Work, Energy and Power

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Work — Force and Displacement:

Work is done when a force causes displacement in the direction of the force. The mathematical formula is W = F × d × cosθ, where F is the force applied, d is the displacement, and θ is the angle between the force direction and the displacement direction. SI unit: Joule (J). Work is a scalar quantity.

Key facts:

• When θ = 0° (force in same direction as motion): W = Fd (maximum)
• When θ = 90° (force perpendicular to motion): W = 0 (e.g., centripetal force in uniform circular motion)
• When θ = 180° (force opposite to motion): W = -Fd (negative — force does work against motion)

Practical example: An NABTEB student carries a 3 kg load of books horizontally for 10 m at constant speed. The work done against gravity = 0 (since displacement is horizontal, not vertical). The work done against friction depends on the force applied — at constant speed, applied force = friction force, and the net work = 0.

⚡ NABTEB Tip: In NABTEB, always identify the force doing work and the displacement in the same direction. Lifting a load vertically: W = mgh. Lowering a load slowly (braked): W = -mgh (work done by gravity is positive, but the student doing the lowering does negative work). Throwing a ball upward: work done by gravity = -mgh (negative, gravity opposes motion).

---

🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Kinetic Energy (KE) and Potential Energy (PE):

KE = ½mv². This is always positive. A lorry of mass 2000 kg travelling at 15 m/s has KE = ½ × 2000 × 225 = 225,000 J.

Gravitational PE = mgh. This depends on the reference level chosen for h = 0. Only changes in PE matter physically — the absolute value is arbitrary.

Energy transformation: When a ball falls from height h, its PE decreases from mgh to 0 while its KE increases from 0 to ½mv². At any point during the fall: ½mv² + mgh’ = mgh (total energy conserved if no air resistance).

Power — Rate of Doing Work:

Power = Work done / Time = Energy transformed / Time. SI unit: Watt (W). 1 Watt = 1 Joule per second.

Average power = Total work / Total time. Instantaneous power = Force × velocity = Fv.

Example: A mining machine lifts 500 kg of ore through 20 m in 10 seconds. Work done = mgh = 500 × 10 × 20 = 100,000 J. Average power = 100,000/10 = 10,000 W = 10 kW. The useful power output is 10 kW. If the motor input power is 12 kW, efficiency = 10/12 × 100% = 83.3%.

⚡ NABTEB Tip: The formula W = Fd cosθ requires that d is the displacement of the point of application of the force. If a force acts at an angle, only the component F cosθ in the direction of displacement does work. This is a very common mistake in NABTEB exams.

---

🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Conservation of Mechanical Energy:

When only conservative forces (gravity, spring force) act, total mechanical energy (KE + PE) is conserved. When non-conservative forces (friction, air resistance, tension with dissipation) act, mechanical energy changes — the difference equals work done by non-conservative forces.

For a pendulum: at angle θ from vertical (string length L), height h = L - L cosθ = L(1 - cosθ). PE = mgL(1 - cosθ). At the bottom (θ = 0): v = maximum, PE = 0, KE = ½mv² = mgL(1 - cosθ_max). Therefore v_max = √(2gL(1 - cosθ_max)). This is derived from energy conservation, not kinematics.

Collisions:

Elastic collision (e = 1): Both momentum AND kinetic energy conserved. For m₁ at u₁ colliding with stationary m₂: v₁ = ((m₁ - m₂)/(m₁ + m₂)) × u₁ v₂ = (2m₁/(m₁ + m₂)) × u₁

Perfectly inelastic (e = 0): Objects stick together. Only momentum conserved — KE is lost. v = (m₁u₁ + m₂u₂)/(m₁ + m₂). Fraction of KE lost = 1 - 1/(1 + m₁/m₂)².

Efficiency of Machines:

Efficiency η = (Useful energy output / Total energy input) × 100%. For a system with multiple energy transformations, overall efficiency = product of individual efficiencies. Example: A generator has efficiency 80% and a motor has efficiency 90%. Overall efficiency = 0.80 × 0.90 = 72%.

⚡ NABTEB Pattern: NABTEB frequently asks: “A machine does 500 J of work in 10 seconds. What is its power?” Answer: P = W/t = 500/10 = 50 W. Or: “Calculate the kinetic energy of a 2 kg ball moving at 3 m/s.” Answer: KE = ½ × 2 × 9 = 9 J. Common errors include forgetting to square the velocity and mixing up units.