Simple Harmonic Motion Calculator

Simple Harmonic Motion

Simple harmonic motion (SHM) describes oscillations where restoring force is proportional to displacement. Mass-spring systems, small pendulum swings, and vibrating molecules all approximate SHM near equilibrium.

Equations of Motion

$$x = A\cos(\omega t + \phi)$$ $$v = -A\omega\sin(\omega t + \phi)$$ $$a = -A\omega^2\cos(\omega t + \phi)$$

$A$ is amplitude, $\omega = 2\pi f$ is angular frequency, $t$ is time, and $\phi$ is the phase constant. Velocity leads displacement by 90° and acceleration is antiparallel to displacement.

Energy Perspective

Total mechanical energy $E = \tfrac{1}{2} k A^2$ is constant in ideal SHM. Kinetic and potential energy exchange continuously while amplitude stays fixed in the absence of damping.

Related tools: Harmonic Wave Equation Calculator and BPM Calculator.

Frequently Asked Questions

What is the difference between frequency and angular frequency?

Frequency $f$ is cycles per second (hertz). Angular frequency $\omega = 2\pi f$ is radians per second and appears directly in the cosine argument.

Why use cosine instead of sine?

Both describe SHM. Cosine with $\phi = 0$ starts at maximum displacement. Sine with $\phi = 0$ starts at zero displacement with maximum speed.

When is acceleration maximum?

Acceleration magnitude is maximum at the turning points where displacement equals $\pm A$. At the equilibrium position, acceleration is zero but velocity is greatest.

What does the phase constant phi control?

Phase constant sets the initial conditions at $t = 0$. It shifts the entire motion along the time axis without changing amplitude or frequency.

Does this model include damping?

No. These equations assume undamped SHM. Real oscillators lose energy to friction and require damped harmonic motion equations.

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Simple Harmonic Motion Calculator