant (angular frequency in rad s-1). Simple geometry will lead us to a useful expression of displacement, amplitude and frequency of oscillation.

Harmonic oscillators are all around us:

They can destroy:

They can cure:

Start with a pendulum: - A +A Displacement s

+A Displacement s time - A Periodic time T

Imagine a pendulum oscillating over an old-fashioned record turntable: Looking from above: start Turntable has moved to here Pendulum bob moves to here

Looking from above: Although the pendulum bob is moving up and down in a complex sinusoidal way, the arrow is moving at a CONSTANT SPEED round the circle.

The moves up and down the vertical line. The arrow goes round the circle anticlockwise. ? The phasor picture

The clock arrow rotates at constant angular frequency  in rad s-1 Answer to ? The clock arrow rotates at constant angular frequency  in rad s-1

Phase angle Deg rad 45 /4 90 /2 135 180  225 270 315 360 2  45 /4 90 /2 135 3 /4 180  225 5 /4 270 3 /2 315 7 /4 360 2  s = A sin   = t Radius is the Amplitude A  =  /t in rad s-1  is the phase angle

Some basics …………. Clock arrow rotates 2 in time period T. Angle  = 2 (t/T) T = 1/f where f = frequency So…… Angle  = 2ft And since  =  /t, so  = 2 f. Displacement s = A sin = A sin 2ft

The language of oscillators: f = 1/T in Hz = 2f in rad s-1 Always work out  first, if your oscillator is harmonic! It’s then easy to find your displacement s: s = A sin 2 ft s = A sin t Note that s = A sin 2 ft when s = 0 when t = 0. If s = A when t = 0, there is a /2 phase difference and s = A cos 2 ft.

3. What will be the displacement after 8 secs? Try this example: A child’s swing oscillates at a frequency of 0.5 Hz. It’s amplitude at the start of its swing is 2 m. 1. What’s its angular frequency? (Don’t be put off by it not moving in a circle!) Answer:  = 2f = 2 x 3.14 x 0.5 = 3.1 rad s-1 2. What will be the displacement of the swing after 3.4 seconds? In which direction? Answer: s = A sin t = 2 x sin  x 3.4 = 0.37 m towards equilibrium. 3. What will be the displacement after 8 secs? Answer: since f = 1/T, T = 2 secs. Displacement will be 0 m.

Two very useful equations (easy to remember) Max velocity in any cycle = A (where A is Amplitude.) Max acceleration in any cycle = 2A

Summary

All harmonic oscillators have these things in common: They are accelerated towards an equilibrium position by a “spring-like” force. This always pulls it back towards the equilibrium position. (F = ma) At the equilibrium position, the velocity of the oscillator is unchanged and at its maximum. So, there is no acceleration here – and no resultant force! Its time trace is sinusoidal, oscillating at the NATURAL FREQUENCY fo of the oscillator.

What about the energy of the oscillator? It stores energy. The energy goes back and forth,being stored by a sort of “spring” (P.E.) at the extremes, and then carried by the motion (K.E.) as it passes through the equilibrium position. Resistive forces gradually drain the oscillator of its energy. Its amplitude gradually decreases until more energy is fed back into it to compensate.