What is a simple harmonic oscillator

[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

<br /> x(t)=A\sin(\omega t)+B\cos(\omega t)<br />

<br /> \omega^2 =\frac{k}{m}<br />

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
<br /> m\frac{d^2 x}{dt^2}=-kx\;,\qquad(1)<br />
where x is the position of the point mass.

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

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

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[APhysicist]

I understand that a sinusoidal oscillator is an object (usually a mass on a spring) that has a position that varies sinusoidally in time. The equation for the motion of this oscillator is given by x(t)=A\sin(\omega t)+B\cos(\omega t). Furthermore, we can derive the frequency of the oscillations as \omega=\sqrt{\frac{k}{m}}.