In simple harmonic motion what fraction of total energy is kinetic energy and what fraction is potential energy when displacement is half of its amplitude?

Arun
25750 Points
5 years ago

Let the length of the pendulum be l, the mass of the pendulum bob be m and the acceleration due to gravity be g. We assume that the pendulum shaft is massless. Let the maximum angular displacement from the vertical be delta radians where delta is a small angle.

Set the reference point for zero potential energy is the point at which the pendulum shaft is vertical, At maximum displacement all of the pendulum energy is potential and is equal to mgl(1-cos(delta)). The small angle approximation for cos(delta) is 1 - (delta^2)/2 so the potential energy is mgl(delta^2/2.

At half displacement the potential energy is mgl(1 - cos(delta/)/2) = mgl(delta/2^2)/2 = mgl(delta^2)/8.

Therefore at half displacement the potential energy is a quarter of the maximum potential energy.

Since the total energy is the sum of the kinetic energy and potential energy, at half displacement the kinetic energy is 3/4 of the total energy.