7.1 Simple Harmonic Motion, Springs, and Pendulums

What defines SHM

Simple harmonic motion (SHM) occurs when a restoring force is directly proportional to displacement and points toward equilibrium: F = -kx. This gives the differential equation d²x/dt² = -ω²x, whose solution is x(t) = Acos(ωt + φ). Here A is amplitude, ω is angular frequency, and φ is the phase constant set by initial conditions.

Springs: mass-spring system

For a mass on a spring, F = -kx gives ω = √(k/m), independent of amplitude. Velocity is v(t) = -Aωsin(ωt+φ) and acceleration a(t) = -Aω²cos(ωt+φ) = -ω²x. Total mechanical energy E = ½kA² stays constant, continuously trading between kinetic (½mv²) and elastic potential (½kx²) energy.

Pendulums

A simple pendulum (small angle approximation, sinθ ≈ θ) has restoring torque -mgLsinθ ≈ -mgLθ, giving ω = √(g/L), independent of mass and amplitude. A physical pendulum with moment of inertia I about the pivot has ω = √(mgd/I), where d is the distance from pivot to center of mass.

Key terms

Simple Harmonic Motion (SHM)

Motion where restoring force is proportional to displacement, F = -kx, producing sinusoidal x(t) = Acos(ωt+φ).

Angular frequency of mass-spring system

ω = √(k/m), independent of amplitude.

Angular frequency of simple pendulum

ω = √(g/L), valid for small angle approximation; independent of mass and amplitude.

Physical pendulum angular frequency

ω = √(mgd/I), where I is moment of inertia about pivot, d is distance to center of mass.

Energy in SHM (spring)

E = ½kA² = ½mv² + ½kx², constant total energy that oscillates between kinetic and potential forms.

Try an AP-style question

A 0.5 kg mass on a spring with k = 50 N/m oscillates. What is its angular frequency?

1. 5 rad/s
2. 10 rad/s
3. 100 rad/s
4. 25 rad/s