# What is a simple harmonic oscillator

1. Jul 24, 2014

### Greg Bernhardt

Definition/Summary

An object (typically a "mass on a spring") which has a position (or the appropriate generalization of position) which varies sinusoidally in time.

Equations

$$x(t)=A\sin(\omega t)+B\cos(\omega t)$$

$$\omega^2 =\frac{k}{m}$$

Extended explanation

According to Hooke's law and Newton's 2nd Law, a point mass of mass $m$ attached to a spring of spring constant $k$ obeys the equation
$$m\frac{d^2 x}{dt^2}=-kx\;,\qquad(1)$$
where $x$ is the position of the point mass.

The solution of equation (1) is given by
$$x(t)=A\sin(\omega t)+B\cos(\omega t)\;,\qquad(2)$$
where A and B are constants that may be chosen so that x(t) satisfies the appropriate initial conditions, and
where
$$\omega=\sqrt{\frac{k}{m}}\;.$$

For example, in terms of the initial position $x_0$ and initial velocity $v_0$, equation (2) can be written as
$$x(t)=\frac{v_0}{\omega}\sin(\omega t)+x_0\cos(\omega t)\;.$$

* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!