Work Done by a Variable Force

Work done by a force can be given by the equation:

W = Fs

Here,

W is the work done

F is the force applied

s is the displacement caused by the force

To perform any kind of work, displacement is very important. If the displacement is zero, the force does not do any work, regardless of the amount of energy it transfers to the object. The transferred energy is absorbed in a different form than work, like potential energy or heat. The above equation clearly shows that the work done is directly proportional to the force applied and the displacement caused.

This means that if the work output is desired to be increased, it can be done by increasing the force or using the same force to traverse a greater distance. In everyday life, we perform work against gravity. This includes lifting objects against the force of gravity or just walking. Since we are doing work against gravity, we are required to work more compared to if the gravitational force didn’t exist.

Theoretically, if a force is applied to an object in a system free of unbalanced forces and the object is allowed to travel in constant motion indefinitely, then the displacement approaches infinity. One could say that the work done is infinite in this case. But this is not true. A more accurate description of work links it with the kinetic energy of the object on which the work is done. This is known as the work-energy theorem and it is one of the most important discoveries in science.

The work-energy theorem states that the work done on an object by an instantaneous force is equal to the kinetic energy gained by the object. So, if we apply ten newtons of force on an object in rest and the object acquires kinetic energy of ten joules, then the work done on the object is ten joules.

Therefore, using this theorem, we can conclusively state that the work done on an object in indefinite constant motion is still finite. This is because the kinetic energy of the object is not changing and it is constant because of the constant motion. The work-energy theorem also helps us in defining positive and negative work. Positive work is done when the change in kinetic energy of an object is also positive.

This means that the applied force should increase the kinetic energy of the object. Similarly, negative work is done when the applied force causes a negative change in the kinetic energy. The total work done by a force is a scalar quantity and can turn out to be positive later on even if it was negative in the initial stages.

If an object is displaced from its position by a conservative force without changing its velocity-- it can be done by moving the object with infinitesimal velocity-- then the work done is zero. The change in energy is instead manifested as the change in potential energy.

Most of the forces we see in our daily lives are non-conservative, which means that the work done by them is dependent on the path taken. Non-conservative forces are generally variable. Variable forces are those that change with position or time. The work done by these forces is calculated by applying path integral on the path which the object took while its kinetic energy changed.

This is done by taking into account the work done at each coordinate of the path taken. All the infinitesimal values are added, with the limit of the size of the component values approaching zero. This expression can be written as follows:

Subsequently, we apply the limit of x approaching zero:

The limit transforms this expression into a path integral, given by the expression:

The above equation is the general equation of work done by a variable force. In all the above equations, the force applied is taken to be dependent on the x-coordinate.