Conservative Force Formula

There are various types of force acting in our surroundings. In physics, force plays a very important role. Any interaction will change the motion of an object when unopposed is understood as force. Forces can be classified into two types of forces: Conservative forces and Non-conservative forces. Most crucial aspect of physics is the Conservative force. The electrostatic force, magnetic force, gravitational force and elastic force are the different types of forces in nature. 

Work done by every force is quite different. These forces are therefore divided into conservative and non-conservative forces. Conservative forces are the forces that depend only on the initial and the final points and not on the path taken by the force. On the other hand, non-conservative forces are the forces that depend on the path taken by the body object under a specific force. So, we can see how conservative forces relate to energy and what would happen if conservative force is applied to the law of conservation of energy.

Before we move forward and understand everything else, we need to learn about the meaning of conservative force. It is a force that possesses the property that whenever there is work done on moving an object from one point to another, it is independent of the path taken by the object and is just dependent on the final and initial points. Moreover, if the work done by the object is in a closed loop, then we say that the conservative force does no work. Now, you might have understood the meaning of conservative force.

Examples of conservative forces are gravitational force and force stored in a spring.

Potential Energy and Conservative Forces

Potential energy can be defined as the energy present in a body due to its shape, configuration, or position. Therefore, it can be considered stored energy that can be recovered entirely. Moreover, we can define potential energy for the work done by any conservative force to reach the end position or configuration. For example, when you wind a spring, some energy gets stored in it, and this energy is entirely recoverable. Therefore, we can say that the energy stored in the spring is conservative energy. Let us now see how conservative force is applied to the law of conservation of energy.

Potential Energy of a Spring

As per Hooke’s law, let us calculate the work done when a spring is stretched and compressed. According to the law, the magnitude of the force that is present in the spring is directly proportional to the deformation that would be encountered, which would be ΔL.  

So, we can say that

F is proportional to ΔL, or by employing a constant, we can have,

F=kΔL

where k is a constant

For the spring, we will be replacing the amount of deformation ΔL with ‘x’, which is the distance by which the spring is stretched or compressed along its length.

So, we can say that the force which is observed after stretching or compressing the spring would be,

F = kx, where again the spring constant is k

Therefore, the force will increase linearly from 0 to the point kx when the spring is fully stretched. So, the average force would be kx/2.

Now, we can calculate the work done when the spring is stretched or compressed,

Ws= Fd

where F is force, and d is distance

Ws= (kx/2)x

which would lead to (kx2)/2

So,

Ws=(kx2)/2

According to the F vs x graph, and as per the kinetic energy and the work-energy theorem, the work done by the force would be (kx2)/2

We can also say that the potential energy of spring would be,

PE= (kx2)/2

Where k is the spring force constant, and x is the displacement that the spring undergoes. In this, the potential energy displays the work done on the spring when the spring is compressed or stretched by a distance. We can see from this that the potential energy depends on the final and initial points and not on the path taken. Let us now find out how conservative force is applied to the law of conservation of energy.

Conservation of Mechanical Energy

To start with the conservative force derivation, we need to find certain aspects such as changes in kinetic and potential energy.

To reach the principle of conservation of energy, we need to take into account the work-energy theorem when only conservative forces are involved in the system. According to the theorem of work energy, it is found that the net work done on a body by a system of forces would be equal to the change observed in the kinetic energy of the body. We can represent it as,

Wnet=mv2 /2-mv02/2  change in energy or ΔKE

But, if we say that only conservational forces are acting, then

Wnet= WC

here, Wc would be the net sum of all the conservative forces acting on the body

Therefore, we can also say that Wc is the change in kinetic energy

However, if there is a spring force or force of gravitation, then the potential energy would be lost by the system, and Wc would be equal to -PE

Or we can say that,

-PE would be equal to change in energy, which is

−ΔPE=ΔKE

We can write it as follows:

ΔKE+ΔPE=0

Therefore, the total kinetic and potential energy is constant when only conservative forces are involved

We can also say that,

KEi+PEi=KEf+PEf

where i represents the initial point, and f is the final point.

This is how a system experiencing only conservative forces will have total energy as constant. This was conservative force derivation.

Conclusion

The action of conservative forces on a system can be a bit different. But we can prove the law of energy conservation when conservative force is applied to the law of conservation of energy. We have seen that the system’s total energy remains constant, and the change in kinetic energy would be negative to that of the potential energy. This means that the total work done by the system would be zero. This helps us establish the law of energy conservation in terms of work done and energy.