# What Is Simple Harmonic Motion?

Many simple oscillators display the characteristic of Simple Harmonic Motion (SHM). Examples are springs, pendula, waves on the sea... all of which move in a predictable way and follow a certain pattern. Oscillators demonstrating SHM have displacement-time graphs that can be related to the graph of or plotted against , in other words regular Sine or Cosine curves.

Oscillators in simple harmonic motion follow several equations. These are related to different factors about the oscillator. For the purposes of this document I will confine these equations to springs, which can easily be used to test the ideas put forward.

An alternative derivation of SHM is also available, which uses a theoretical experiment to obtain by logical progression the proportionality contained in the equation.

From Hooke's Law we know that Force exerted by a spring on a mass attached to it is directly proportional to the extension of the spring, or:

Where *F* is
the force exerted on the spring and *x* is the extension of the
spring from its normal length. From this we can see that the equation
relating these two vectors is:

Where *k* is
Hooke's constant.

From this equation,
and therefore from the slope of a Force against Displacement graph,
we can calculate *k*.

With a mass with
weight *W* Newtons attached but at rest, the spring stretches by
distance . This then
means that:

as the force provided by the spring must be equal to the downward force provided by the mass.

If we pull the weight
down a further distance *x*, the restoring force, *F* provided
by the spring in an upwards direction acting on the mass is *F+W*.
As at this position:

Applying Newton's
second Law of Motion, *F=ma* gives us:

As we know that
from Hooke's Law for *W* the spring extends a certain distance:

We can then say:

Which can then be cancelled to give:

or

This then relates a spring's Hooke's constant to the acceleration acting on a certain mass.

From one of the equations for SHM we can derive another equation:

comparing this with
the above equation derived from *F=ma*:

As the period, *T*,
of the motion of the mass on the spring is given by the general equation:

we can substitute to give:

where *m* is,
in this case, the mass on the spring and *k* is the stiffness of
the spring.