Mass on a Spring

When a spring with spring constant k and length l has a mass m attached to the bottom it extends by an extension e, this is called the static extension and is the new equilibrium point. The tension in the spring is balanced by the weight. We can represent this as:

If the mass is pulled down by a displacement x and released it will undergo SHM.

The net upwards force will be:

This can be multiplied out to become:

Since this can become:

It simplify to:

Since both and (for SHM) the equation now becomes:

This simplifies to:

Rearranging for f gives us:

And since the equation becomes:

Time is measured in seconds, s