Work Done by a Variable Force with Integration Process

Work is physical quantity and is defined as the dot product of force and displacement. This concept is valid when the applied force is constant. If the applied force is variable, we shall calculate work done for each part of the force by multiplying it with displacement and adding that entire small works together. This can be done with a mathematical process called integration.

If the applied force is variable, we shall integrate the force and displacement equation to get the total work done in the process.

We can derive mathematical form of work done with respect to variable force, using the definition of kinetic energy. We know the mathematical form of kinetic energy. Let us differentiate the equation with respect to time. Being mass of a physical quantity is constant, we can write it outside the integration and we need to integrate the square of velocity with respect to time.

By simplifying the equation further, we can get that change in kinetic energy is the product of force and change in time. This is only for a small magnitude of kinetic energy in a small interval of time.

By competing the required steps, we can get the equation for change in kinetic energy. We also know that work done is nothing but change in kinetic energy. It means that the change in kinetic energy is stored in the form of work done.

To get the total work done, we shall integrate the equation.It is shown in detail in the following video lesson.