Work done in raising Simple Pendulum to Vertical angle

We need to do certain amount of work in lifting a simple pendulum to a certain height from the reference position. We are shifting the bob of the simple pendulum and hence the string of the pendulum shifts to a certain angle from the vertical position. This work done is further stored in the form of energy and this energy is called potential energy as it is with respect to mean position.

Simple pendulum consists of heavy and point sized mass suspended from a rigid support suspended via a in extensible string.

Let us consider a bob of mass m suspended with the help of a in extensible string of length L. Let a horizontal force is applied on the bob so that it shift to a new position. With reference to the bottom most position of the bob, it shifts to new height h and the work done is stored in the form of potential energy.

We would like to represent the work done in the form of the vertical angle. We can express the height as the difference between the length of the pendulum and the parallel portion of the height lifted with reference to mean position. It can be expressed in terms of COS angle as shown in the video below.

Here in the case of simple pendulum entire mass of the system is concentrated in the bob itself and hence work done is direct answer is what we have derived. Suppose the body that is lifted to a vertical angle is a uniform thin rod, its mass is not at the bottom rather it is distributed over the entire rod. We shall consider center of mass concept and we know that for a uniform body center of mass is at the geometrical centre. So the work done in this case is only half of the previous case.

All this is explained and derived in detailed in the below video lesson.