Question: Comets move around the sun in a highly elliptical orbital. The gravitational force on the comet due ...

Question

Comets move around the sun in a highly elliptical orbital. The gravitational force on the comet due to the Sun is not normal to the comet’s velocity in general. Yet the work done by the gravitational force over every complete orbit of the comet is zero. Why?

Answer 1

Solution

Gravitational force is a conservative force. The total work done by a conservative force is zero. Therefore, the work done by the gravitational force over every complete orbit of the comet is zero.

Step by step answer:
Work done by gravitational force: ${W_g} = - mg(\Delta h)$
Where ${W_g}$ is the work done by gravitational force and is expressed in Joules $(J)$ , $m$ is the mass of the body and is expressed in kilograms $(kg)$ , $g$ is the acceleration due to gravity and is approximately $9.8m/{s^2}$ and $\Delta h$ is the change I height and is expressed in meter $(m)$ .
Conservative force is defined by a force whose work done is independent of the path taken. If this force acts on a particle which is travelling in a closed path, the total work done is zero; that is it is conserved. These forces conserve mechanical energy.
Gravitational force acting between two bodies is conserved. That is no matter what path it takes to act between two bodies in a system, when it comes to the end of it or reaches the same place it started from (closed loop), the work done will ultimately be zero. The work done by a conservative force is equal to the negative change in potential energy during the process.
In case of gravitational force, the work done depends on the change in height.
Mathematically it is represented as: $Wg = - mg(\Delta h)$ .
This shows that gravitational force does not depend on the path taken by the two bodies in a system, but rather on the initial and final positions.
This is the reason why the work done by the gravitational force over every complete orbit of the comet is zero.

Note: Like other conservative forces gravitational force is path independent. But gravitational force depends on change in height rather than being totally path independent.