Oscillations
Core ideas
Simple harmonic motion (SHM) occurs when restoring force is proportional to displacement: F = -kx. The period of a mass-spring system is T = 2π√(m/k); a simple pendulum has T = 2π√(L/g). Notably, neither depends on amplitude for ideal SHM.
Candidates must connect position, velocity, and acceleration in SHM: at maximum displacement, velocity is zero and acceleration is maximum (toward equilibrium); at equilibrium, speed is maximum and acceleration is zero. Energy oscillates between potential (½kx²) and kinetic (½mv²), with total mechanical energy constant.
Graphs of x(t) are sinusoidal; velocity leads position by a quarter cycle and acceleration is opposite in sign to displacement.
Traps
- Believing larger amplitude means longer period - false for ideal SHM.
- Confusing maximum speed (at equilibrium) with maximum acceleration (at endpoints).
- Using pendulum period for large angles - the formula assumes small angles.
- Forgetting that a mass-spring period depends on mass, while a pendulum's does not.
Rule of thumb
Use energy to find speed at any position: ½kA² = ½kx² + ½mv². Use T formulas only for period; they are independent of amplitude.
Worked example
A 0.5 kg mass on a spring with k = 50 N/m has T = 2π√(0.5/50) = 2π√0.01 ≈ 0.63 s. At x = A/2, speed follows from ½kA² = ½k(A/2)² + ½mv², giving v = A√(3k/4m).