# Conservative Force and Non-conservative Forces

## A comparative study between Non-Conservative and Conservative force

Here we will discuss two types of forces namely conservative forces and non-conservative forces. So how do they differ? Consider the following two situations as shown in the figure

In figure 1, when the mass comes to rest, the spring gets compressed by a distance ‘x’. The spring then comes to its normal length and mass attains a velocity ‘v’. This compression and elongation continue and each time with a maximum value of ‘x’. So we can infer that the total mechanical energy of the system (Block + Spring) remains constant. While in the second case also there will be compression and elongation but the minimum value will keep on decreasing. So we can say that the total mechanical energy of the system is not conserved in this case.

From our observations in the above case, we can now define conservative and non-conservative forces. A conservative force is a force that does zero work done in a closed path. If only these forces act then the mechanical energy of the system remains conserved. Examples of conservative force: Gravitational force, spring force etc.

On the other hand, non-conservative forces are those forces which cause loss of mechanical energy from the system. In the above case friction is the non-conservative force. But as we know energy can neither be created nor destroyed hence these forces convert mechanical energy into heat, sound, light etc.

Now, Conservative force has one more property that works done by it is independent of the path taken.

## Work was done by a conservative force

Consider the following scenario

Work done by the conservative force in a closed path is zero.

In figure one we know work done by conservative force in a closed path is zero.
W1, A, B + W2, B, A = 0

W1, A, B = – W2, B, A

We also have,

W2, B, A = – W2, A, B

Using the above two equations we get,

W1, A, B = W2, A, B

The above equation shows that work done to move a particle from point A to B through path 1 and 2 as shown in figure 2 will take the same amount of work done. But this statement is not valid for non-conservative forces.

### Conservative force formula

Conservative force is defined as the force such that the work done is independent of the path taken and is dependent only on the initial and final position. Conservative force is applied to the law of conservation of energy. The basic understanding of conservation of energy holds good for conservation of kinetic energy and is given as:

 $\Delta KE=\int F.dr$