Work

When we push or pull a body at rest, it may or may not be set into motion. When the body is set into motion, we say that some work is done. We may apply a large amount of force on a wall and try to displace it. Since the wall does not move or get displaced, we say work is not done. From these examples, it is clear that whenever force is applied on a body and the body gets displaced, work is said to be done. Sometimes, instead of the total force applied on a body, only a part of it may be responsible to bring the body into motion from rest. In such a situation also, work is said to be done.

‘Work is said to be done when a net force acting on the body, displaces the body in the direction of the force.

Units of Work

The CGS unit of work is ‘erg’ ( which is derived from the Greek word ‘Energia’ meaning in work ) and the SI unit of work is joule ( J ) (in honor of the English scientist James Prescott Joule).

1 erg = 1 dyne × 1 cm

Hence, one erg is defined as the work done when a net force of one dyne displaces a body through one centimeter in its direction; Similarly, 1 joule = 1 newton × 1 meter.

Hence, one joule is defined as the work done when a net force of one newton displaces it through one meter in its direction.

Work done formula derivation

The work done on a body is proportional to the net force acting on the body and the displacement produced by the force on the body.

W∝F and W∝s

‘W; ‘F’ and ‘s’ are work done, applied force and displacement of the body in the direction of force, respectively.

From the above, we get

W∝Fs

where ‘k’ is a proportionality constant and the units of force and work are defined in such a way that k = 1.

Hence,                         W = Fs

Thus, when one unit force applied on a body produces a displacement of one unit in the direction of force, one unit of work is said to be done.

Real life example of work done

[caption id="attachment_21824" align="aligncenter" width="300"] FIGURE 1.1[/caption]

Consider a lawn roller being pulled as shown in Fig. 1.1. The direction of applied force is in the direction of displacement. Hence, the total force applied is utilized and is responsible for the displacement of the roller. Thus, we can write work done, W = Fs.

[caption id="attachment_21825" align="aligncenter" width="300"] FIGURE 1.2[/caption]

But when the direction of applied force makes an angle ‘θ’ with the direction of displacement as shown in Fig. 1.2, the total force applied is not responsible for the movement of the roller.

Only a part or a component of force which is equal to ‘Fcosθ’ is responsible for the displacement of the roller.

Hence, in this case, work done is given as,

W = (F cosθ) s or W = Fs cosθ

So, in general we can express the work done as the product of displacement of a body and the component of force responsible for the displacement of the body; The component of force responsible for the displacement will be in the direction of displacement and generally it is ‘F cos θ’ where θ is the angle between the directions of force and displacement. Hence, the general way of expressing work is W = Fs cosθ.

The three cases of work

Case ( i ) If θ < 90°

W = Fs cosθ ⇒ work done is positive as cosθ > 0 when 0°<θ< 90 °

Case ( ii ) If θ > 90° ⇒ work done is negative as cosθ < 0 when 90°< θ < 180°

Case (iii) If θ = 90 ° ⇒ work done is equal to zero as cos 90º = 0

Example: In uniform circular motion , work done by a centripetal force is equal to zero since centripetal force and displacement of the body are perpendicular to each other.

In the case of pulling or pushing a lawn roller, work is done on the lawn roller.

When we use a pressure cooker, the steam produced in the cooker due to pressure pushes up the weight kept on the lid where work is done by steam. Work is a scalar quantity.

Work Done by the Force of Gravity

All bodies are attracted towards the centre of Earth do to force of gravity. if a body moves short distance horizontally over the surface of the earth or in a circular path around the earth, like satellite, work done by the force of gravity zero because the displacement of the body at any instant is perpendicular to the direction of force of gravity. if our body of mass ‘m’ Falls vertically down through a height ‘h’,  then the work done will be equal to mgh, i.e., the change in the gravitational potential energy.

Dimensional Formula of Work

Work = (Force) × (displacement)

               W = Fs

    [ W ] = [ M1 L1 T-2 ] · [ L1 ] = M1 L2 T-2

Dimensional formula of work is [ M1 L2 T-2]