Simple Harmonic Motion

Key Facts

• Period of a spring-mass system: $T = 2\pi\sqrt{\frac{m}{k}}$
• Period of a simple pendulum: $T = 2\pi\sqrt{\frac{L}{g}}$ - independent of mass and amplitude (for small angles)
• Frequency and period: f = 1/T
• Maximum velocity at equilibrium: $v_{\text{max}} = A\omega$ (where $\omega = 2\pi f$)
• Maximum acceleration at turning points: $a_{\text{max}} = A\omega^2$
• Total energy: $E = \frac{1}{2}kA^2 = \frac{1}{2}mv_{\text{max}}^2$

Vocabulary

Amplitude (A)

Maximum displacement from the equilibrium position.

Period (T)

Time for one complete oscillation cycle, measured in seconds.

Frequency (f)

Number of complete oscillations per second, measured in hertz (Hz).

Angular frequency (ω)

Rate of oscillation in radians per second: ω = 2πf.

Restoring force

The force that always pushes or pulls the object back toward equilibrium; proportional to displacement in SHM.