Work, Energy and Power
'Energy is the currency of the universe.' — Richard Feynman
1. Chapter Overview
WORK, ENERGY, and POWER are INTERCONNECTED concepts that provide an ALTERNATIVE approach to solving mechanics problems. While Newton's laws use FORCE as the central quantity, this chapter uses ENERGY — a SCALAR quantity that often simplifies complex problems. The WORK-ENERGY THEOREM and the LAW OF CONSERVATION OF ENERGY are among the MOST POWERFUL tools in physics.
2. Work
Definition (Physics)
- Work is done when a FORCE causes a DISPLACEMENT
- W = F·d = Fdcosθ (dot product of force and displacement)
- θ is the angle between F and d
Special Cases
| θ | cosθ | Work | Example |
|---|---|---|---|
| 0° | 1 | Positive (F and d in same direction) | Pushing a box |
| 90° | 0 | ZERO | Carrying a bag while walking (force up, displacement forward) |
| 180° | -1 | Negative | Friction opposing motion |
Key Points
- Work is a SCALAR quantity
- SI unit: joule (J) = N·m
- Work done by a VARIABLE force: W = ∫F·dx
3. Energy
- Energy: The CAPACITY to do work (scalar, SI unit: joule)
- Kinetic Energy (KE): Energy due to motion
- K = ½mv²
- Potential Energy (PE): Energy due to position or configuration
- Gravitational PE: U = mgh (near Earth's surface)
- Elastic PE: U = ½kx² (spring, where k = spring constant, x = displacement)
Work-Energy Theorem
- Net work done = Change in Kinetic Energy
- W_net = ΔK = K_f — K_i = ½mv² — ½mu²
- This holds TRUE for both constant and variable forces
Worked Problem
Q: A 2 kg block slides down a 5 m long incline at 30° (μ_k = 0.2). Find speed at bottom using work-energy. A: W_net = mgssinθ — μ_k mgcosθ × s = 2×10×5×½ — 0.2×2×10×√3/2×5 = 50 — 17.3 = 32.7 J ΔK = ½mv² = 32.7 → v² = 32.7 → v = 5.72 m/s
4. Conservation of Mechanical Energy
Statement
For a system with ONLY CONSERVATIVE forces (gravity, spring force), total mechanical energy is CONSERVED:
- E = K + U = CONSTANT
- K₁ + U₁ = K₂ + U₂
Conservative vs Non-conservative Forces
| Conservative | Non-conservative |
|---|---|
| Work path-independent | Work path-dependent |
| Energy is recoverable | Energy dissipates |
| Examples: gravity, spring, electrostatic | Examples: friction, air resistance, viscous drag |
Applications
- Simple pendulum: KE ↔ PE interconversion
- Vertical circular motion
- Spring-mass system
- Freely falling body
Worked Problem
Q: A 0.5 kg pendulum bob is released from rest at 30° to vertical. Find speed at lowest point. (Length = 1 m) A: Height raised = L(1 — cosθ) = 1(1 — cos30°) = 1(1 — 0.866) = 0.134 m Energy conservation: mgh = ½mv² → v = √(2gh) = √(2×10×0.134) = √2.68 = 1.64 m/s
5. Power
- Power: Rate of doing work
- Average Power: P̄ = W/t
- Instantaneous Power: P = dW/dt = F·v (dot product of force and velocity)
- SI unit: watt (W) = J/s
- 1 horsepower (hp) = 746 W
- Kilowatt-hour (kWh): Unit of ENERGY, not power. 1 kWh = 3.6 × 10⁶ J
6. Collisions
Types
| Type | Momentum Conserved? | KE Conserved? | Examples |
|---|---|---|---|
| Elastic | YES | YES | Billiard balls, atomic collisions |
| Inelastic | YES | NO (some KE lost) | Car crash, clay ball sticking |
| Perfectly Inelastic | YES | NO (maximum loss) | Two objects STICK together |
Elastic Collision (1D) — Final Velocities
- v₁' = [(m₁ — m₂)/(m₁ + m₂)]u₁ + [2m₂/(m₁ + m₂)]u₂
- v₂' = [2m₁/(m₁ + m₂)]u₁ + [(m₂ — m₁)/(m₁ + m₂)]u₂
Special Cases
- Equal masses (m₁ = m₂): Velocities EXCHANGE (v₁' = u₂, v₂' = u₁)
- m₁ >> m₂ (heavy hitting light): v₁' ≈ u₁, v₂' ≈ 2u₁ — u₂
- m₁ << m₂ (light hitting heavy): v₁' ≈ -u₁ (rebounds), v₂' ≈ u₂
7. Common Mistakes
- Work can be ZERO even if force is applied: If displacement is zero or perpendicular
- Conservative force work around closed path is ZERO: This is the DEFINING property
- Power is NOT the same as energy: Power is RATE of energy transfer
- Momentum is conserved in ALL collisions, KE only in elastic: Don't assume KE conservation
- Negative work is not 'bad': It just means force opposes displacement
8. CBSE Exam Focus
- Work-energy theorem derivation (3-mark)
- Conservation of mechanical energy (pendulum, free fall problems)
- Power and velocity relation — numericals
- Elastic and inelastic collision problems (5-mark)
- Potential energy of a spring (½kx² derivation)
9. Key Formulas
- W = Fdcosθ
- K = ½mv²
- U_g = mgh
- U_s = ½kx²
- W_net = ΔK
- P = Fv (for constant force in direction of velocity)
- Coefficient of restitution: e = (v₂' — v₁')/(u₁ — u₂). e = 1 (elastic), e = 0 (perfectly inelastic)
10. Self-Test (5+ Q&A)
Q1: A force F = (2î + 3ĵ) N displaces a particle by d = (4î — ĵ) m. Find work done. A: W = F·d = 2×4 + 3×(-1) = 8 — 3 = 5 J.
Q2: A spring (k = 200 N/m) is compressed by 5 cm. Find energy stored. A: U = ½kx² = ½×200×(0.05)² = 0.25 J.
Q3: A 1000 kg car accelerates from rest to 20 m/s in 10 s. Find average power. A: Work = ΔK = ½×1000×400 = 200000 J. P = W/t = 200000/10 = 20000 W = 20 kW.
Q4: In an elastic collision, a 2 kg ball moving at 4 m/s hits a 4 kg ball at rest. Find final velocities. A: Using elastic formulas: v₁' = [(2-4)/(6)]×4 = -4/3 m/s, v₂' = [4/(6)]×4 = 8/3 m/s.
Q5: Can KE be negative? Can PE be negative? A: KE is ALWAYS ≥ 0 (½mv²). PE CAN be negative (reference-dependent). Gravitational PE is negative when object is BELOW reference level.
11. Conclusion
Work, energy, and power give you a SCALAR approach to mechanics — often much simpler than force analysis. The WORK-ENERGY theorem and CONSERVATION OF ENERGY are universal principles that apply far beyond mechanics (thermodynamics, electromagnetism). Collisions show you how momentum and energy interact in the real world. Master these concepts — they are among the most frequently tested in competitive exams.
