The complete lecture — every idea comes alive in the live panel on the right as you read. Scroll down; the animation keeps pace, and in the pendulum and friction sections the panel swaps in real 3D apparatus you can play with yourself.
1 — What physics calls work: W = Fd cos θ
Holding a heavy bag while standing still feels like hard work — but in physics it is zero work, because nothing moved. Work is done only when a force produces a displacement.
- Work — W = Fd cos θ, where θ is the angle between the force and the displacement. It is the scalar (dot) product W = F · d.
- Joule (J) — 1 J = 1 N × 1 m; dimensions [ML²T⁻²]. Work is a scalar — magnitude and sign, no direction.
worked — rope at 60°
50 N rope at 60°, crate dragged 8 m.
W = 50 × 8 × cos 60° = 50 × 8 × 0.5 = 200 J
2 — Positive, zero & negative work
The factor cos θ gives work its sign, and the sign says whether energy flows into the body or out of it.
| Case | θ | Work | Example |
|---|
| Positive | 0° ≤ θ < 90° | W > 0 | horse pulling a cart; gravity on a falling mango |
| Zero | θ = 90° | W = 0 | coolie walking with a load on his head; centripetal force on a satellite |
| Negative | 90° < θ ≤ 180° | W < 0 | friction on a sliding ball; gravity on a rising ball |
Exam point: the normal reaction and centripetal force never do work — both stay perpendicular to the motion.
3 — Work against gravity & against friction
To lift a mass m through height h at steady speed you push with mg, so the work against gravity is W = mgh — and only the height counts, never the path: stairs, ramp or lift give the same mgh.
To drag a crate at steady speed across a rough floor you must balance friction f = μmg, so the work against friction is W = f·d — and this work is not stored; it leaves as heat. A rickshaw dragging its brakes warms the drums for exactly this reason.
Two kinds of “work against”against gravity: W = mgh — stored as PE, path-independent
against friction: W = f × d — lost as heat, path-dependent
worked — ramp vs lift
A 20 kg box: lifted 1.5 m, or dragged 6 m up a ramp with 30 N friction.
gravity: mgh = 20 × 9.8 × 1.5 = 294 J either way · friction adds 30 × 6 = 180 J of heat
4 — Kinetic energy: KE = ½mv² (derivation)
Energy is the ability to do work; kinetic energy is the energy of motion. Start a mass m from rest with force F over distance d, reaching speed v:
Derivationv² = 0 + 2ad → d = v²/2a
W = Fd = (ma)(v²/2a) = ½mv² → KE = ½mv²
KE grows with the square of speed: double a car's speed and its braking energy — and roughly its braking distance — becomes four times larger.
worked — car KE
1200 kg car at 20 m/s, then 40 m/s.
½ × 1200 × 400 = 2.4 × 10⁵ J → at double speed 9.6 × 10⁵ J (4×)
5 — PE = mgh & conservation of mechanical energy
Potential energy is stored by position: lifting m through h banks PE = mgh (from a chosen reference level). Release the body and gravity pays it back as KE — with the total constant. For a 2 kg ball from 20 m (g = 9.8):
| Height | v = √(2g·fall) | PE | KE | PE + KE |
|---|
| 20 m (top) | 0 | 392 J | 0 J | 392 J |
| 10 m (midway) | 14 m/s | 196 J | 196 J | 392 J |
| 0 m (ground) | 19.8 m/s | 0 J | 392 J | 392 J |
Work–energy theorem & free fallW(net) = ΔKE = ½mv² − ½mu²
mgh = ½mv² → v = √(2gh) — independent of mass
6 — Energy interchange: the pendulum (KE ↔ PE)
A swinging bob is the cleanest energy converter in physics. At each extreme it pauses — all its energy is PE (mgh above the lowest point). Sweeping through the lowest point it is fastest — all the energy is KE. In between, the sum stays fixed.
The pendulum bookkeepingat extremes: E = mgh, v = 0 · at the bottom: E = ½mv², h = 0
v(bottom) = √(2gh) — same rule as the falling ball
With air resistance and pivot friction, a little mechanical energy leaks to heat every swing — that is why a playground swing dies away unless you pump it (your muscles feed chemical energy back in).
Try it: swing the 3D bob on the right and watch where it moves fastest and where it stops for an instant.
7 — Power: the rate of doing work
Two porters lift identical trunks to the same floor — equal work — but the one who does it in half the time has twice the power.
PowerP = W / t = energy per second · unit: watt (W) = J/s
P = Fd/t = Fv (constant force along motion)
1 kW = 10³ W · 1 MW = 10⁶ W · 1 hp = 746 W
worked — P = Fv
Car engine: 2000 N driving force at a steady 25 m/s.
P = Fv = 2000 × 25 = 50 kW ≈ 67 hp
8 — Efficiency
No real machine turns all its input energy into useful output — friction and heat always take a cut.
Efficiencyefficiency = (useful output / total input) × 100%
electric motor ≈ 70–90% · petrol engine ≈ 25–30% · human muscle ≈ 25%
worked — pump with 70% efficiency
A pump lifts 200 kg of water per minute through 7 m; efficiency 70%. Input power?
useful = mgh/t = 200×9.8×7/60 ≈ 228.7 W → input = 228.7/0.70 ≈ 327 W
Energy is conserved even here: the “lost” 30% is not destroyed — it becomes heat and sound. Efficiency measures usefulness, not survival.
9 — The kilowatt-hour & energy at full scale
Your electricity bill charges per unit = 1 kilowatt-hour: the energy a 1 kW appliance uses in 1 hour.
kW·h is ENERGY, not power1 kWh = 1000 W × 3600 s = 3.6 × 10⁶ J
energy (kWh) = power (kW) × time (h)
- Tarbela Dam (hydel) — sun lifts sea water (solar → PE) → rain fills the reservoir → falling water (PE → KE) spins turbines (→ work) → ≈ 4888 MW of electrical power lights the country.
- Escalator — electrical energy → mgh per passenger + friction losses: 30 people of 60 kg through 5 m per minute ≈ 1.5 kW useful.
- Rickshaw — petrol (chemical) → KE + heat; at ~25% efficiency, of every 4 rupees of fuel, 3 warm the Karachi air.
worked — the AC bill
1.5 kW AC, 6 h/night, 30 nights.
E = 1.5 × 180 = 270 kWh = 270 × 3.6 × 10⁶ = 9.72 × 10⁸ J
10 — Worked numericals & exam recap
numerical — catching a cricket ball
0.5 kg ball at 30 m/s, hands give 0.25 m.
KE = ½ × 0.5 × 900 = 225 J → F = 225/0.25 = 900 N — draw your hands back!
numerical — speed by energy conservation
Mango falls 4.9 m.
v = √(2gh) = √(2 × 9.8 × 4.9) = 9.8 m/s (mass cancels)
numerical — heater in kWh
2 kW heater for 90 min.
E = 2 × 1.5 = 3 kWh = 1.08 × 10⁷ J
- W = Fd cos θ; joule = N·m; positive / zero (θ=90°) / negative work.
- Against gravity W = mgh (path-free); against friction W = f·d (heat).
- KE = ½mv² (derived from v² = 2ad); PE = mgh from a reference level.
- Work–energy theorem: W(net) = ΔKE.
- Free fall & pendulum: PE + KE constant; v = √(2gh).
- P = W/t = Fv; 1 hp = 746 W; efficiency = useful/total × 100%; 1 kWh = 3.6 × 10⁶ J.