Work, Energy & Power

Work is done when a force moves an object through a distance.

Formula: W = F × s × cos θ, where θ is the angle between force and displacement.

• Maximum work when F is along motion (θ = 0°).
• Zero work when F is perpendicular to motion (θ = 90°) — like the centripetal force on a satellite.
• Negative work when F opposes motion (friction).

Unit: joule (J) = 1 N × 1 m. 1 kJ = 1000 J. 1 calorie ≈ 4.18 J. 1 kWh = 3.6 × 10⁶ J.

Energy is the capacity to do work. Energy units = work units (joule).

Forms of energy:

• Kinetic energy (KE) = ½ mv². The energy a body has by virtue of its motion.
• Potential energy (PE) at height h above ground = mgh.
• Elastic PE of a stretched spring = ½ kx².
• Heat, electrical, chemical, nuclear, sound, light — all are forms of energy.

Law of conservation of energy: energy cannot be created or destroyed, only transformed. A falling stone converts PE → KE. A bulb converts electrical → light + heat.

Mechanical energy (PE + KE) is conserved when only conservative forces (like gravity, spring) act. With friction, mechanical energy → heat (a non-conservative force converts energy to a "spread out" form).

Power = rate of doing work = work / time.

• Formula: P = W/t. Also P = F × v (force times velocity for steady motion).
• Unit: watt (W) = 1 J/s. 1 horsepower (HP) ≈ 746 W.

Efficiency = useful output / total input × 100%. A 60 W bulb that emits 5 W of light has only ~8% efficiency; the rest is heat.

Work-energy theorem: Work done by net force on a body = change in its kinetic energy.
W_net = KE_final − KE_initial.

Example 1 — Power calculation:
A man lifts a 50 kg box 2 m in 5 seconds. Power exerted?
Method: Work = mgh = 50 × 10 × 2 = 1000 J. Power = 1000 / 5 = 200 W.

Example 2 — Falling object:
An object falls from a height of 80 m. Find its velocity just before impact (ignore air resistance, g = 10).
Method: Use v² = u² + 2gs → v² = 0 + 2×10×80 = 1600 → v = 40 m/s.
Equivalently: PE at top = KE at bottom: mgh = ½mv² → v = √(2gh) = √1600 = 40 m/s.

Example 3 — Electrical work:
An electric kettle is rated 1500 W. Energy consumed in 6 minutes?
Method: Energy = power × time = 1500 × 360 s = 540,000 J = 540 kJ ≈ 0.15 kWh.

Example 4 — Spring:
A spring with constant k = 200 N/m is stretched by 0.05 m. Elastic PE stored?
Method: PE = ½kx² = ½ × 200 × 0.0025 = 0.25 J.

Example 5 — Power of a car:
A car of mass 1000 kg accelerates from 0 to 20 m/s in 5 s. Average power developed?
Method: KE gained = ½ × 1000 × 20² = 200,000 J. Time = 5 s. Avg power = 200000/5 = 40,000 W = 40 kW.

Quick-recall energy values:

• A 60 W bulb running for 1 hour uses 60 × 3600 = 216,000 J = 0.06 kWh of energy.
• 1 unit of electricity (1 kWh) costs about ₹6–8 in Indian residential tariffs.
• A typical human's resting power consumption: ~80–100 W (basal metabolic rate).

RRB favorite: questions where a body slides down a frictionless incline. Use energy conservation: mgh = ½mv² → v depends only on height, not on slope angle.

Trap: confusing power (rate of energy use) with energy (total used). A 1000 W appliance uses 1000 J/s; in 1 hour, it consumes 1 kWh of energy.