Class 11 Physics Notes Chapter 6 (Work; energy and power) – Physics Part-I Book

Alright class, let's delve into Chapter 6: Work, Energy, and Power from your NCERT Class 11 Physics textbook. These concepts are fundamental not just for your Class 11 understanding but form the bedrock for many topics in physics and are frequently tested in various government examinations. Pay close attention to the definitions, units, conditions, and the conservation principles.

Chapter 6: Work, Energy and Power - Detailed Notes

1. Introduction

• These concepts deal with the causes and effects of motion, focusing on energy transfer and transformation.
• Work links force with displacement.
• Energy represents the capacity to do work.
• Power describes the rate at which work is done or energy is transferred.

2. Work (W)

• Definition: In physics, work is said to be done by a force when the body is displaced actually through some distance in the direction of the applied force or a component of the force.
• Work Done by a Constant Force:
  • If a constant force F acts on an object, causing a displacement d, the work done is given by the scalar (dot) product of the force and displacement vectors:
    W = F ⋅ d = F d cos θ
  • Where θ is the angle between the force vector F and the displacement vector d.
• Nature of Work Done:
  • Positive Work (θ < 90°): When cos θ is positive (e.g., force applied in the direction of motion, lifting an object upwards by an upward force). Work done by the force increases the energy of the system (usually kinetic).
  • Negative Work (90° < θ ≤ 180°): When cos θ is negative (e.g., work done by friction, work done by gravity on an object lifted upwards). Work done by the force decreases the energy of the system.
  • Zero Work (θ = 90° or d = 0 or F = 0):
    • Force is perpendicular to displacement (e.g., work done by centripetal force in uniform circular motion, work done by gravity on an object moving horizontally).
    • No displacement occurs (e.g., pushing against a rigid wall).
    • No force is applied.
• Units and Dimensions:
  • SI Unit: Joule (J) (1 J = 1 N m)
  • CGS Unit: erg (1 erg = 1 dyne cm)
  • Relation: 1 J = 10⁷ erg
  • Dimension: [ML²T⁻²] (Same as energy)
• Work Done by a Variable Force:
  • If the force is not constant, we consider the work done over a small displacement Δx, where the force F(x) is approximately constant. ΔW ≈ F(x) Δx.
  • The total work done in moving the object from an initial position xᵢ to a final position x<0xE1><0xB5><0xA3> is the sum (integral) of these small works:
    W = ∫<0xE2><0x82><0x99>ᵢ<0xE1><0xB5><0xA3> F(x) dx
  • Graphical Interpretation: Work done by a variable force is numerically equal to the area under the Force-Displacement (F-x) graph.

3. Energy

• Definition: Energy is defined as the capacity or ability of a body to do work.
• It is a scalar quantity.
• Units and Dimensions are the same as Work ([ML²T⁻²], Joule).
• Forms of Energy: Mechanical (Kinetic & Potential), Heat, Light, Sound, Electrical, Chemical, Nuclear, etc. This chapter focuses on Mechanical Energy.

4. Kinetic Energy (K or KE)

• Definition: The energy possessed by a body by virtue of its motion.
• Formula: For an object of mass 'm' moving with velocity 'v':
  K = ½ mv²
• Derivation: Consider a body of mass m starting from rest, acted upon by a constant force F, achieving velocity v over displacement d.
  • Work done W = Fd = (ma)d
  • Using v² = u² + 2as (here u=0), we get v² = 2ad, or ad = v²/2.
  • Substituting, W = m(v²/2) = ½ mv². This work done is stored as Kinetic Energy.
• KE is always non-negative (m is positive, v² is positive or zero).
• It depends on the frame of reference (as velocity does).

5. Work-Energy Theorem

• Statement: The work done by the net force acting on a body is equal to the change in its kinetic energy.
  W<0xE2><0x82><0x99><0xE2><0x82><0x8F><0xE1><0xB5><0x9C> = K<0xE1><0xB5><0xA3> - Kᵢ = ΔK
• Significance: This theorem provides a direct link between the work done by all forces (conservative and non-conservative) and the change in the object's speed.
• It is a very general theorem, applicable even if the force is variable and for non-conservative forces.

6. Potential Energy (U or PE)

• Definition: The energy possessed by a body by virtue of its position or configuration (shape/size) in a force field.
• Potential energy is defined only for Conservative Forces.
• Conservative Force: A force is conservative if:
  • The work done by the force in moving a particle from one point to another depends only on the initial and final positions and not on the path taken.
  • The work done by the force in moving a particle around any closed path is zero.
  • Examples: Gravitational force, Electrostatic force, Elastic spring force.
  • Mathematically, for a conservative force F, the change in potential energy ΔU is defined as the negative of the work done by that conservative force:
    ΔU = U<0xE1><0xB5><0xA3> - Uᵢ = - W<0xE2><0x82><0x9C> = - ∫<0xE2><0x82><0x99>ᵢ<0xE1><0xB5><0xA3> F ⋅ dr
  • Alternatively, F = - dU/dr (Force is the negative gradient of potential energy).
• Non-Conservative Force: A force is non-conservative if the work done depends on the path taken. Work done in a closed loop is not zero.
  • Examples: Friction, Air resistance, Viscous force.
  • Work done by non-conservative forces usually dissipates energy as heat or sound.
• Gravitational Potential Energy (near Earth's surface):
  • The work done by gravity (a conservative force) when an object of mass 'm' is lifted vertically by height 'h' is W<0xE1><0xB5><0x8D> = -mgh.
  • The change in gravitational potential energy is ΔU = -W<0xE1><0xB5><0x8D> = +mgh.
  • If we choose the reference level (where PE=0) at the ground (h=0), then the potential energy at height 'h' is:
    U(h) = mgh
  • The choice of the reference level (zero PE) is arbitrary; only the change in PE is physically significant.
• Potential Energy of a Spring:
  • According to Hooke's Law, the restoring force exerted by a spring when stretched or compressed by a displacement 'x' from its equilibrium position is F<0xE2><0x82><0x9B> = -kx, where 'k' is the spring constant (a measure of stiffness).
  • The work done by the spring force (conservative) when stretched from 0 to x is W<0xE2><0x82><0x9B> = ∫₀ˣ (-kx) dx = -½ kx².
  • The potential energy stored in the spring is ΔU = -W<0xE2><0x82><0x9B> = +½ kx².
  • Assuming U=0 at x=0 (equilibrium position):
    U(x) = ½ kx²
  • This is also called Elastic Potential Energy. It's always non-negative.

7. Conservation of Mechanical Energy

• Mechanical Energy (E): The sum of kinetic energy (K) and potential energy (U) of a system.
  E = K + U
• Principle of Conservation of Mechanical Energy: If only conservative forces are doing work on a system, its total mechanical energy remains constant.
  If W<0xE2><0x82><0x99><0xE1><0xB5><0x8E><0xE1><0xB5><0x84> (work by non-conservative forces) = 0, then ΔE = ΔK + ΔU = 0
  or Kᵢ + Uᵢ = K<0xE1><0xB5><0xA3> + U<0xE1><0xB5><0xA3>
• Derivation: From the Work-Energy Theorem, W<0xE2><0x82><0x99><0xE2><0x82><0x8F><0xE1><0xB5><0x9C> = ΔK. The net work is the sum of work done by conservative forces (W<0xE2><0x82><0x9C>) and non-conservative forces (W<0xE2><0x82><0x99><0xE1><0xB5><0x8E><0xE1><0xB5><0x84>).
  W<0xE2><0x82><0x9C> + W<0xE2><0x82><0x99><0xE1><0xB5><0x8E><0xE1><0xB5><0x84> = ΔK.
  Since W<0xE2><0x82><0x9C> = -ΔU, we have -ΔU + W<0xE2><0x82><0x99><0xE1><0xB5><0x8E><0xE1><0xB5><0x84> = ΔK.
  Rearranging, W<0xE2><0x82><0x99><0xE1><0xB5><0x8E><0xE1><0xB5><0x84> = ΔK + ΔU = ΔE.
  If W<0xE2><0x82><0x99><0xE1><0xB5><0x8E><0xE1><0xB5><0x84> = 0, then ΔE = 0, meaning E is conserved.
• Examples:
  • A freely falling body (neglecting air resistance): As it falls, PE decreases, KE increases, but K + U remains constant.
  • Oscillation of a simple pendulum (neglecting air resistance): Energy transforms between KE (max at mean position) and PE (max at extreme positions), but total mechanical energy is conserved.
• Presence of Non-Conservative Forces: If forces like friction or air resistance are present, W<0xE2><0x82><0x99><0xE1><0xB5><0x8E><0xE1><0xB5><0x84> is usually negative (dissipative), leading to a decrease in total mechanical energy (ΔE < 0). The lost mechanical energy typically appears as heat.

8. Power (P)

• Definition: The rate at which work is done or energy is transferred.
• Average Power (P<0xE2><0x82><0x90><0xE1><0xB5><0x9D><0xE2><0x82><0x9D>): Total work done (W) divided by the total time taken (t).
  P<0xE2><0x82><0x90><0xE1><0xB5><0x9D><0xE2><0x82><0x9D> = W / t
• Instantaneous Power (P): The limiting value of the average power as the time interval approaches zero. It's the rate of doing work at a particular instant.
  P = dW / dt
• Relation with Force and Velocity: Since dW = F ⋅ dr,
  P = dW/dt = F ⋅ (dr/dt) = F ⋅ v
  P = F v cos θ
  Where θ is the instantaneous angle between the force F and velocity v.
• Units and Dimensions:
  • SI Unit: Watt (W) (1 W = 1 J/s)
  • Common Unit: Horsepower (hp) (1 hp ≈ 746 W)
  • Commercial Unit of Energy: kilowatt-hour (kWh) (1 kWh = 1000 W × 3600 s = 3.6 × 10⁶ J). Note: kWh is a unit of energy, not power.
  • Dimension: [ML²T⁻³]

9. Collisions

• Definition: An event in which two or more bodies exert relatively strong forces on each other for a relatively short time. Physical contact is not necessary (e.g., Rutherford scattering).
• Conservation Law: In any collision (elastic or inelastic), the total linear momentum of the system of colliding bodies is conserved, provided there is no external force acting on the system.
• Types of Collisions:
  • Elastic Collision: A collision in which both the total momentum and the total kinetic energy of the system are conserved. (Forces involved are conservative).
  • Inelastic Collision: A collision in which the total momentum is conserved, but the total kinetic energy is not conserved. Some KE is lost (converted to heat, sound, deformation).
  • Perfectly Inelastic Collision: A type of inelastic collision where the bodies stick together after the collision and move with a common velocity. Maximum loss of KE occurs (consistent with momentum conservation).
• Coefficient of Restitution (e): A measure of the elasticity of a collision. For a direct collision of two bodies,
  e = (Relative velocity of separation after collision) / (Relative velocity of approach before collision)
  e = |v₂ - v₁| / |u₁ - u₂|
  • For a perfectly elastic collision, e = 1.
  • For a perfectly inelastic collision, e = 0.
  • For all other inelastic collisions, 0 < e < 1.
• Elastic Collision in One Dimension:
  • Consider two masses m₁ and m₂ with initial velocities u₁ and u₂ colliding elastically and moving with final velocities v₁ and v₂ along the same line.
  • Conservation of Momentum: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
  • Conservation of KE: ½m₁u₁² + ½m₂u₂² = ½m₁v₁² + ½m₂v₂²
  • Solving these equations gives the final velocities:
    v₁ = [(m₁ - m₂) / (m₁ + m₂)]u₁ + [2m₂ / (m₁ + m₂)]u₂
    v₂ = [2m₁ / (m₁ + m₂)]u₁ + [(m₂ - m₁) / (m₁ + m₂)]u₂
  • Special Cases (Important for Exams):
    • If m₁ = m₂: v₁ = u₂ and v₂ = u₁ (Velocities are exchanged).
    • If m₂ is at rest (u₂ = 0):
      v₁ = [(m₁ - m₂) / (m₁ + m₂)]u₁
      v₂ = [2m₁ / (m₁ + m₂)]u₁
    • If m₂ is at rest and m₁ << m₂ (light body hits heavy body): v₁ ≈ -u₁ and v₂ ≈ 0 (Light body rebounds with nearly the same speed, heavy body barely moves).
    • If m₂ is at rest and m₁ >> m₂ (heavy body hits light body): v₁ ≈ u₁ and v₂ ≈ 2u₁ (Heavy body continues almost unaffected, light body moves off with twice the speed of the heavy body).
• Inelastic Collision in One Dimension: Only momentum is conserved. If it's perfectly inelastic (bodies stick), m₁u₁ + m₂u₂ = (m₁ + m₂)v<0xE1><0xB5><0x84><0xE1><0xB5><0x92><0xE1><0xB5><0x90>, where v<0xE1><0xB5><0x84><0xE1><0xB5><0x92><0xE1><0xB5><0x90> is the common final velocity.
• Collisions in Two Dimensions (Oblique Collision): Momentum is conserved vectorially. Apply conservation of momentum separately along two perpendicular axes (usually x and y). If the collision is elastic, KE is also conserved.

Key Formulas Summary

• Work (Constant Force): W = Fd cos θ
• Work (Variable Force): W = ∫ F(x) dx
• Kinetic Energy: K = ½ mv²
• Work-Energy Theorem: W<0xE2><0x82><0x99><0xE2><0x82><0x8F><0xE1><0xB5><0x9C> = ΔK
• Potential Energy Change: ΔU = - W<0xE2><0x82><0x9C> (for conservative force)
• Force from PE: F = -dU/dr
• Gravitational PE: U = mgh (near surface, relative to h=0)
• Spring PE: U = ½ kx² (relative to x=0)
• Mechanical Energy: E = K + U
• Conservation of Mech. Energy: Kᵢ + Uᵢ = K<0xE1><0xB5><0xA3> + U<0xE1><0xB5><0xA3> (if only conservative forces work)
• Work by Non-conservative Force: W<0xE2><0x82><0x99><0xE1><0xB5><0x8E><0xE1><0xB5><0x84> = ΔE
• Average Power: P<0xE2><0x82><0x90><0xE1><0xB5><0x9D><0xE2><0x82><0x9D> = W/t
• Instantaneous Power: P = dW/dt = F ⋅ v = Fv cos θ
• Coefficient of Restitution: e = |v₂ - v₁| / |u₁ - u₂|
• Elastic Collision (1D, m₂ at rest): v₁ = [(m₁ - m₂) / (m₁ + m₂)]u₁ ; v₂ = [2m₁ / (m₁ + m₂)]u₁

Multiple Choice Questions (MCQs)

1. A cyclist comes to a skidding stop in 10 m. During this process, the force on the cycle due to the road is 200 N and is directly opposed to the motion. The work done by the road on the cycle is:
   a) +2000 J
   b) -2000 J
   c) Zero
   d) +200 J

2. An object of mass 'm' is moving in a circular path of radius 'r' with constant speed 'v'. The work done by the centripetal force during one complete revolution is:
   a) ½ mv²
   b) mv²/r
   c) 2πr × (mv²/r)
   d) Zero

3. If the kinetic energy of a body increases by 300%, its momentum will increase by:
   a) 100%
   b) 150%
   c) 200%
   d) 400%

4. Which of the following forces is non-conservative?
   a) Gravitational force
   b) Electrostatic force
   c) Force of friction
   d) Elastic spring force

5. A body of mass 5 kg is lifted vertically to a height of 10 m by a force of 100 N. The work done by the applied force is:
   a) 500 J
   b) 1000 J
   c) -500 J
   d) -1000 J

6. The potential energy of a system increases if work is done:
   a) Upon the system by a non-conservative force
   b) By the system against a conservative force
   c) By the system against a non-conservative force
   d) Upon the system by a conservative force

7. A light body and a heavy body have the same kinetic energy. Which one has greater momentum?
   a) The light body
   b) The heavy body
   c) Both have equal momentum
   d) Data insufficient

8. An engine pumps water continuously through a hose. Water leaves the hose with a velocity 'v' and 'm' is the mass per unit length of the water jet. What is the rate at which kinetic energy is imparted to water?
   a) ½ mv²
   b) ½ mv³
   c) mv³
   d) mv²

9. In a perfectly inelastic collision, which of the following is conserved?
   a) Only kinetic energy
   b) Only momentum
   c) Both momentum and kinetic energy
   d) Neither momentum nor kinetic energy

10. A ball is dropped from height 'h' onto a floor. If the coefficient of restitution is 'e', the height to which the ball rebounds after the first bounce is:
    a) eh
    b) e²h
    c) h/e
    d) h/e²

Answers to MCQs:

1. b) -2000 J (Force is 200 N, displacement is 10 m, angle is 180°, W = Fd cos 180° = 200 × 10 × (-1) = -2000 J)
2. d) Zero (Centripetal force is always perpendicular to the instantaneous displacement/velocity, so cos 90° = 0)
3. a) 100% (K = p²/2m. If K becomes 4K (increase by 300%), then p² becomes 4p², so p becomes 2p. Increase in momentum = (2p - p)/p × 100% = 100%)
4. c) Force of friction (Work done by friction depends on the path length)
5. b) 1000 J (Work done by applied force = F_applied × d × cos 0° = 100 N × 10 m × 1 = 1000 J. Note: Work done by gravity would be -mgh = -5×9.8×10 = -490 J)
6. b) By the system against a conservative force (Work done by the system against a conservative force is -W<0xE2><0x82><0x9C>. Since ΔU = -W<0xE2><0x82><0x9C>, if the system does work against the conservative force (e.g., lifting an object against gravity), the potential energy increases.)
7. b) The heavy body (K = p²/2m, so p = √(2mK). If K is constant, p ∝ √m. The heavier body (larger m) has greater momentum.)
8. b) ½ mv³ (Mass flowing out per second = (mass/length) × (length/time) = m × v. KE imparted per second (Power) = ½ (mass/sec) v² = ½ (mv) v² = ½ mv³)
9. b) Only momentum (By definition, KE is not conserved in inelastic collisions, but momentum is conserved if no external force acts.)
10. b) e²h (Velocity just before impact v = √(2gh). Velocity just after rebound v' = ev = e√(2gh). Height reached after rebound h' = v'²/2g = (e² * 2gh) / 2g = e²h.)

Make sure you understand the definitions, the conditions under which work is positive, negative, or zero, the Work-Energy Theorem, the concept of conservative forces, and the principles of conservation of energy and momentum. These are crucial for problem-solving. Go through the solved examples in your NCERT book as well. Let me know if any specific part needs further clarification.