A pendulum initially is at rest in vertical position. The bob is pulled slowly towards right. Find the change in gravitational potential energy of the pendulum bob of mass m as the function of x.

Explanation

The change in potential energy of a particle depends upon the vertical displacement of the particle with respect to initial position. It increases for upward displacement and decrease for downward displacement. Here the change in potential energy of the bob at any angular position `theta` is

`DeltaU=mgl(1-costheta)` (i)
`cos theta=(sqrt(l^2-x^2))/(l)` (ii)
Using Eqs (i) and (ii), we get
`U=mgl(1-sqrt(1-x^2/l^2))`
For writing potential energy of point mass, we take earth surface as reference point where we take potential energy zero.
For a regular-shaped body where mass is distributed uniformly, we take as mass concentrated at its geometrical counter. This point is called center of gravity. We can replace the body a point mass placed at the centre of gravity.