Simple Harmonic Motion (SHM) Cheat Sheet

Key Facts

• Simple harmonic motion occurs when the restoring force is proportional to displacement and opposite in direction, so $F = -kx$ for a spring.
• Displacement in SHM can be modeled by $x(t) = A\cos(\omega t + \phi)$ or $x(t) = A\sin(\omega t + \phi)$.
• Angular frequency is related to frequency and period by $\omega = 2\pi f = \frac{2\pi}{T}$.
• Velocity in SHM has maximum magnitude $v_{\max} = A\omega$ and can be written as $v(t) = -A\omega\sin(\omega t + \phi)$ for cosine displacement.
• Acceleration in SHM is proportional to displacement and opposite in direction, so $a = -\omega^2 x$.
• The period of a mass-spring oscillator is $T = 2\pi\sqrt{\frac{m}{k}}$.
• For small angles, the period of a simple pendulum is $T = 2\pi\sqrt{\frac{L}{g}}$.
• Total mechanical energy in an ideal spring oscillator is constant and equals $E = \frac{1}{2}kA^2$.

Vocabulary

Amplitude

Amplitude is the maximum displacement from equilibrium, represented by $A$.

Equilibrium Position

The equilibrium position is the point where the net restoring force is zero.

Period

Period is the time for one complete cycle of motion, represented by $T$.

Frequency

Frequency is the number of cycles per second and is given by $f = \frac{1}{T}$.

Angular Frequency

Angular frequency measures how quickly the oscillator moves through its cycle and is given by $\omega = 2\pi f$.

Restoring Force

A restoring force is a force that acts toward equilibrium and often has the form $F = -kx$.