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:
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.