# Work, Energy and Power II

DIFFERENT CASES OF WORK DONE BY CONSTANT FORCE
Case i) Force and displacement are in same direction
θ = 0
Since,            W = Fs Cos θ
Therefore      W = Fs Cos 0
or,                  W = Fs
Ex - Coolie pushing a load horizontally
Case ii) Force and displacement are mutually perpendicular to each other
θ = 90
Since,           W = Fs Cos θ
Therefore     W = Fs Cos 90
or,                 W = 0
Ex - coolie carrying a load on his head & moving horizontally with constant velocity. Then he applies force vertically to balance weight of body & its displacement is horizontal.
Case iii) Force & displacement are in opposite direction
θ = 180
Since,       W = Fs Cos θ
Therefore W = Fs Cos 180
or,            W = - Fs
Ex - Coolie carrying a load on his head & moving vertically down with constant velocity. Then he applies force in vertically upward direction to balance the weight of body & its displacement is in vertically downward direction.

ENERGY

Capacity of doing work by a body is known as energy.
Note - Energy possessed by the body by virtue of any cause is equal to the total work done by the body when the cause responsible for energy becomes completely extinct.

TYPES OF ENERGIES

There are many types of energies like mechanical energy, electrical, magnetic, nuclear, solar, chemical etc.

MECHANICAL ENERGY
Energy possessed by the body by virtue of which it performs some mechanical work is known as mechanical energy.
It is of basically two types-
(i) Kinetic energy
(ii) Potential energy

KINETIC ENERGY

Energy possessed by body due to virtue of its motion is known as the kinetic energy of the body. Kinetic energy possessed by moving body is equal to total work done by the body just before coming out to rest.
Consider a body of man (m) moving with velocity (vo).After travelling through distance (s) it comes to rest.
KINETIC ENERGY IN TERMS OF MOMENTUM
POTENTIAL ENERGY
Energy possessed by the body by virtue of its position or state is known as potential energy. Example:- gravitational potential energy, elastic potential energy, electrostatic potential energy etc.

GRAVITATIONAL POTENTIAL ENERGY
Energy possessed by a body by virtue of its height above surface of earth is known as gravitational potential energy. It is equal to the work done by the body situated at some height in returning back slowly to the surface of earth.
Consider a body of mass m situated at height h above the surface of earth. Force applied by the body in vertically downward direction is
F = mg
Displacement of the body in coming back slowly to the surface of earth is
s = h
Hence work done by the body is
W = FsCosθ
or, W = FsCos0
or, W = mgh
This work was stored in the body in the form of gravitational potential energy due to its position. Therefore
G.P.E = mgh

ELASTIC POTENTIAL ENERGY
Energy possessed by the spring by virtue of compression or expansion against elastic force in the spring is known as elastic potential energy.

Spring

It is a coiled structure made up of elastic material & is capable of applying restoring force & restoring torque when disturbed from its original state. When force (F) is applied at one end of the string, parallel to its length, keeping the other end fixed, then the spring expands (or contracts) & develops a restoring force (FR) which balances the applied force in equilibrium.
On increasing applied force spring further expands in order to increase restoring force for balancing the applied force. Thus restoring force developed within the spring is directed proportional to the extension produced in the spring
FR α x
or, FR = kx (k is known as spring constant or force constant)

If x = 1, FR = k
Hence force constant of string may be defined as the restoring force developed within spring when its length is changed by unity.

But in equilibrium, restoring force balances applied force.
F = FR = k x
If x = 1, F = 1

Hence force constant of string may also be defined as the force required to change its length by unity in equilibrium.

Mathematical Expression for Elastic Potential Energy

Consider a spring of natural length ‘L’ & spring constant ‘k’ its length is increased by XO.
Elastic potential energy of stretched spring will be equal to total work done by the spring in regaining its original length.
If in the process of regaining its natural length, at any instant extension in the spring was x then force applied by spring is
F = kx
If spring normalizes its length by elementary distance dx opposite to x under this force then work done by spring is
dW = F. (-dx) . Cosθ
(force applied by spring F and displacement –dx taken opposite to extension x are in same direction)
dW = -kxdx
Total work done by the spring in regaining its original length is obtained in integrating dW from XO to 0

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