7.1 Period of Simple Harmonic Motion

What makes motion simple harmonic

Simple harmonic motion (SHM) occurs when the restoring force on an object is directly proportional to its displacement from equilibrium and points toward equilibrium (F = -kx). This produces smooth, repeating oscillations. Springs and small-angle pendulums are classic examples where the motion repeats with a constant period, independent of amplitude.

Period of a mass-spring system

For a mass m on a spring with spring constant k, the period is T = 2π√(m/k). Period depends only on mass and spring stiffness — not amplitude. A stiffer spring (larger k) or lighter mass gives a shorter period, meaning faster oscillations.

Period of a simple pendulum

For a pendulum of length L undergoing small-angle oscillations, the period is T = 2π√(L/g), where g is gravitational acceleration. Period depends only on length and gravity — not mass or amplitude. Longer pendulums and weaker gravity both increase the period.

Key terms

Simple harmonic motion (SHM)

Oscillatory motion where the restoring force is proportional to displacement and directed toward equilibrium (F = -kx).

Period of a mass-spring system

T = 2π√(m/k); depends on mass and spring constant, not amplitude.

Period of a simple pendulum

T = 2π√(L/g); depends on length and gravity, not mass or amplitude (for small angles).

Restoring force

A force that always acts to push or pull an object back toward its equilibrium position.

Does amplitude affect period in SHM?

No — for ideal SHM (springs, small-angle pendulums), the period is independent of amplitude.

Try an AP-style question

The mass on a spring is quadrupled. How does the period change?

1. It quadruples
2. It stays the same
3. It doubles
4. It halves