Detailed Notes for JEE Exam

Work and Energy

Basic Concepts Related to Constant and Variable Forces; Kinetic Energy

1. Concept of Work

• Work done by a constant force:

  • Definition: Work is done when a force acts on an object and causes it to move in the direction of the force.
  • Formula: $$W = F * d * cosθ$$
    • W represents the work done (in joules, J).
    • F represents the magnitude of the constant force applied (in newtons, N).
    • d represents the displacement of the object (in meters, m).
    • θ represents the angle between the force vector and displacement vector.

• Work done by a variable force:

• Definition: When the magnitude or direction of the force changes during the displacement, the force is called a variable force.

• Formula: In such cases, work done can be calculated using integration: $$W = ∫ F(x) dx$$ where F(x) represents the variable force as a function of position x.

2. Work-Energy theorem

• Defintion: The work done on an object is equal to its change in kinetic energy.
• Formula: $$W = ΔK = K_f - K_i$$
  • W represents the work done on the object (in joules, J).
  • ΔK represents the change in kinetic energy (in joules, J).
  • Kf is the final kinetic energy of the object.
  • K_i is the initial kinetic energy of the object.

3. Kinetic Energy

• Definition: Kinetic energy is the energy possessed by an object due to its motion.
• Formula: $$KE = (1/2)mv^2$$
  • KE represents the kinetic energy (in joules, J).
  • m represents the mass of the object (in kilograms, kg).
  • v represents the velocity of the object (in meters per second, m/s).

4. Work-Energy Principle for Variable Force:

• Defintion: The net work done on an object by all the forces acting on it is equal to the change in its kinetic energy.
• Formula: $$W_{net} = ΔK = K_f - K_i$$
• W_net represents the net work done on the object (in joules, J).

5. Conservation of Energy:

• Defintion: In an isolated system, the total energy remains constant, although it may be converted from one form to another.
• Formula: $$E_i = E_f$$
  • E_i represents the initial total energy (in joules, J).
  • Ef represents the final total energy (in joules, J).

6. Potential Energy

• Defintion: Potential energy is the energy stored in an object due to its position or configuration.

• Gravitational potential energy: Energy stored due to position in a gravitational field. $$U_g = mgh$$

  • U_g represents the gravitational potential energy (in joules, J).
  • m represents the mass of the object (in kilograms, kg).
  • g represents acceleration due to gravity (9.8 m/s^2 on Earth).
  • h represents the height of the object above a reference point (in meters, m).

• Elastic potential energy: Energy stored in a deformed elastic object, like a stretched spring. $$U_e = (1/2)kx^2$$

  • U_e represents the elastic potential energy (in joules, J).
  • k is the spring constant (in newtons per meter, N/m).
  • x represents the displacement from the spring’s equilibrium position (in meters, m).

• Spring potential energy: Energy stored in a stretched or compressed spring. $$U_s = (1/2)kx^2$$

  • U_s represents the spring potential energy (in joules, J).
  • k represents the spring constant (in newtons per meter, N/m).
  • x represents the displacement from the spring’s equilibrium position (in meters, m).

7. Power

• Definition: Power is the rate at which work is done or energy is transferred.
• Formula: $$P = W/t$$
• P represents power (in watts, W).
• W represents the work done (in joules, J).
• t represents the time taken to do the work (in seconds, s).

8. Efficiency

• Definition: Efficiency is the ratio of useful output energy to the energy input.
• Formula: $$η = W_{out} / W_{in}$$
  • η represents efficiency (no units).
  • W_{out} represents the useful output energy (in joules, J).
  • W_{in} represents the energy input (in joules, J).

Reference books:

• NCERT Physics Class 11
• NCERT Physics Class 12

By mastering these subtopics and practicing relevant problems, you can solidify your understanding of the fundamental concepts in Work and Energy and improve your chances of success in the JEE exam.