Energy and Speed in SHM

We will now calculate the total energy stored in an object oscillating on a spring. Bu the result we obtain will to a certain extend be a general one that is valid for all oscillators, i.e. all objects in simple harmonic motion (SHM).

Remember that the potential energy stored in a steched or compressed is

U =

k x²

Thus the total energy is given by:

E = K + U =

m v² +

k x².

At the maximum displacement we have x = A and v = 0:

E =

m 0² +

k A² =

k A².

Thus the energy of an object undergoing SHM is proportional to the amplitude squared:

E =

k A²

This is the first important result on this page: the total energy stored in an oscillator is proportional to the square of the amplitude.

We can now express the velocity as a function of position:

E = K + U

=> k A² = m v² + k x²

=> v² = (k/m)(A²-x²)

=> v = ±[(k/m)(A²-x²)]^1/2

This is the general formula for the velocity, v, as a function of the displacement, x, from equilibrium.

At x = 0, v is at its maximum

v_max = A (k/m)^1/2