Conservative Force

Introduction

What will be the potential energy stored in an object of mass ‘m’ lying on the surface of earth and then raised to a height ‘h’ in a direction perpendicular to the surface (as shown in figure 1) of earth? Your answer will be - Potential energy = (m.g.h).

The reason behind the expression of the potential energy stored in the object - that we have assumed that the potential energy at the surface of earth is zero (assumed as datum) and the work done to bring that object from the surface to a height ‘h’ against the gravitational force which is pulling the object towards the center of the earth with the acceleration equals to the acceleration due to gravity ‘g’.

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As we know that,

Work done = $\mathrm{\overrightarrow{F}.\overrightarrow{S}}$

Where $\mathrm{\overrightarrow{F}}$ = Force vector and $\mathrm{\overrightarrow{S}}$ = Displacement vector

So, in this case

Work done = (mg). h. Cos180°

= -(mg).h

Here, the angle between the force and displacement vector will be 180 because we are moving in the opposite direction to the applied force (Gravitational force = mg). The magnitude of this work done will be stored as potential energy in the object.

Now let us take the second case in which the object is raised through a distance ‘h’ again but by the following a zig-zag path as shown in Figure - 2 given below. Now think about the potential energy stored in the object at the height ‘h’?? It will be the same = m.g.h. You may think why? Because the object has traveled more distance, but you forget that the potential energy stored in the body is due to the work done to bring the object to that height and the work done depends on the displacement not on the distance covered. So the same amount of potential energy will be stored.

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It means that in this case the work done does not depend on the path opted by the object. It just depends on the initial and final point of the object. It is due to the conservative nature of Gravitational force. So, a force is said to be conservative when the work done by or on the force for moving an object depends only on the initial and final position of the object.

Examples of Conservative Force

• Gravitational force: All types of gravitational force, not only the gravitational force by the earth.

• Elastic force: In a stretched or compressed elastic materials

• Electrostatic force Forces between two electrically charged body

Properties of Conservative Force

• The work done by the conservative forces does not depend on the path taken, it only depends on the initial and final position.

  Let us take an example to understand it better -

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Suppose we have to move an object from A to B with the help of a conservative force via paths 1, 2 and 3. What do you think? Yes, the work done to bring the object from A to B will be the same through each path if we are using conservative force to do this work. So the work done by the conservative force will be independent of the path followed.

• The net work done by the conservative force will be zero in any closed path. It means that if the net displacement of the object is zero then the net work done will also be zero if the force is of conservative nature.

• The work done by the conservative force is reversible.

Conservative Force and Potential Energy

Potential energy is the energy by the virtue of the position or configuration. It means that if we can regain the original position or configuration then we can recover the potential energy. Also we know that the work done by the conservative force does not depend on the path followed and only depends on the final and initial position. So we can say that the potential energy can only be defined for the conservative forces. So if we are changing the configuration by applying the conservative force then the potential energy stored in the object will only depend on the final and initial configuration of the object rather than the path followed to achieve that configuration. So the potential energy is defined for Gravitational force, electrostatic force, elastic force and all the conservative forces.

Let us study about these potential energies in brief -

• Gravitational potential energy (GPE): In the beginning of this tutorial, we have discussed this in brief. So this potential energy depends on the initial and final position. The GPE is not an absolute quantity, it is always a relative quantity. The GPE is assumed to be zero at infinity because the gravitational force will be zero at infinity (datum) and we are calculating the potential energy at any point or position with respect to the datum.

• Elastic potential energy: The elastic force is also conservative in nature. For example, if we are compressing a spring with a force, then that spring is storing the potential energy in it due to change in its configuration. No matter how slow or fast we are compressing the spring to that particular position, it will store the same amount of potential energy with respect to its position.

• Electrostatic potential energy: This potential energy will arise due to the change in the position of a charged particle in an electric field. So the amount of work done to change the position of that charged particle will be stored in the form of electrostatic potential energy.

So through this tutorial we have completed all the aspects of the Conservative forces.

FAQs

Q1. What is conservative force?

Answer- Conservative force is the force which is having the property that the work done by that force is depending on the initial and final position only not on the path followed.

Q2. Give some examples of conservative forces.

Answer - Gravitational force, electrostatic force and elastic force are some examples of conservative force.

Q3. If the object is moving from A to B by a conservative force then the work done will be -

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a) Maximum via path 3

b) Maximum via path 2

c) Maximum via path 1

d) Will be equal via all the path

Answer - The work done by the conservative force is independent of the path followed. So the work done will be the same via all the paths. So the correct option will be (d).

Q4. Why is gravitational potential energy zero at infinity?

Answer - The gravitational force on any object at infinity is zero because it will have no influence of gravitational field intensity around them from any other object. So the gravitational potential energy is zero at infinity.

Q5. Potential energy is defined for which type of forces?

Answer - Potential energy is defined only for conservative force. This is because the potential energy is due to the virtue of position.