In this article, we shall study the concept of work and power.

Work:

When a force is applied to a body and there is the displacement of the body in the direction of the force or along the direction of the component of force, then work is said to be done by the force.

Work is defined as the product of the force applied and the displacement of the body in the direction of the force. Both force and displacement are vector quantities but work is a scalar quantity.

W   =   F . S

Where W = Work done

F = Force applied

S = Displacement of the body in the direction of the force.

In vector form, the formula can be written as

W = F. S

Dimensions of work are [L²M¹T^-2]

Sign of Work:

• Work can be zero, positive and negative
• When force is applied to the body and there is no displacement of the body, then work done by the force is zero. e.g. Consider a body suspended in the air using thread. The gravitational force pulls the body down but there is no displacement of the body in the direction of gravitational force. In this case, the work done by the gravitational force is zero.
• When applied force and the displacement of the body are perpendicular to each other then the work done by the force is zero. e.g. The moon revolves around the earth in a stable orbit. The earth’s gravitational force acts on it and pulls the moon towards its centre, but the moon moves in the direction perpendicular to the direction of gravitational force. Thus there is no displacement of the moon in the direction of gravitational force. Thus work done by the gravitational force is zero.
• When the displacement of the body is in the direction of force causing the displacement, the work done by the force is positive. e.g. Consider a freely falling body. A gravitational force acts on it and pulls downward. Thus the displacement of the body is in the direction of gravitational force. Hence the work done by the gravitational force is positive.
• When the displacement of the body is in the opposite direction to that of force causing the displacement, the work done by the force is negative. e.g. Consider a body which is being lifted up. The gravitational force pulls the body down, but the body moves up i.e. in the opposite direction to that of gravitational force. Thus work done by the gravitational force is negative.

Unit of Work:

When the applied force and the displacement of the body are in the same direction, work done is given by

Work  =  Force  ×   Displacement

Unit work  =  Unit force  ×   Unit displacement

Definition of Unit Work: Unit work is said to be done when the unit force produces a unit displacement in its own direction.

S.I. unit of work is joule (J): 1 J  =  1 N  ×  1 m

When a force of 1 newton acting on a body produces a displacement of 1 metre in the direction of force, then work done by the force is called 1 joule.

C.G.S. unit of work is erg: 1 erg  =  1 dyne  x   1 cm

When a force of 1 dyne acting on a body produces a displacement of 1 centimetre in the direction of force, then work done by the force is called 1 erg.

Derivation of  Expression for the Work Done by the Force:

Suppose the force produces a displacement in the direction making an angle θ with the direction of the force. The component of the force along the direction of displacement is F.Cos θ.

Now, Work done = Component of force in the direction of displacement × displacement

∴  W = (F.Cos θ)(s)

∴  W = F. s. Cos θ

W = F. S

Thus the work done is a scalar product of force and displacement. Thus work done is a scalar quantity.

When the displacement is  in the direction of the force

In such a case, θ = 0°

W = F. S . Cos 0°

W = F. S (1)

W = F. S

Thus when the displacement is in the direction of force the work done, is

equal to the product of magnitudes of force and the displacement.

When the displacement is perpendicular to force

In such a case, θ = 90°

W = F. s . Cos 90°

W = F. s (0)

W = 0

Thus when the displacement is perpendicular to the direction of force the work done is zero.

Work done by the Gravitational Force of the Earth on the Moon:

Work is defined as the product of the force applied and the displacement of the body in the direction of the force. The moon revolves around the earth in a stable circular orbit. The earth’s gravitational force acts on it and pulls the moon towards its centre, but the moon moves in the direction perpendicular to the direction of gravitational force. Thus there is no displacement of the moon in the direction of gravitational force. Thus work done by the gravitational force is zero.

In this case, θ = 90°

W = F. s . Cos 90°

W = F. s (0)

W = 0

Power:

The rate at which work is done is called power. As work and time are scalar quantity power is also a scalar quantity.

P = W / t

Unit of Power:

By the definition of power, unit of power = unit of work / unit of time = 1 J /  1s = 1 W

In S.I. system of units the unit of power is watt. Its symbol is ‘W’. Thus the power is said to be 1 watt if the rate of doing work is 1 joule per second.

In C.G.S. system of units the unit of power erg/s. Thus the power is said to be 1 erg/s if the rate of doing work is 1 erg per second.

But in practice, the unit power may be used with some prefixes.

1 kW  =  1000 W

1 MW = 1000000 W

1 horsepower = 746 W

Relation Between the Power and Velocity of a Body:

Suppose a force F acts a body which causes a displacement of s in the direction of the force in the body in ‘t’ seconds.

Then work don is given by W  =  F  .  S

By definition of power  P = W / t

∴   P = F  .  S/ t

∴   P = F  .  (S/ t)

∴  P = F  . v

Where v is the magnitude of the instantaneous velocity.

Thus the power is the product of magnitudes of the force acting on the body and velocity of the body.

Horsepower:

A horsepower is a unit of power used in the engineering. Its symbol is hp. Its relation with watt is as follows

1 horsepower (hp) = 746 watts

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