Simple Harmonic Motion Explorer

SHM occurs when a restoring force is proportional to displacement. For a spring this is Hooke's law.

The resulting motion is sinusoidal.

where $\omega_0 = \sqrt{k/m}$ is the natural angular frequency, $A$ is the amplitude, and the period is $T = 2\pi/\omega_0$.

Adding a damping force proportional to velocity gives

The damping ratio $\zeta = \frac{b}{2\sqrt{mk}}$ determines the behavior.

• Underdamped ($\zeta < 1$) oscillates with decaying amplitude
• Critically damped ($\zeta = 1$) returns to equilibrium fastest without oscillating
• Overdamped ($\zeta > 1$) returns slowly without oscillating

For small angles, a pendulum follows the same math as a mass-spring system with an effective spring constant $k_{\text{eff}} = mg/L$.

The period depends only on length and gravity, not on mass or amplitude (for small swings). A longer pendulum swings more slowly.

A phase portrait plots velocity vs displacement. For undamped SHM, the trajectory is an ellipse.

• No damping traces a closed ellipse (energy is constant)
• Underdamped spirals inward toward the origin
• Critically/overdamped approaches the origin without looping

The phase portrait shows the system's entire state (position and velocity) at a glance. Points far from the origin have high energy; points near the origin have low energy.