Circular Motion and Simple Harmonic Motion

A-Level Physics · Further Mechanics and Thermal Physics

Circular motion

An object moving in a circle at constant speed is accelerating, because its direction (and so velocity) is constantly changing. This requires a centripetal force directed towards the centre.

Angular speed: ω = 2π ÷ T = 2πf (rad/s).
Linear speed: v = ωr.
Centripetal acceleration: a = v² ÷ r = ω²r, directed towards the centre.
Centripetal force: F = mv² ÷ r = mω²r.

The centripetal force is provided by another force (gravity, tension, friction), not a separate force. There is no outward "centrifugal" force in this model.

Simple harmonic motion (SHM)

SHM occurs when the restoring force (and acceleration) is proportional to the displacement and directed towards the equilibrium position:

a = −ω²x

The minus sign shows acceleration is always opposite to the displacement. Examples: a mass on a spring, a simple pendulum (small angles).

Key SHM relationships

Displacement: x = A cos(ωt) (starting at maximum) — A is the amplitude.
Maximum speed: v_max = ωA (at the equilibrium position).
Maximum acceleration: a_max = ω²A (at maximum displacement).
Period is independent of amplitude (isochronous).
Mass–spring: T = 2π√(m/k); simple pendulum: T = 2π√(L/g).

Energy in SHM

Energy continually transfers between kinetic (max at equilibrium) and potential (max at the extremes). The total energy stays constant (ignoring damping).

Damping and resonance

Damping — resistive forces (e.g. air resistance) reduce the amplitude over time. Light damping decays slowly; critical damping returns to equilibrium fastest without oscillating.
Resonance — when a system is driven at its natural frequency, the amplitude grows very large. Important in bridges, buildings and tuning circuits; damping reduces resonant amplitude.

Worked example

A mass on a spring has period 0.5 s and amplitude 0.02 m. Find its maximum speed.

ω = 2π/T = 2π/0.5 = 12.6 rad/s.
v_max = ωA = 12.6 × 0.02 = 0.25 m/s. ✓

Common mistakes

Thinking there's an outward (centrifugal) force — the net force is centripetal, towards the centre.
Forgetting the minus sign in a = −ω²x (defines SHM).
Saying period depends on amplitude — for SHM it doesn't.

Exam tips

Learn centripetal formulas (a = v²/r = ω²r, F = mv²/r) and where the force comes from.
Recognise SHM from a = −ω²x; use v_max = ωA, a_max = ω²A.
Explain resonance as driving at the natural frequency and the role of damping.

Key facts to remember

Circular motion needs a centripetal force to the centre: F = mv²/r = mω²r, a = v²/r; v = ωr, ω = 2πf.
SHM: a = −ω²x (restoring force ∝ displacement); period independent of amplitude; v_max = ωA.
Energy swaps KE ↔ PE (total constant); damping reduces amplitude; resonance at the natural frequency gives large amplitude.