In everyday language we often use term Work. A teacher teaching a class, a student preparing for exams, lifting an object against the Earth's gravitation, driving a car up a hill all are said to be working.
Work is said to be done when a force is applied on the body and body get displaced through some distance in the direction of the applied force.
However, when there is no displacement in the direction of the applied force, no work is said to be done, that is work done is zero, when displacement of the body in the direction of the force is zero.
Infact, work done by the force is the product of component of force in the direction of the displacement and the magnitude of the displacement.
Suppose a constant force F acting on a body produces a displacement s in the body along the positive x-direction.
If θ is the angle which F makes with the positive x-direction of the displacement, then the component of F in the direction of displacement is (Fcosθ).
W = (Fcosθ) s …… (1)
If displacement is in the direction of force applied,
θ = 0°.
From (1),
W = (F cos0°) s = F s
Equation (1) can be written as
W = F • s ……… (2)
Thus, work done by a force is the dot product of force and displacement.
Work is a scalar quantity, i.e. it has magnitude only and no direction.
However, work done by a force can be positive or negative or zero.

## Dimension and units of work

As we know,
work = force × distance
Therefore,
W = [M1 L1 T-2] × L
W = [M1 L2 T-2]

## The units of work are of two types:

Absolute units and Gravitational units.

• ### Absolute units

• Joule is The absolute unit of work on SI.

• Work done is said to be one joule, when a force of one Newton actually moves a body through a distance of one meter in the direction of applied force.
From W = F s cosθ
1 joule = 1 Newton × 1 meter × cos0° = 1 N-m

• Erg is the absolute unit of work on cgs system.

• Work done is said to be one erg, when a force of one dyne actually moves a body through a distance of one cm in the direction of applied force.
From W = F s cosθ
1 erg = 1 dyne × 1 cm × cos0°

### Relation between joule and erg

As, 1 joule = 1 N × 1 m × cos0°
Therefore,
1 J = 105 dyne × 102 cm × 1
1 joule = 107 erg

• ### Gravitational units

• These are also called the practical units of work.

• Kilogram –meter (kg-m) is the gravitational unit of work on SI.

• Work done is said to be one kg-m, when a force of 1 kg f moves a body through a distance of 1 m in the applied force.
From W = F s cos θ
1 kg-m = 1 kg f × 1 m × cos0° = 9.8 N × 1 m = 9.8 joule
1 kg –m = 9.8 J

• Gram-centimeter (g-cm) is the gravitational unit of work on cgs system.

• Work done is said to be one g-cm, when a force of 1 g f moves a body through a distance of 1 cm. in the direction of the applied force.
From W = F s cosθ
1 g-cm = 1 g f × 1 cm × cos0°
1 g-cm = 980 dyne × 1 cm × 1
1 g-cm = 980 ergs

### Relation between kg-m and g-cm

1 kg-m= 103 g × 102 cm
1 kg-m = 105 g-cm

## Nature of work done

Although, work done is a scalar quantity its value may be positive, negative or even zero.

### Positive work

As, W = F • S = F s cosθ
Therefore when θ is acute (< 90°), cosθ is positive.
Hence, work done is positive.
Some examples

• When a body falls freely under the action of gravity (θ = 0°, cos θ = cos 0° = + 1). Therefore, work done by gravity on body falling freely is positive.
• When lawn roller is pulled up applying a force along the handle at an acute angle, work done by the applied force is positive.
• When a gas filled in a cylinder fitted with a movable piston is allowed to expand, work done by the gas is positive. This is because force due to gaseous pressure and displacement of position are in the same direction.
• When a spring is stretched, work done by the stretching force is positive.

• ### Negative work

As W = F • s = F s cos θ
Therefore, when θ is obtuse (>90°), cos θ is negative. Hence, work done is negative.
Some examples

• When a body is thrown up, its motion is opposed by gravity. The angle θ between gravitational force F and the displacement s is 180°. As cos θ = cos 180° = - 1, therefore, work done by gravity on the body moving upwards is negative.
• When a body is moved over a rough horizontal surface, the motion is opposed by the force of friction. Hence, work done by frictional force is negative.
• When brakes are applied on a moving vehicle, work done by the braking force is negative.

• ### Zero work

When force applied F or the displacement s or both are zero, work done W = F s cos θ is zero. Again, when angle θ between F and s is 90°, cos θ = cos 90° = 0. Therefore, work done is zero.
Some examples

• When you push hard against a wall, the force you exert on the wall does no work, because s = 0. However, in this process our muscles are contracting and relaxing alternatively and internal energy is being used up. That is why you do get tired.
• A weightlifter holding a 100 kg mass steadily on his shoulder for 30 seconds does no work on the load this time, because s = 0.
• Tension in the string of a simple pendulum is always perpendicular to displacement of the bob. Therefore, work done by tension is always zero.

• ## Conservation and Non–conservation forces

### Conservation forces

A force is said to be conservative if work done by or against the force in moving a body depends only on the initial and final positions of the body, and not on the nature of path followed between the initial and the final positions.
This means, work done by or against a conservative force in moving a body over any path between fixed initial and final positions will be the same.
For example,
Force in an elastic spring, electrostatic force between two electric charges, magnetic force between two magnetic poles, gravitational force are conservative force.
Properties of conservative force

• Conservative force depends only on the initial and final position of the body.
• It is always zero in closed path.
• It does not depend upon the nature of path followed by the body.

• ### Non–conservative force

A force is said to be non-conservative, if work done by or against the force is moving a body from one position to another, depends on the path followed between these two positions.
For example,
Force of friction, Viscous force are non–conservative forces.