7.3 Pendulums and Springs

Simple pendulum period

A simple pendulum oscillates with period T = 2π√(L/g), where L is string length and g is gravitational acceleration. This only holds for small angles (approximately less than 15°), so the motion approximates simple harmonic motion (SHM). Notice: T depends only on L and g — not on mass or amplitude.

Mass-spring system period

A block on a spring oscillates with period T = 2π√(m/k), where m is mass and k is the spring's force constant. This is true for any amplitude, since the restoring force F = -kx is always linear (ideal Hooke's law spring). T increases with more mass and decreases with a stiffer spring.

Energy in oscillators

Both systems trade kinetic and potential energy while total mechanical energy stays constant (no friction). For springs, PE = ½kx²; for pendulums, PE = mgh. Maximum speed occurs at equilibrium (x=0 or bottom of swing); maximum PE occurs at maximum displacement, where velocity is zero.

Key terms

Period of simple pendulum

T = 2π√(L/g); depends only on length and gravity, valid for small angles.

Period of mass-spring system

T = 2π√(m/k); depends on mass and spring constant, valid for any amplitude (ideal spring).

Small-angle approximation

For angles less than about 15°, sin(θ) ≈ θ, making pendulum restoring force approximately linear (SHM).

Energy exchange in oscillators

Kinetic and potential energy continuously convert into each other while total mechanical energy remains constant (no friction).

Where is speed maximum in SHM?

At the equilibrium position, where displacement and potential energy are zero and kinetic energy is maximum.

Try an AP-style question

A 2 kg mass hangs at rest from a spring, stretching it 0.1 m. What is k? (g ≈ 10 m/s²)

1. 20 N/m
2. 2 N/m
3. 0.2 N/m
4. 200 N/m