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# Unit of Work

## Definition

Work is the force acting per unit displacement. It is a measure of displacement. Work is a scalar quantity as it is a dot product of two vector quantities. Some examples of work done are – riding a bicycle on a hill, carrying a trolley from one place to another, running from one end of a road to another, etc. Remember running across a field may result in zero work done as the displacement is zero in this case. The person running will return to his original position resulting in zero displacements.

Similarly, the work done by a weightlifter is zero as there is no change in displacement taking place. Therefore, work is mechanical energy used to lift, move, or do other activities with the object. The formula for the work done is given by-
W = F. d cos θ
Where F = force applied on the body
d = displacement
θ = angle formed between force and displacement

## Types of work

There are two types of work – positive work and negative work.

Positive work
The equation W = F describes the work. d CosӨ.
When Ө is acute, i.e., Ө< 90°, work is positive.

## Examples

• When a lawn roller is pulled by applying force along with the handle. The work done by the applied force F is positive.
• Work done by the stretching force in spring is positive.

Negative work
It happens when Ө is obtuse, i.e., Ө> 90°.
The angle between the force F and displacement S is 180°. Therefore, the work done by gravity on moving a body upwards is negative.

## Examples

• Rocket propulsion
• Work done in pulling the load.
• Pulling the water from the well.
• Flying an airplane.

## Real-Life Examples of Work

• The work done against the gravitational pull to escape an object from the earth.
• When the road is busy, the force applied by a vehicle to slow down the speed is work.
• The work done by the pendulum to move to and fro when pulled from its rest position.
• The work done by the earth to rotate about its polar axis and around the Sun.
• A motorman riding a bike from one city to another is work.

## How to measure work done?

Every day we work differently on different tasks or activities. However, the amount of energy spent, labor done, and toil, are all synonyms of work. The work done is calculated in Joules. It is the amount of a unit force applied per unit displacement. In other words, 1N of force applied to cause a displacement of 1m. Therefore, the unit of work is Newton-meter (N-m).

Work in measured in gases as-
W = P ΔV
Here, W is the work done in expanding the volume of the gas in a piston.

Units of work
In SI, the base unit is Kg m² s⁻²
Kg m² s⁻² is the MKS unit.
The MKS stands for meter-kilogram-second.
The CGS (centimeter-gram-second) unit for work is dyne-cm or erg.

Units of work are represented as-

 SI unit N-m Joule CGS unit Dyne-cm Erg Dimensional formula ML2T-2

Some other units of work include horsepower, erg, foot-pound, kilowatt-hour, liter-atmosphere, foot-poundal, etc. Work has the same physical dimension as that of heat. Therefore, units like BTU, calorie, therm, etc. are also units of work done. Conversions of units are given below-

 Units Equivalent unit in Joules 1 erg 1.0E-7 J 1 horsepower-hour 2684519.54 J 1 Newton-meter 1 Joule 1 foot-pound 1.35582 J 1 kilowatt-hour 3.6e+6J 1 BTU 1055.06 J
###### Example 1 Ques: A box is pulled with a force of 25 N to produce a displacement of 15 m. If the angle between the force and displacement is 30˚, find the work done by the force.

Solution
Force, F = 25 N
Displacement, dr = 15 m
Angle between F and dr, θ = 30o
Work done, W = Fdr cosθ
W = 25 x 15 x cos 30 = 25 x 15 x √3 /2 = 324.76 J

##### Example 2

A moving body of 30kg has 60 J of KE. Calculate the speed.

Solution: KE = 0.5 mv2
60 = 0.5 x 30 x v2
60 = 15 v2
V = 2m/s

##### Example 3

Calculate the energy possessed by a stone of mass 10kg kept at a height of 5m If 196 x102 J of energy were used to raise a 40kg boy above the ground, how high would he be raised?

Solution: The energy possessed by a stone of mass 10kg kept at a height of 5m = PE
PE = mgh = 10 x 5 x 9.8 = 490 j
196 x102 J = 40 x 9.8 x h => h = 50m

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