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Work, Energy and Power

Part of the ECAT (Engineering College Admission Test) study roadmap. Physics topic phy-4 of Physics.

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Work, Energy and Power

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your ECAT Physics paper.

Work is the scalar product of force and displacement: W = F·d·cosθ, where θ is the angle between the applied force and the displacement vector. Work is measured in joules (J), and the sign tells the story — positive when force aids motion, negative when it opposes. Energy is the capacity to do work, split into kinetic (KE = ½mv²) and potential (gravitational mgh, elastic ½kx²) forms. Power measures the rate at which work is done: P = W/t (watts), or for moving vehicles P = F·v·cosθ.

  • Work-Energy Theorem: net work on a body equals its change in KE, W = ΔKE = ½m(v_f² − v_i²)
  • Conservation: KE + PE stays constant when only conservative forces act (no friction, no drag)
  • Efficiency = (useful output / total input) × 100%

🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months before ECAT.

Core Definitions and Sign Convention

Work quantifies energy transferred by a force over a displacement. Three cases dominate ECAT questions: W > 0 when θ is acute (force helps motion), W = 0 when force is perpendicular to displacement, and W < 0 when θ exceeds 90° (force opposes motion — friction during sliding is the classic example). Energy is the stored capacity for doing work; the Work-Energy Theorem ties the two ideas together: net work done on a body equals the change in its kinetic energy, W_net = ½mv_f² − ½mv_i².

Potential Energy and Conservative Forces

Gravitational PE = mgh is measured from a chosen reference, so only changes in h matter. Spring PE = ½kx² stores energy in a stretched/compressed spring with stiffness k. Conservative forces (gravity, spring force, electrostatics) allow mechanical energy to be conserved: KE + PE = constant along the path. Non-conservative forces (friction, air drag) dissipate mechanical energy into heat, so total mechanical energy decreases.

Power and Efficiency

Average power P = W/t; instantaneous power for a moving body is P = Fv·cosθ. The horsepower unit (1 hp = 746 W) occasionally appears in numerical MCQs. Efficiency η = (W_output / E_input) × 100% is always ≤ 100% by the second law of thermodynamics.

QuantityFormulaUnits
WorkW = Fd cosθJ
Kinetic EnergyKE = ½mv²J
Gravitational PEPE = mghJ
Spring PEPE = ½kx²J
Average PowerP = W/tW
Instantaneous PowerP = Fv cosθW
  • A 2 kg block moving at 3 m/s has KE = 9 J; doubling speed quadruples KE.
  • Lifting a 5 kg mass by 2 m against gravity stores 98 J of gravitational PE (g = 9.8 m/s²).

🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline before ECAT.

Variable Forces and the Area-Under-Curve Method

ECAT sometimes tests work done by a variable force. When force varies with position, W = ∫F·dx, and on a graph of F vs x the area under the curve equals the work. A spring obeys Hooke’s law F = −kx, so work done stretching it from 0 to x is W = ½kx², recovering the elastic PE expression.

Edge Cases and Common Traps

Watch for these examiner favourites: applying KE = ½mv² using final velocity alone (the theorem demands v_f² − v_i²); mixing up W = Fd·cosθ by ignoring θ; treating power as energy rather than rate; forgetting that on an inclined plane the gravity component along the slope is mg sinθ, not mg. When friction is present, mechanical energy drops by exactly the work done against friction: ΔE_mech = −W_friction.

Worked Micro-Example

A 0.5 kg ball is thrown upward at 10 m/s. Ignoring air drag, find its maximum height. Using KE_i = ½(0.5)(100) = 25 J, and at the top KE = 0, so all 25 J converts to PE: mgh = 25 → h = 25/(0.5 × 9.8) ≈ 5.10 m. Power at launch if delivered over 0.2 s: P = W/t = 25/0.2 = 125 W.

ECAT Strategy

This topic contributes about 4% of the ECAT Physics paper — roughly 1–2 MCQs. Prioritise numericals on the Work-Energy Theorem, energy conservation on inclines and springs, and instantaneous power for vehicles. Budget 90 seconds per question and always carry the units through.

  1. Spring-mass and incline problems appear every year — master energy conservation.
  2. Conceptual MCQs on conservative vs non-conservative forces test conceptual clarity.

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