How to Calculate Simple Harmonic Motion

Describes a mass bouncing on a spring. The angular frequency is √(spring constant ÷ mass), which fixes the period on its own: the amplitude makes no difference to it. The greatest speed and the total energy of the motion are also given.

A mass bouncing on a spring performs simple harmonic motion: a restoring force always acts, proportional to how far it has moved from rest.

\omega = \sqrt{\dfrac{k}{m}}, \quad T = \dfrac{2\pi}{\omega}

$k$ is the spring constant, $m$ the mass, $\omega$ the angular frequency and $T$ the period. The greatest speed is $A\omega$ and the total energy is $\frac{1}{2}kA^2$ , where $A$ is the amplitude.

The period does not depend on the amplitude

Notice that $A$ does not appear in the formula. Swinging wide or narrow takes exactly the same time for one cycle. A wider swing covers more ground, but it also moves faster, and the two effects cancel. Pendulum clocks keep time because of this.

Example

A 0.2 kg mass on a 20 N/m spring swings with an amplitude of 0.1 m. The angular frequency is $\sqrt{20 \div 0.2} = 10$ rad/s, the period is $2\pi \div 10 = 0.628$ seconds and the frequency is 1.59 Hz. The greatest speed is $0.1 \times 10 = 1$ m/s and the total energy is $\frac{1}{2} \times 20 \times 0.1^2 = 0.1$ J.

Notes

Tripling the amplitude triples the greatest speed but multiplies the energy by nine. Only the energy goes with the square of the amplitude.

A simple pendulum has a period of $2\pi\sqrt{\ell \div g}$ , which does not involve the mass at all. The same kind of motion, settled by different quantities.