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 Î¸ 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 (FcosÎ¸).

W = (FcosÎ¸) s …… (1)

If displacement is in the direction of force applied,

Î¸ = 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 = [M

^{1}L^{1}T^{-2}] × L
W = [M

^{1}L^{2}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 cosÎ¸**

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 cosÎ¸

1 erg = 1 dyne × 1 cm × cos0°

### Relation between joule and erg

As, 1 joule = 1 N × 1 m × cos0°

Therefore,

1 J = 10

^{5}dyne × 10^{2}cm × 1
1 joule = 10

^{7}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 Î¸

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 cosÎ¸

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= 10

^{3}g × 10^{2}cm
1 kg-m = 10

^{5}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 cosÎ¸

Therefore when Î¸ is acute (< 90°), cosÎ¸ is positive.

Hence, work done is positive.

**Some examples**

### Negative work

As W = F • s = F s cos Î¸

Therefore, when Î¸ is obtuse (>90°), cos Î¸ is negative. Hence, work done is negative.

**Some examples**

### Zero work

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

**Some examples**

## 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**

### 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.

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