Simple Harmonic Systems — Quick Summary

Mr Toogood's Physics · Periodic motion

AQA 3.6.1.3
T = 2π√(l/g)
Simple pendulum
T = 2π√(m/k)
Mass–spring
a = −g sinθ
Restoring accel.
θ ≤ 10°
Small-angle limit

The simple pendulum

Forces on a simple pendulum, weight resolved into components along and perpendicular to the swing

Weight resolves into a component along the path (restoring force) and one perpendicular to it.

The weight's component along the swing, −mg sinθ, is the restoring force. By Newton's 2nd law:
a = −g sinθ
For small angles (θ ≤ 10°), sinθ ≈ s/L, and arc length s ≈ x, giving:
a = −gx/L = −(2πf)²x
Comparing with T = 1/f gives the pendulum period:
T = 2π√(l/g)
Finding g: squaring gives T² = 4π²(l/g) — a graph of T² against l is a straight line through the origin with gradient 4π²/g.

Mass–spring systems

The same logic applies to a mass on a spring, combining three equations:

F = ma    F = ke    a = −(2πf)²x
Substituting gives:
−ke = −m(2πf)²x
Extension e and displacement x are both lengths and cancel, rearranging to:
T = 2π√(m/k)
Key difference: unlike the pendulum, the spring's period depends on mass (not gravity) — this is why mass–spring oscillators can be used to measure mass in space, where weighing scales don't work.

Quick contrast

  • Pendulum: T depends on l and g only — independent of mass and amplitude (for small angles).
  • Spring: T depends on m and k only — independent of g and amplitude.

Exam essentials

Key equations

  • T = 2π√(l/g) — pendulum.
  • T = 2π√(m/k) — mass–spring.
  • Small-angle condition: θ ≤ 10° (≈0.17 rad).

What each depends on

  • Pendulum: length & g only — not mass or amplitude.
  • Spring: mass & spring constant only — not g or amplitude.

Common slips

  • Pendulum formula only holds within the small-angle approximation.
  • On a T²–l graph, the gradient is 4π²/g, not g itself.
  • Don't confuse extension e with displacement x — equal in magnitude here, but conceptually different quantities.