Question

A bob of a mass $m$ of a simple pendulum can be raised to a maximum height $h$ . Find the work done by gravity to move the bob from one extreme to another.
A) $2mgh$
B) $mgh$
C) $3mgh$
D) zero

Answer 1

Work done by gravity will be the dot product of the force due to gravity and the displacement that occurs due to this force. As the bob moves from one extreme to another no vertical displacement occurs.

Formula used:
Work done by the force is given by, $W = F \cdot d$ where $F$ is the force acting on the object and $d$ is the resulting displacement.

Complete step by step answer.
Step 1: List the parameters given in the questions.
The mass of the bob of the simple pendulum is $m$ .
Thus the force due to gravity which is also known as the weight of the bob will be $F = mg$ and it is directed downwards.
The maximum height up to which the bob can be raised is $h$ .

Step 2: Calculate the work done by the force of gravity.
Work done by the force is given by, $W = F \cdot d$ where $F$ is the force acting on the object and $d$ is the resulting displacement.
The bob swings from one extreme position to the other extreme position. Here, the displacement due to the weight of the bob is zero i.e., there is no vertical displacement
Thus, the work done will be zero.

Hence, the correct option is D.

Note: Alternate method
At both the extreme positions, the kinetic energy of the bob is zero i.e., $K.E = 0$.

It is the potential energy of the bob that exists at the extreme positions but it is equal at both the extremes and so the change in potential energy will be zero i.e., $\Delta P.E = 0$

Now, the work done can also be defined as the change in mechanical energy i.e., $W = \Delta E$

The mechanical energy is given by, $E = K.E + P.E$
So, the work done is essentially the change in potential energy i.e., $W = \Delta P.E$ and it is zero as $\Delta P.E = 0$ .