Simple harmonic motion

MDCAT Physics · Physics · Waves and oscillations. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

In simple harmonic motion (SHM) a body moves to and fro about a mean position. Its acceleration is proportional to its displacement and always points back towards the mean position: a = −ω²x. So displacement and acceleration are always opposite in direction.

At the mean position the speed and kinetic energy are greatest and the acceleration is zero. At the extreme positions the speed is zero and the acceleration and potential energy are greatest. The total energy stays constant.

Damping removes energy through friction, so the amplitude slowly falls. Resonance happens when a body is driven at its own natural frequency: its amplitude then becomes very large.

Rules to remember

Condition for SHM: a = −ω²x, with ω = 2πf = 2π/T
displacement: x = x₀ sin ωt, where x₀ is the amplitude
speed: v = ω√(x₀² − x²); greatest speed = ωx₀ at the mean position
greatest acceleration = ω²x₀ at the extreme positions
Energy: PE = ½kx², KE = ½k(x₀² − x²), total = ½kx₀²
Mass–spring system: T = 2π√(m/k), does not depend on g or on the amplitude
Simple pendulum: T = 2π√(l/g), does not depend on the mass or on a small amplitude

Common mistake

Students think a heavier bob makes a pendulum swing more slowly. The period of a pendulum depends only on its length and g; it is the mass–spring period that depends on mass.

Worked example

A 0.50 kg mass on a spring of spring constant 200 N m⁻¹ oscillates with an amplitude of 0.10 m. Find its period, greatest speed and total energy.

ω = √(k/m) = √(200 ÷ 0.50) = √400 = 20 rad s⁻¹.
T = 2π/ω = 2π ÷ 20 = 0.31 s.
Greatest speed = ωx₀ = 20 × 0.10 = 2.0 m s⁻¹.
Total energy = ½kx₀² = ½ × 200 × 0.10² = 1.0 J.

Answer: T = 0.31 s; greatest speed 2.0 m s⁻¹; total energy 1.0 J.

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