How to calculate the work done by a variable force?

To calculate the work done by a variable force, we need to integrate the force with respect to displacement.

When a force acts on an object, it can cause the object to move and do work. The amount of work done by a constant force is given by the formula W = Fd, where W is the work done, F is the force applied, and d is the displacement of the object. However, if the force is not constant, we need to use calculus to find the work done.

Let's consider a force F(x) that varies with displacement x. To find the work done by this force over a displacement from x1 to x2, we need to integrate the force with respect to displacement:

W = ∫x1x2 F(x) dx

This formula tells us that the work done is equal to the area under the force-displacement curve between x1 and x2. To evaluate the integral, we need to know the function F(x). If F(x) is given as a formula, we can simply substitute it into the integral and evaluate it using integration techniques.

For example, suppose a force of F(x) = 2x + 3 acts on an object over a displacement of 4 metres. The work done by the force is:

W = ∫0^4 (2x + 3) dx
W = [x^2 + 3x]0^4
W = (4^2 + 3(4)) - (0^2 + 3(0))
W = 28 J

Therefore, the work done by the force is 28 J.