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Waves · 15.1

Simple Harmonic Motion

A mass on a spring, a swinging pendulum, a vibrating string — all repeat the same motion. Simple harmonic motion is the model, and every wave carries it from place to place.

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01

Build the model

Connect the measurement to the mechanism.

Motion is simple harmonic when the restoring force — and so the acceleration — is proportional to the displacement and points back toward equilibrium. That single condition produces sinusoidal motion whose period, for ideal simple harmonic motion, does not depend on amplitude.

Simple definition
Simple harmonic motion is oscillation in which acceleration is proportional to displacement and always directed toward the equilibrium position.
Example
Pull a mass on a spring down twice as far and the restoring force doubles — it oscillates with the same period, just a larger swing.
Defining conditiona = −ω²x

Acceleration mirrors displacement with a minus sign: the further from centre, the harder it is pulled back.

ω = 2πf = 2π/T

Displacementx = A sin(ωt)

The position traces a sine curve in time — A is the amplitude, the largest displacement reached.

Or A cos(ωt), depending on the start

Mass–spring and pendulumT = 2π√(m/k), T = 2π√(L/g)

Heavier masses and softer springs swing slower; longer pendulums swing slower, and amplitude appears in neither expression. For the pendulum that holds only while the swing stays small — at large angles the period creeps up.

For ideal SHM the period is independent of amplitude. A simple pendulum behaves approximately as SHM for small angular displacements

01

Restoring force

At equilibrium the net force is zero. Displace the object and a force appears pulling it back — overshooting equilibrium keeps the cycle going.

02

Energy exchange

Kinetic energy peaks at the centre; potential energy peaks at the extremes. The total stays constant in the ideal, undamped case.

03

Why it matters for waves

In a mechanical wave — sound, a string, a ripple — each particle of the medium performs SHM about its own equilibrium, and the wave is the pattern of those oscillations passing energy along. An electromagnetic wave needs no medium: there the electric and magnetic fields oscillate instead, but the same sinusoidal mathematics applies.

02

Change one variable at a time

Make the relationship visible.

Watch the acceleration mirror the displacement with opposite sign — that is the SHM condition a = −ω²x.

xt

Displacement x0 cm

Velocity-15.7 cm/s

Acceleration0 cm/s²

ω = 2π/T3.14 rad/s

03

Catch the common trap

Explain before calculating.

A pendulum's length is quadrupled. What happens to its period?

Choose an answer to test the model.

04

Worked examples

State the rule, substitute, then check units.

EasyA pendulum completes 20 swings in 30 s. Find its period and frequency.
  1. T = 30 ÷ 20 = 1.5 s.
  2. f = 1/T ≈ 0.67 Hz.

AnswerT = 1.5 s, f ≈ 0.67 Hz

MediumA 0.50 kg mass on a spring of constant k = 20 N/m oscillates. Find the period and frequency.
  1. T = 2π√(m/k) = 2π√(0.50/20).
  2. T = 2π × 0.158 ≈ 0.99 s.
  3. f = 1/T ≈ 1.0 Hz.

AnswerT ≈ 0.99 s, f ≈ 1.0 Hz

HardA 0.20 kg mass on a spring oscillates with amplitude 5.0 cm and period 0.80 s. Find its maximum speed and maximum acceleration.
  1. ω = 2π/T = 7.85 rad/s.
  2. v(max) = Aω = 0.05 × 7.85 ≈ 0.39 m/s (at the centre).
  3. a(max) = Aω² = 0.05 × 61.7 ≈ 3.1 m/s² (at the extremes).

Answerv(max) ≈ 0.39 m/s; a(max) ≈ 3.1 m/s²

ChallengingA 'seconds pendulum' (T = 2.000 s) is moved to a mountain where g is 0.20% smaller. How much time does the clock lose per day?
  1. T ∝ 1/√g: fractional period change = ½ × 0.20% = 0.10%.
  2. Slower tick by 1.001 factor → loses 0.10% of 86 400 s.
  3. ≈ 86 s — nearly a minute and a half per day.

Answer≈ 86 s/day slow