Work: In physics work is said to be done when a force acting on a body displaces it through a certain distance. The work by a constant force is measured by the dot product of force will displacement.

i.e. W = F.S or W = F S cosq …(1)

where F = force and S = displacement

And q = angle between F and S.

From eq.(1), we have W=F cosq x S. Therefore, work done by a force is equal to the product of the component of the force in the direction of displacement of the body and the distance through which the body moves.

As work is the dot or scalar product of two vectors (force and displacement), therefore, it is a scalar quantity.

Special cases:

CaseI: When the body moves in the direction of the force. In this case, q = 0^o.

from eq (i), we have

W = FScoso^o

i.e. W = FS …(ii)

i.e. W = product of the force and the distance through which the body moves in the direction of the force.

This type of work is said to be positive. Therefore, work done by a force is positive is q < 90^o

Examples of Positive Work:

(1) When a spring is stretched, the work done by the stretching force is positive as the force acts in the direction of displacement of the spring.

(2) When a body falls freely under gravity, the work done by gravity is positive as force (gravity) and displacement of the body are in the same direction.

(3) When a gas taken in a cylinder expands, work done by the force is positive as the displacement is in the direction of the force.

Case II:. When the force acting on a body and its displacement are in the opposite direction

In this case, q = 180^o. Therefore, from eqn (i), we have

W = FS cos180^o = FS(-1)

Or W = -FS …. (iii). This type of work is said to be negative. Therefore, work done by a force is negative if q > 90^o.

Examples of Negative Work:

(i) The work done against friction is negative as the frictional force is always in direction opposite to displacement.

(ii) Work done in lifting a body is negative as gravitational force and displacement are in the opposite direction.

Case III:. When force and displacement are at right angle to each other.

In This case q = 90^o

From eqn (i), we have

W = FS cos90^o = FS . 0 = 0

Therefore, no work is done by a force which acts at right angle to the direction of displacement of the body.

Examples of Zero Work:

(i) The work done by the centripetal force acting on a body moving in a circle with uniform speed is zero as it is always at right angle to the direction of motion of the body.

W = FS cos90^o

= FS (0)

= 0

(ii) A person carrying a load on his hand and walking on a horizontal road does no work as the force is acting at right angle to displacement.

Case IV: If S = 0, then from eqn. (i), we have

W = f x 0 x cosq = 0

Therefore, no work is done by because if it can not displace a body on which it acts.

Note: Work done by a variable force in displacing a body from its initial position x_i to the final position x_f is given by

W =  = Area under the curve between x_i and x_f.

Units of Work: We know that:

W = FS cosq = F x S; if q = 0^o i.e. when displacement is in the direction of the force.

Therefore, units of work – Units of force x Units of Distance

There are to types of units of work:

(a) Absolute Units and (b) Gravitational or Practical Unites.

(a) Absolute Units:

(i) In the CGS System, the absolute unit of work is erg.

When F = 1 dyne, S = 1 cm, then W = 1 erg

1 erg = 1 dyne x 1 cm

1 erg is defined as the amount of work done when a force of 1 dyne moves a body through a distance f 1 cm in its own direction.

(ii) In the MKS System or SI Units, the absolute unit of work is joule (J).

When F = 1 newton, S = 1 m, then W = 1 joule

1J = 1N x 2 m

1 joule is defined as the amount of work done when a force of 1 newton moves a body through a distance of 1 metre in its own direction.

Relationship between joule and erg:

1J = 1N x 1m = 10⁵ dyne x 100 cm

1J = 10⁷ (1dyne x 1 cm) = 10⁷ erg

1J = 10⁷ erg

(b) Gravitational Units:

(i) In the CGS System, the gravitational unit of work is gram centimeter (g cm):

One gram centimeter is defined as the amount of work done when a body of mass one gram is raised vertically upwards through a distance of one centimeter.

Or

One gram centimeter may also be defined as the amount of work done when a force of one gram weight (gram force) moves a body through a distance of one centimeter in its own direction.

i.e. 1 g cm = 1 g f x 1 cm = 981 dyne x 1 cm = 981 (1 dyne x 1 cm)

or 1 g cm = 981 erg or 980 erg

(ii) In the MKS System or SI Units, the gravitational unit of work in kilogram metre (Kg m): One kilogram metre is defined as the amount of work done when a body of mass one kg is raised vertically upwards through a distance of one metre.

Or

One kilogram metre may also be defined as the amount of work done when a force of one kilogram weight (kilogram force) moves a body through a distance of one metre in its own direction i.e. 1 kg m = 1 kgs x 1 m = 9.81 N x 1 m = 9.81 (1N x 1m) = 9.8 J

1kgm = 9.81 J

Absolute Unit: Work done is said to be one absolute units if unit absolute force displaces the body through unit distance in the direction of work done is said to be one gravitational unit.

Important Note: (i) Work is a scalar quantity

(ii) The unit of work in nuclear energy is electron-volt (eV). One electron is defined as the energy gained by an electron when it is accelerated through a potential difference of one volt.

1 eV = 1.6 x 10^-14 J

(iii) The commercial unit of electrical energy in kilowatt hour (kWh).

1 kWh = 3.6 x 10⁶ J