Energy and SHM

When a spring is stretched or compressed it has elastic potential energy. When a mass is moving it has kinetic energy. In all shm motions energy "sloshes" between potential and kinetic energy. the potential energy may be elastic, gravitational or even electrical.

The graph below shows how these different kinds of energy vary with displacement for an undamped oscillator. Notice that the total energy at any time remains constant and is equal to the sum of the other two.

As v = A ω cos ω t   we get   v_max = A ω  and so K.E.max = 1/2 m A2ω2

E.P.E. can be found using 1/2 k x²    and so    E.P.E.max = 1/2 k A²  

Damping

The graph below shows how the displacement of a mass on a spring changes with time. Notice how the amplitude decays exponentially with time. (but the period stays the same)

The amplitude decreases every cycle due to damping. The system is initially given a certain amount of energy and every cycle a certain amount of energy is lost due to friction (e.g. air resistance).

Damping may be light, moderate or heavy depending on the amount of friction. Engineers often aim for "critical damping" where the energy of the system is dissipated as quickly as possible.