Potential Energy

Energy of a body is its capacity to do work. There are two basic forms of energy.

(i)                Kinetic energy               (ii)       potential energy

The kinetic energy is possessed by a body due to its motion and is given b the formula

Where m is  the body moving with velocity V.

The potential energy is possessed by a body because of its position in a force field e.g. gravitational field or because of its constrained state the potential energy due to gravitational field near the surface of the earth at a height H is given by the formula

P.E. = mgh                ……………..                 (4.8)

This is called gravitational potential energy. The gravitational P.E. is always determined relative to some arbitrary position which is assigned the value of zero P.E. in the present case, this reference level is the surface of the earth as position of zero P.E. in some cases a point at infinity from the earth can also be chosen as zero reference level.

The energy stored in a compressed spring is the potential energy possessed by the spring due to its compressed or stretched state. This form of energy is called the elastic potential energy.

Work-energy principle

Whenever wor is doen on a body, it increases its energy. For example a body of mass m is moving with velocity Vi. A force F acting through a distance d increases the velocity to Vi then from equation of motion

Where the left hand side of the above equation gives the work done on the body and rith hand side gives the increase or change in kinetic energy of the body. Thus

Work done on the body equals the change in its kinetic energy.

This is known as work –energy principle. If a body is raised up from the earth’s surface, the work done changes the gravitational potential energy similarly, if a spring is compressed, the work done on it equals the increase In its elastic potential energy.

Absolute potential energy:

The absolute gravitational potential energy of an object at a certain position is the work done by the gravitational force in displacing the object from that position to infinity where the force of gravity becomes zero.

The relation for the calculation of the work done by the gravitational force of potential energy = mgh, is true only near the surface of the earth where the gravitational force is nearly constant. But if the body is displaced through a large distance in space from, let, point 1 to N ( Fig. 4.10) in the gravitational field, then the gravitational force will not remain constant, since it varies inversely to the square of the distance.

In order to overcome this difficulty, we divide the distance between points 1 and N into small steps each of length ∆r so that the value of the force remains constant for each small step. Hence, the total work done can be calculated by adding the work done during all these steps. If r1 and r2 are the distances of points 1 and 2 respectively, from the center O of the earth (fig. 4.10), the work done during the first step i.e., displacing a body from point 1 to point 2 can be calculated as below.

The distance between the centre of this step and the center of the earth will be

The gravitational force F at the centre of this step is

Where  m = mass of an object  ,  M = mass of the earth and          G = gravitational constant.

Squaring Eq. 4.12

Do You know?

As this force is assumed to be constant during the interval ∆r, so the work done is

Absolute potential energy:

The absolute gravitational potential energy of an object at a certain position is the work done by the gravitational force in displacing the object from that position to infinity where the force of gravity becomes zero.

The relation for the calculation of the work done by the gravitational force of potential energy = mgh, is true only near the surface of the earth where the gravitational force is nearly constant. But if the body is displaced through a large distance in space from, let, point 1 to N ( Fig. 4.10) in the gravitational field, then the gravitational force will not remain constant, since it varies inversely to the square of the distance.

In order to overcome this difficulty, we divide the distance between points 1 and N into small steps each of length ∆r so that the value of the force remains constant for each small step. Hence, the total work done can be calculated by adding the work done during all these steps. If r1 and r2 are the distances of points 1 and 2 respectively, from the center O of the earth (fig. 4.10), the work done during the first step i.e., displacing a body from point 1 to point 2 can be calculated as below.

The distance between the centre of this step and the center of the earth will be

The gravitational force F at the centre of this step is

Where  m = mass of an object  ,  M = mass of the earth and          G = gravitational constant.

Squaring Eq. 4.12

Do You know?

As this force is assumed to be constant during the interval ∆r, so the work done is

Escape velocity:

It is our daily life experience that an object projected upward comes back to the ground after rising to a certain height. This is due to the force of gravity acting downward. With increased initial velocity, the object rises to the greater height before coming back. If we go on increasing the initial velocity of the object, a stage comes when it will not return to the ground.

It will escape out of the influence of gravity. The initial velocity of an object with which it goes out of the earth’s gravitational field, is known as escape velocity.

The escape velocity corresponds to the initial kinetic energy gained by the body, which carries it to an infinite distance from the surface of earth.

We know that the work done in lifting a body from Earth’s surface to an infinite distance is equal to the increase in its potential energy.

Where M and R are the mass and radius of the Earth respectively. The body will escape out of the gravitational field if the initial K.E. of the body is equal to the increase in P.E. of the body in lifting it up to infinity. Then

Interconversion of Potential Energy and Kinetic Energy

Consider a body of mass m at rest, at a height h above the surface of the earth as shown in Fig. 4.11.

At position A, the body has P.E. – mgh and K.E. = 0. We release the body and as it falls, we can examine how kinetic and potential energies associated with it interchange.

Fig. 4.11

Let us calculate P.E. and K.E. at position B when the body has fallen through a distance x, ignoring air friction.

 

Velocity at B, can be calculated from the relation,

Total energy at B = P.E. + K.E.

                        = mg (h – x) + mgx = mgh     …………..                  (4.18)

At position C, just before the body strikes the Earth, P.E. = 0

Thus at point C, kinetic energy is equal to the original value of the potential energy of the body. Actually when a body falls, its velocity increases i.e., the body is being accelerated under the action of gravity. The increase in velocity results in the increase in its kinetic energy. On the other hand, as the body falls, its height decreases and hence, its potential energy also decreases. Thus we see (Fig. 4.12) that,

Loss in P.E. = Gain in K.E.

Where V1 and V2 are velocities of the body at heights h1 and h2 respectively. This result is true only when frictional force is not considered.

If we assume that a frictional force f is present during the downward motion, then a part of P.E. is used in doing work against friction equal to f h. the remaining P.E. = mgh – fh is converted into K.E.

Thus,

Loss in P.E. = Gain in K.E. + work done against friction.

Conservation of Energy

The kinetic and potential energies are both different forms of the same basic quantity, i.e. mechanical energy. Total mechanical energy of a body is the sum of the kinetic energy and potential energy may change into kinetic energy and vice versa, but the total energy remains constant. Mathematically,

Total energy = P.E. + K.E. = constant

This is a special case of the law of conservation of energy which states that:

Energy cannot be destroyed. It can be transformed from one kind into another, but the total amount of energy remains constant.

This is one of the basic laws of physics. We daily observe many energy transformations from one form to another. Some forms, such as electrical and chemical energy, are more easily transferred than others, such as heat. Ultimately all energy transfers result in heating of the environment and energy is wasted. For example, the P.E. of the falling object changes to K.E., but on striking the ground, the K.E. changes into heat and sound. If it seems in an energy transfer that some energy has disappeared, the lost energy is often converted into heat. This appears to be the fate of all available energies and is one reason why new sources of useful energy have to be developed.

Example 4.3: A brick for mass 2.0 kg is dropped from a rest position 5.0 m above the ground. What is its velocity at a height of 3.0 m above the ground?