JEE Physics: Work, Energy and Power – Complete Guide

The chapter on Work, Energy and Power is vital in the JEE Physics syllabus. Understanding these concepts deeply is essential for solving many mechanics problems efficiently. This guide covers fundamental definitions, important formulas, work-energy theorem, types of energy, power and efficiency, and numerous solved examples to strengthen your preparation.

1. What is Work in Physics?

Work is said to be done when a force causes displacement of a body in the direction of the force. It is a scalar quantity measured in Joules (J).

1.1 Mathematical Definition of Work

If a constant force \( \vec{F} \) acts on a body causing displacement \( \vec{d} \), work done \( W \) is:

\( W = \vec{F} \cdot \vec{d} = F d \cos \theta \)

where \( \theta \) is the angle between the force and displacement vectors.

1.2 Work Done by a Variable Force

When force varies with position, work done is the integral:

\( W = \int_{x_i}^{x_f} F(x) \, dx \)

1.3 Sign of Work Done

Positive work: Force and displacement in the same direction (\( 0^\circ \leq \theta < 90^\circ \)).
Negative work: Force and displacement in opposite direction (\( 90^\circ < \theta \leq 180^\circ \)).
Zero work: Force perpendicular to displacement (\( \theta = 90^\circ \)), e.g., centripetal force in uniform circular motion.

2. Energy

Energy is the capacity to do work. It exists in multiple forms but JEE focuses mainly on mechanical energy forms.

2.1 Kinetic Energy (KE)

The energy possessed by a body due to its motion is called kinetic energy.

\( KE = \frac{1}{2} m v^2 \)

where \( m \) is mass and \( v \) is velocity.

2.2 Potential Energy (PE)

The energy possessed by a body due to its position or configuration.

2.2.1 Gravitational Potential Energy

For an object of mass \( m \) at height \( h \) above a reference level:

\( PE = m g h \)

2.2.2 Elastic Potential Energy

Energy stored in a stretched or compressed spring:

\( PE_{spring} = \frac{1}{2} k x^2 \)

where \( k \) is the spring constant and \( x \) is displacement from equilibrium.

3. Work-Energy Theorem

This theorem relates the net work done on a body to the change in its kinetic energy:

\( W_{net} = \Delta KE = \frac{1}{2} m v_f^2 - \frac{1}{2} m v_i^2 \)

It means net work done on a body changes its kinetic energy.

4. Power

Power is the rate of doing work or the rate of transfer of energy. It is a scalar quantity measured in Watts (W).

4.1 Formula for Power

\( P = \frac{W}{t} \)

where \( W \) is work done and \( t \) is time taken.

4.2 Instantaneous Power

When force \( F \) acts on a body moving with velocity \( v \) in the direction of force:

\( P = \vec{F} \cdot \vec{v} = F v \cos \theta \)

4.3 Horsepower

1 horsepower (hp) = 746 Watts.

5. Conservative and Non-Conservative Forces

5.1 Conservative Forces

A force is conservative if work done by it on an object moving between two points is independent of the path taken. Examples include gravitational and elastic spring forces.

5.2 Non-Conservative Forces

Forces like friction and air resistance, where work done depends on the path and dissipates mechanical energy as heat.

5.3 Conservation of Mechanical Energy

In absence of non-conservative forces, total mechanical energy (sum of KE and PE) remains constant:

\( KE_i + PE_i = KE_f + PE_f \)

6. Elastic Collisions and Energy Considerations

Elastic collisions conserve both momentum and kinetic energy, while inelastic collisions do not conserve kinetic energy.

7. Applications and Examples

Example 1: Work done by force on a moving block

A force of 10 N acts on a block moving 5 m in the same direction. Find work done.

Solution:

\( W = F \times d = 10 \times 5 = 50 \, J \)

Example 2: Calculating kinetic energy

A 2 kg mass moves with velocity 3 m/s. Find kinetic energy.

Solution:

\( KE = \frac{1}{2} m v^2 = \frac{1}{2} \times 2 \times 3^2 = 9 \, J \)

Example 3: Power of a motor lifting a load

A motor lifts a 50 kg load to height 10 m in 5 seconds. Calculate power.

Solution:

Work done = \( mgh = 50 \times 9.8 \times 10 = 4900 \, J \)

Power = Work / time = \( 4900 / 5 = 980 \, W \)

8. Efficiency

Efficiency measures the effectiveness of a machine or system in converting input energy/work into useful output.

\( \text{Efficiency} = \frac{\text{Useful Work Output}}{\text{Total Work Input}} \times 100\% \)

9. Conservation of Energy in Practical Situations

Problems involving roller coasters, pendulums, and spring systems often use energy conservation principles.

10. Tips for JEE Work, Energy and Power

Memorize key formulas and understand when and how to apply them.
Practice problems on variable forces using integrals.
Work on conceptual questions differentiating work done by different forces.
Understand energy transformations and power calculations clearly.
Use energy conservation for efficient problem-solving instead of force analysis where possible.
Draw clear diagrams showing forces, displacements, and energy changes.

11. Summary Table of Important Formulas

Concept	Formula	Remarks
Work done by force	\( W = F d \cos \theta \)	Scalar quantity
Kinetic energy	\( KE = \frac{1}{2} m v^2 \)	Energy due to motion
Potential energy (gravity)	\( PE = m g h \)	Energy due to position
Elastic potential energy	\( PE = \frac{1}{2} k x^2 \)	Energy stored in spring
Power	\( P = \frac{W}{t} \)	Rate of work done
Work-Energy theorem	\( W_{net} = \Delta KE \)	Net work equals change in kinetic energy

      Mastering the concepts of Work, Energy, and Power will help you solve complex JEE problems with ease and build a strong foundation for advanced physics topics.
    