Oscillations

Topic-by-topic explanations

1. Introduction

Periodic motion repeats after a fixed time T; frequency is f=1/T and angular frequency ω=2πf. Oscillation is to-and-fro motion about equilibrium, but not all oscillations are SHM.

2. Periodic and oscillatory motions

An oscillating body moves about its equilibrium position. Its period is the time for a full cycle and its amplitude is the maximum displacement from equilibrium.

3. Simple harmonic motion

Simple harmonic motion has restoring force proportional to displacement and opposite in direction: F=−kx and a=−ω²x. Displacement can be written x=Acos(ωt+φ).

4. Simple harmonic motion and uniform circular motion

A projection of uniform circular motion onto a diameter follows SHM with the same angular frequency; the projection gives sinusoidal displacement.

5. Velocity and acceleration in simple harmonic motion

For x=Acosωt, v=−Aωsinωt and a=−ω²x. Speed is maximum at equilibrium and zero at extremes; acceleration magnitude is maximum at extremes.

6. Force law for simple harmonic motion

A linear spring obeys Hooke's law F=−kx. With mass m, ω=√(k/m), giving T=2π√(m/k) if damping is negligible.

7. Energy in simple harmonic motion

Energy alternates between kinetic energy and spring potential energy U=½kx². Total energy E=½kA²; kinetic energy is maximum at equilibrium and potential energy at extremes.

8. The Simple Pendulum

For a simple pendulum at small angles, T=2π√(l/g), independent of bob mass. The approximation assumes small amplitude and negligible damping.

Formula sheet

Use SI units, state assumptions, show your working and check units before finalising an answer.

Worked example

Common errors and revision checklist

A linear restoring force is required for SHM. Use the pendulum period only for small angles. Velocity and acceleration peak at different positions.

• Write the governing law before substituting values.
• Show units and signs/directions.
• Check whether assumptions are satisfied.