Simple harmonic motion (SHM) is the most basic form of oscillation: a system oscillates when the acceleration is proportional to the negative displacement from equilibrium. Many real systems behave approximately like SHM for small oscillations, making it a central model in mechanics, waves, and circuits.
Equation of motion
A defining form is
x¨ + ω^2 x = 0
For a mass–spring system (Hooke's law F = -kx):
m x¨ + k x = 0
ω = sqrt(k/m)
Solution
x(t) = A cos(ω t + φ)
v(t) = -A ω sin(ω t + φ)
a(t) = -ω^2 x(t)
The motion is sinusoidal with amplitude A and phase φ.
Period and frequency
T = 2π/ω
f = 1/T
Larger mass reduces ω (slower oscillation); larger k increases ω (faster oscillation).
Energy picture
For a spring–mass SHM:
U = 1/2 k x^2
K = 1/2 m v^2
E = K + U = 1/2 k A^2 (constant)
Energy continuously swaps between kinetic and potential while the total stays constant (no damping).
Phase space
In the (x, v) plane, SHM traces an ellipse. This provides a geometric view of periodicity and conservation.
Where SHM appears
- Mass–spring oscillators
- Pendulum under small-angle approximation
- LC circuits (charge/current oscillations)
- Any system linearized near a stable equilibrium
Limits of the model
For large amplitudes or strong nonlinearities, the period can depend on amplitude, waveforms deviate from sinusoids, and more complex behavior (including chaos) can appear.