Consider a mass on a frictionless horizontal surface. Connect a spring to it and set the system into motion and it will oscillate back and forth. How can we fully describe the motion?

Start with the free-body diagram. The normal force is cancelled by the force of gravity, and the only horizontal force is the spring force. Therefore:

ΣF = ma

-kx = ma

The acceleration is the second derivative of the position with respect to time. This gives:

-kx

= m

d2x dt2

d2x dt2

= -

k m

x

Let's guess at a solution: x(t) = A cos(ωt)

Here A represents the amplitude of the oscillation, and ω is called the angular frequency. Also, a sine works just as well as a cosine.

Taking two derivatives of x(t) gives:

v(t)

=

dx dt

= -Aω sin(ωt)

a(t)

=

d2x dt2

=

dv dt

= -Aω2 cos(ωt)

a = -ω² x

Compare this to the equation we derived for the mass on the spring:

a

= -

k m

x

This tells us that the angular frequency is given by:

ω2

=

k m

Motion described by an equation like x(t) = A cos(ωt) is known as simple harmonic motion.

Given that a = -ω² x, any time you find the acceleration to be proportional to the negative of the displacement you'll have a system experiencing simple harmonic motion, with the angular frequency being the square root of whatever is multiplying -x.