Simple Harmonic Motion Simulator — Spring and Oscillation

Simple harmonic motion (SHM) is the most fundamental type of oscillation in physics. It occurs whenever a system is displaced from equilibrium and experiences a restoring force proportional to the displacement. Springs, pendulums (for small angles), sound waves, and even atoms vibrating in a crystal lattice all exhibit SHM.

Understanding SHM is essential for waves, optics, quantum mechanics, and electrical circuits — because oscillation is everywhere.

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The Physics of SHM

Equation of Motion

For a mass $m$ on a spring with spring constant $k$, Newton's second law gives:

$m\ddot{x} = -kx \quad \Rightarrow \quad \ddot{x} = -\omega_0^2 x$

where the natural angular frequency is:

$\omega_0 = \sqrt{\frac{k}{m}}$

The solution is sinusoidal:

$x(t) = A\cos(\omega_0 t + \phi)$

where $A$ is the amplitude and $\phi$ is the initial phase.

Key Quantities

Quantity	Formula	Unit
Angular frequency	$\omega_0 = \sqrt{k/m}$	rad/s
Period	$T = 2\pi/\omega_0 = 2\pi\sqrt{m/k}$	s
Frequency	$f = 1/T$	Hz
Max velocity	$v_{max} = A\omega_0$	m/s
Max acceleration	$a_{max} = A\omega_0^2$	m/s²

Damping

Real oscillators lose energy over time. Adding a damping term $\gamma\dot{x}$ gives:

$\ddot{x} + \gamma\dot{x} + \omega_0^2 x = 0$

The damping ratio $\zeta = \gamma/(2\omega_0)$ characterises the decay:

• $\zeta = 0$: undamped — oscillates forever
• $0 < \zeta < 1$: underdamped — decays but oscillates
• $\zeta = 1$: critically damped — fastest return to zero without oscillation

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Worked Examples

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