Work and Energy : Conservative Forces

If a body of mass m is displaced by an infinitesimal displacement  dx  as a result of the application of a force  F(x), then the infinitesimal work done  dW  is defined as follows

The total work done can be obtained by integrating the above equation

where the limits of the integration are initial and final positions of the body. If the value of this integration depends only on the initial and final positions of the body and not on the path traversed by the body in-between, then the force acting on the body is said to be conservative. The reason we call these forces conservative is that the total mechanical energy (K.E. + P.E.) is found to remain conserved throughout the process (where such forces are involved). This will be demonstrated here.

Some general conclusions can be made about work done by conservative forces. It is possible to express the limits of the above integration in terms of the initial and final velocities of the body. When we do so we get the following result : 

That is,

 If the force is conservative then the work done is equal to the 

difference in the final and initial kinetic energies of the body. 

This is a very general conclusion and applies to all processes where forces are conservative. 

There is one more way we can evaluate the above integral when the forces are conservative. There is a general theorem which states that if a force (field) is conservative it can be expressed as derivative of a scalar function in the following way 

Here, V is referred to as the potential energy. If we proceed with the work integral with this definition of force, we get the following :

 That is, 

If the force is conservative the the work done is equal to the 

difference in the initial and final potential energies of the body. 

Notice the order in which this difference is taken for kinetic energy and potential energy. Again, this conclusion is very general and applies to all cases where conservative forces are involved. From the above two results we have :

 This can be re-written as

 

This is the mathematical expression for conservation of mechanical energy. Hence the forces for which the work integral does not depend on the path are known as conservative forces. These relations between work and kinetic and potential energies come in handy while solving problems on conservative forces.

On the other hand, if the work integral depended on the path traversed by the system between the initial and final positions then the total mechanical energy of the system would not have remained conserved. Consequently, such forces are known as non-conservative forces. Usually in such process there is either dissipation or addition of energy to the total mechanical energy of the system. Frictional force is an example of such kind of forces.