Question

A simple pendulum has a mass of 0.25 kg and a length of 1 m. It is displaced through an angle of 30 degrees and then released. After a time, the maximum angle of swing is only 10 degrees. How much energy has been lost to friction?

Answer 1

Consider that at the maximum angle, the pendulum has no velocity, and therefore, no kinetic energy.

All of its energy is gravitational potential energy = mgh

Relative to the low point, the height is initially 1m(1 - cos 30);

afterwards, the maximum height is only 1m(1 - cos 10).

The change in maximum h is
(h-final) - (h-initial) = 1m(1-cos10) - 1m(1-cos30) = 1m(cos30 - cos10).

The loss of energy, then, is:
mgh = (0.25 kg)(9.8 m/s^2)1m(cos30-cos10)

       = - 0.291 J