Potential Energy

Work Power And Energy of Class 11

When we throw a ball upwards with an initial velocity, it rises to a certain height and becomes stationary for a moment. What happens to the lost kinetic energy? We know with our experience that the ball returns back in our hands with a speed equal to its initial value. The initial kinetic energy is somehow stored and is later fully recovered in the form of kinetic energy. The ball must have something at the new height that it does not have at the previous level. That something by virtue of its position is Potential Energy.

Potential energy is the energy associated with the relative positions of two or more interacting particles.

Potential energy fits well the idea of energy as the capacity to do work. For example, the gravitational potential energy of an object raised off the ground can be used to compress or expand a spring or to lift another weight. As a coil spring unwinds, or a straight spring returns to its natural length, the stored potential energy can be used to do work. For example, if a block is attached to a compressed spring, the elastic potential energy can be converted into kinetic energy of the block as shown in figure(8.15).

Potential Energy

In the above discussion we have seen that in the case of gravity and elastic spring the kinetic energy imparted initially is stored as potential energy for a short time which is regained, later on. But this is not true in all cases.

For example, consider block placed at rest on a rough horizontal surface. If we impart it some initial kinetic energy, it starts sliding on the surface, the frictional force does negative work on the block, decreasing its kinetic energy to zero. But it does not come back to our hand no matter how long we wait! The frictional force has used up the kinetic energy in a non - reversible way.

The forces, such as gravity and spring force, which does work in a reversible manner is called a conservative force. In contrast, the force, such a frictional force, which does work in an irreversible manner is called an non - conservative force.

Important

1.The work done by a conservative force is independent of path. It depends only on the initial and final positions. In contrast, the work done by a non-conservative force depends on the path (see figure 8.17)

2.The work done by a conservative force around any closed path is zero
(fig. 8.16).

Potential Energy

The potential energy is defined only for conservative forces.

The change in potential energy as a particle moves from point A to point B is equal to the negative of the work done by the associated conservative force

ΔU = UB- UA= -WC

Using definition of work

UB– UA= Potential Energy (8.12)

From equation (8.12), we see that starting with potential energy UAat point A, we obtain a unique value UBat point B, because WChas the same value for all paths. When a block slides along a rough floor, the work done by the force of friction on the block depends on the length of the path taken from point A to point B. There is no unique value for the work done, so one can not assign unique values for potential energy at each point. Hence, non-conservative force can not have potential energy.

When the forces within a system are conservative, external work done on the system is stored as potential energy and is fully recoverable.

Note that the potential energy is always defined with respect to a reference point.

Gravitational Potential Energy (Near the Earth’s Surface)

The work done by gravity on a particle of mass m whose vertical coordinate changes from yAto yBis

Wg= -mg(yB– yA)

From equation (8.12), we have Wg= -ΔUg = -(UB– UA).

Thus gravitational potential energy at the point B near the surface of the earth is given by

UB= UA+ mgh

If we assume potential energy at the point A to be zero, then potential energy at the point B is given by

UB= 0 + mgh = mgh

Spring Potential Energy

The work done by the spring force when the displacement of the free end changes from xito xfis given by equation (8.8).

WS = - Potential Energy

By definition WS= -ΔUS= -(Uf– Ui), therefore,

Uf= Ui+ Potential Energy

US= Potential Energy (8.13)

If we assume the potential energy stored in the spring at equilibrium is zero and all the displacements are measured from equilibrium, then xi= 0 and Ui= 0.

Thus, final energy stored in the spring is

Uf= Potential Energy (xf = x)

The potential energy function for an ideal spring is a parabolic function as shown in figure (8.18).

Potential Energy

Example 8.8

A uniform rod of mass M and length L is held vertically upright on a horizontal surface as shown in the figure. Find the potential energy of the rod if the zero potential energy level is assumed at the horizontal surface.

Potential Energy

Solution

Since the parts of the rod are at different levels with respect to the horizontal surface, therefore, we have to use the integration to find its potential energy. Consider a small element of length dy at a height y from the horizontal.

Mass of the element is

dm = M/L dy

Its potential energy is given by

dU = (dm)gy

ordU = M/L gydy

On integrating, we get

Potential Energy