5.3.2 Energy of a simple harmonic oscillator

Okay so this is nice and short. It's just to do with the energy transfers involved in SHM.

As we know, energy must be conserved. Provided there are no frictional forces (e.g damping) the total energy for an object moving in SHM will remain constant:

• At the amplitude the pendulum has zero kinetic energy (it is very briefly stationary). All its energy is in the form of potential energy (GPE in the case of a pendulum, EPE in the case of a horizontal string)
• As it moves through its equilibrium position it has maximum kinetic energy (as it has maximum velocity) and no potential energy.

This graph nicely illustrates how the total energy remains constant:

Credit: Kerboodle OCR A Physics A level textbook

We also need to be able to interpret graphs. In this case I will use an example of a spring on a horizontal glider:

• EPE is given by the equation EP = 0.5kx² (k is the force constant of the spring. This means that a graph of EP-x will be parabolic
• The EPE is always positive and varies from 0 to 0.5kA²
• When at amplitude the glider will be momentarily stationary (so it has no kinetic energy) meaning the total energy of the oscillator equates to 0.5kA²
• The kinetic energy of the glider is the difference between the total energy and the EPE:
• EK = 0.5kA²- 0.5kx²= 0.5k(A²-x²)