Short Answer:

Work is defined as the product of the force applied on a body and the displacement produced in the direction of that force. It is a measure of energy transferred when a force causes an object to move.

In simple words, work is said to be done only when a body moves under the action of an external force. If there is no movement or the displacement is perpendicular to the force, no work is done. The SI unit of work is Joule (J), and mathematically, it is expressed as W = F × s × cosθ, where θ is the angle between force and displacement.

Detailed Explanation:

Work

In mechanics, the concept of work is very important for understanding how forces cause motion or energy transfer. Whenever a force acts on a body and produces a displacement in the direction of the applied force, mechanical work is said to be done.

The idea of work connects force, motion, and energy. It helps explain how energy is transferred from one body to another or converted from one form to another. If there is no displacement, or if the force acts perpendicular to the displacement, then no work is done.

Definition of Work

Work is defined as the product of force and displacement in the direction of that force.

Mathematically,

Where,

•  = Work done (Joules, J)
•  = Force applied (Newtons, N)
•  = Displacement of the body (meters, m)
•  = Angle between force and displacement

From the above equation, we can understand that:

• When θ = 0° (force and displacement are in the same direction), maximum work is done.

• When θ = 90° (force and displacement are perpendicular), no work is done.

Hence, work depends not only on the magnitude of force and displacement but also on the angle between them.

Units and Dimensions of Work

1. SI Unit:
   The SI unit of work is the Joule (J).
   1 Joule = 1 Newton × 1 meter = 1 N·m

Therefore,

1. CGS Unit:
   • The CGS unit of work is erg.
   • 1 erg = 10⁻⁷ Joules
2. Dimensional Formula:

Conditions for Work to be Done

For work to be done by a force on a body, the following conditions must be satisfied:

1. Force Must Act on the Body:
   If no force is applied, no work can be done.
2. Displacement Must Occur:
   The body must move due to the applied force.
3. Force and Displacement Must Have a Component in the Same Direction:
   If the force acts perpendicular to displacement, such as in uniform circular motion, no work is done.

Types of Work

Work can be classified into three main types based on the direction of force and displacement:

1. Positive Work:
   When the force and displacement are in the same direction, the work done is positive.
   Example:
   • Lifting a weight upward against gravity.
   • Pushing a car forward.
     Mathematically, when θ < 90°,

1. Negative Work:
   When the force and displacement are in opposite directions, the work done is negative.
   Example:
   • Work done by gravity when an object is moving upward.
   • Frictional force acting on a moving object.
     Mathematically, when θ > 90°,

1. Zero Work:
   When the force acts perpendicular to the direction of displacement, no work is done.
   Example:
   • Work done by centripetal force in circular motion.
   • Carrying a heavy bag horizontally (force acts vertically, displacement is horizontal).
     Mathematically, when θ = 90°,

Work Done by a Variable Force

When the force applied is not constant, such as a spring or a changing load, the work done is found by integrating the force over the distance moved.

Mathematically,

This means that the total work done is the area under the force-displacement curve. For example, in the case of a spring, the force is proportional to displacement (Hooke’s Law), and the work done in stretching or compressing it is given by:

where  is the spring constant and  is the displacement.

Work-Energy Relationship

Work and energy are closely related. The work done on a body results in a change in its energy. This is expressed as the Work-Energy Theorem, which states that:

Thus, if a body gains kinetic energy, positive work is done; if it loses kinetic energy, negative work is done.

This relationship helps in analyzing the motion and energy of objects in mechanics.

Examples of Work in Real Life

1. Positive Work:
   • Lifting a book upward.
   • Accelerating a car.
2. Negative Work:
   • Applying brakes to stop a vehicle.
   • Lowering a load slowly using a rope.
3. Zero Work:
   • Holding a stationary object in place.
   • Motion of a planet around the sun (centripetal force does no work).

These examples help illustrate how direction and motion influence whether work is positive, negative, or zero.

Applications of Work Concept

1. Engineering Design:
   Used in calculating power and energy requirements in machines.
2. Energy Conversion Systems:
   Helps in understanding how energy is transferred or transformed in engines and mechanical systems.
3. Structural Analysis:
   Used in determining deformation and stress due to work done by external loads.
4. Physics and Thermodynamics:
   The concept of work helps analyze mechanical energy transfer in gases and fluids.

Conclusion

In summary, work is done when a force applied to a body causes displacement in the direction of that force. It serves as a measure of energy transfer due to mechanical action. The amount of work depends on the magnitude of force, the distance moved, and the angle between them. Work can be positive, negative, or zero depending on the direction of the force relative to displacement. The concept of work is fundamental in mechanics, as it links force and motion with energy, forming the basis of mechanical and structural analysis.