Simple harmonic motion

An object is moving with simple harmonic motion if the following two points apply:

  • The acceleration of the object is directly proportional to the displacement from equilibrium position.
  • The acceleration is always directed towards the equilibrium position.

Frequency and period

The frequency of an oscillating object is the reciprocal of the period:

f = \frac{1}{T}

Spring force

The force F of a spring is proportional to the extension x from equilibrium, with k as the spring constant (measured in newtons-per-meter, \N\m^{-1}). The force is negative because it points back towards equilibrium, not outwards along the axis of extension.

F = -kx

The equation fails beyond the limit of proportionality, which is the extension past which the spring is damaged, being permanently deformed.

Angular velocity

The angular velocity \omega of a spring is the speed that the spring will bounce, and is calculated from the period T of the harmonic motion:

\omega = \frac{2\pi}{T} = 2\pi f

Angular velocity can also be calculated from the mass m and the spring constant k (given in \N\m^{-1}):

\omega = \sqrt\frac{k}{m}

Extension over time

We can model the current extension x of a spring relative to the current time t, given the maximum extension x_0, the angular velocity \omega, and the phase offset \phi. This won’t always use \sin, it depends on the extension of the spring at t = 0. The graph shifts to the left when \phi is positive, and to the right when \phi is negative (same as with offsets to x in regular functions).

x = x_0 \cdot \sin(\omega t + \phi)

Restoring force

The restoring force of a spring is the force that pulls back in an attempt to restore equilibrium. The force always points towards the equilibrium, towards the spring. a_x is the restorative acceleration of the spring.

F = ma_x = -kx

Conservation of energy

Springs conserve kinetic energy. The spring mass slows down at the limits, the lost speed becomes potential energy stored in the spring, and as that potential energy causes acceleration in the other direction the mass regains kinetic energy.

At any point in time, total energy is the sum of the kinetic and potential energy.

E = E_k + E_p
E = \frac{1}{2}mv^2 + mgh

The potential energy stored in a stretched spring is proportional to the extension x of the spring, where F = kx:

U_s = \frac{1}{2}Fx = \frac{1}{2}kx^2

Pendulums

The period T of a pendulum is given by the length L and the gravitational acceleration g:

T \approx 2\pi \sqrt\frac{L}{g}