# Energy

Energy can be defined as the capacity or ability to do work and is measured by the amount of work a body can do. So, energy is measured in the same units as work, namely, joule. Like work, energy is a scalar quantity.Energy can exist in various forms, such as heat energy, electrical energy, sound energy, light energy, chemical energy, nuclear energy, mechanical energy, etc. We will be mainly concerned with mechanical energy. Mechanical energy is of two types, kinetic and potential.

# Kinetic Energy: Energy due to Motion

A moving object can do work on another object when it strikes it. In other words, an object in motion has the ability to do work and, by definition, has energy. The energy possessed by a body by virtue of its motion is called kinetic energy.All initially motionless body can move and acquire a velocity only if a force acts on it. The work done by the force in causing the body to move measures the kinetic energy (written as KE) of the moving body, i.e.

KE = W

Consider a body of mass m at rest at time t = 0; its initial velocity u = 0.

Let a constant force F act on it for time t.

The body acquires a velocity Ï… and suffers a displacement S in the direction of the force. Newton's second law gives

F = ma

Where a is the constant acceleration. The distance S moved is given by

2aS = Ï…

^{2 }- u

^{2 }

since u = 0, we have

2as = Ï…

^{2 }

The work done is given by

Thus the kinetic energy of a moving body is given by one-half the product of its mass and the square of its velocity.

Although this result has been obtained for a constant force, it also holds if the force is variable. Let a variable force F act on a body and let S be the displacement produced in the direction of the force. Then the work done by the force (which measures the kinetic energy of the body) is given by

=

=

=

=

=

or KE =

If the velocity of the body increases from zero to a value Ï…, we have

which agrees with the expression for KE obtained above for a constant force.

**Note:** The expression has been derived for a constant force and for a force having a variable magnitude. It may be remarked that this expression holds even for a more general case in which the force varies both in magnitude and direction. This expression, therefore, holds regardless of the way the body reaches its final velocity.

# Work-Energy Principle

Suppose a body of mass m moves with an initial velocity u. A force F acts on it, as a result of which it acquires a final velocity Ï…. The work done by the force is given by= final KE - initial KE

= change in KE

Thus, the work done by a force in displacing a body measures the change in its kinetic energy. This is the work-energy principle.

Thus, when a force does work on a body, its kinetic energy increases; the increase in kinetic energy being equal to the amount of work done. The converse of this is also true. When the kinetic energy of body is decreased by a retarding force, the decrease is equal to the work done by the body against the retarding force. Thus kinetic energy and work are equivalent quantities and are, therefore, measured in the same units, namely, joule.

# Potential Energy

**Energy due to position or Configuration**

An object can have energy not only by virtue of its motion, but also because of its position or configuration. The energy possessed by a body owing to its position or configuration is called potential energy. For example, a wound watch spring has potential energy on account of its wound state or the coils. As the spring unwinds, it does work to move the hands of the watch. Thus, a wound spring has the potentiality to do work.

**Gravitational potential Energy**

An object held at a position above the surface of the earth has potential energy by virtue of its position. When it falls from that position, it can do work. The potential energy of an object held above the earth is called gravitational potential energy. To calculate the energy stored in a body which has been lifted above the earth's surface against the gravitational force, we have to calculate the amount of work done in carrying it there.

Consider a body of mass m. It is lifted vertically to a height h above the earth by applying a force F vertically upward. The force F must be just enough to overcome the gravitational attraction, i.e.

F = mg

Where g is the acceleration due to gravity at that place. For bodies not too far above the surface of the earth, the value of g is practically constant. Hence the work done by a constant force F in displacing a body by a height h can be calculated by the product F h = mgh. Thus gravitational potential energy of a body of mass m at a height h above the surface of the earth is mgh. The gravitational potential energy on the surface of the earth is taken to be zero.

**Potential Energy of a Spring**

Consider a perfectly elastic spring. One end of the spring is fixed to a rigid wall and other end is fixed to a block which is placed on a frictionless horizontal surface as shown in figure (a).

We assume that the mass of the spring is negligible compared to the mass of the block.

If we stretch the spring by a distance x, the spring will exert a force on us during stretching. This force is due to the reaction of the spring and is called the restoring force which is proportional to the displacement x acts in a direction opposite to the displacement, i.e.

F Î± - x

F = - kx

Where k is the force constant of the spring. The negative sign indicates that the force acts in a direction opposite to displacement. To stretch a spring by a displacement x, we must exert a force F on it, equal but opposite to the force F exerted by the spring on us. Therefore, the applied force is

F = -F = kx

Notice that F is a variable force as it depends on x. Therefore, the work done by the applied force in stretching the spring through a distance x is given by

=

=

We could also evaluate this work by finding the area under the force-displacement curve from x = 0 to x = x. This area is shown shaded in Fig.6.6 (b). Note that the graph of F against x is a straight line (since F = kx, where k is the force constant). The work done by the applied force in stretching the spring through a distance x is

W = area of the triangle of base x and altitude kx

=

=

W =

It is evident that the work done in compressing the spring by an amount x is also given by

The work done in stretching or compressing a spring is stored in it in the form of potential energy which is due to the changed configuration of the coils of the spring. Hence the potential energy of a mass less elastic spring of force constant k when it is stretched or compressed by an amount x is given by