Work done by a Variable Force

We already know how to find work done by a constant force.

But what if force becomes stronger or weaker as the object moves. Let us take an example.

Assume you are inspecting an airplane engine and someone turns it on!

Now, you will notice that you would fly away with more force when you are closer to the engine.

And force on you decreases as you move away from the engine.

Force is a vector quantity . So, even if it has same magnitude but direction changes,it would be a “Variable Force”.

Like, while skateboarding along an arc:

Here, angle between gravitational force and displacement changes but magnitude of force remains same.

Let us see how we deal with such cases!

We break our interval into very small portions where force can be considered constant.

Let us look at this situation graphically to understand this more clearly.

Consider the following plot, it shows a variable force acting on a body.

Now, we divide the total displacement into small parts.

Force acting on the small rectangle can be considered constant for a small variation in force.

Thus, the small amount of work done for a very small displacement will be:

So, if we add up the work done calculated in all such small parts , we will get the total work done.

Now, if the displacement approaches zero then total work done can be expressed as:

Thus in general total work is given by:


Force is considered variable when its magnitude or direction changes.

Work done by a variable force is given by:

The End