Chapter in a nutshell: Work is done only when a force produces displacement ($W=Fs\cos\theta$). Energy is the capacity to do work — chiefly kinetic ($\tfrac12mv^2$) and potential ($mgh$) — and is conserved, only changing form. Power is the rate of doing work ($P=W/t=Fv$). This chapter ties force, energy and power together with their units.
1. Work
In physics, work is done only when an applied force makes a body move (produces displacement). A man pushing a wall that does not move does no work; a coolie standing still with a load on his head does no work (no displacement). $$W = F\,s\cos\theta$$ where θ is the angle between the force and the displacement. Work is a scalar.- It equals (i) force × displacement-component along the force, or (ii) force-component along displacement × displacement.
- Depends on: magnitude of F, magnitude of displacement s, and cos θ.
- For a variable force, work = area under the force–displacement graph (e.g. $\tfrac12 F\,s$ for a triangular graph).
2. Special Cases of Work
| Case | Angle θ | Work | Example |
|---|---|---|---|
| Positive | 0° (cos = +1), $W=Fs$ | +ve | free fall (gravity ∥ motion); coolie raising a load = $mgh$ |
| Zero | 90° (cos = 0) | 0 | coolie walking with load on head; centripetal force in circular motion |
| Negative | 180° (cos = −1), $W=-Fs$ | −ve | friction; gravity on a body thrown up = $-mgh$; fielder catching a ball |
3. Work Done by the Force of Gravity ($W=mgh$)
When a body of mass m moves down a vertical height h, gravity does work $W = mgh$ — independent of the path (stairs, slope or lift give the same value, since only the vertical drop h matters). Raising the body does work $-mgh$ by gravity (or $+mgh$ by the person against gravity).4. Units of Work
| System | Unit | Definition |
|---|---|---|
| SI | joule (J) | work by 1 N over 1 m in its own direction; $1\,\text{J}=1\,\text{N m}$ |
| CGS | erg | work by 1 dyne over 1 cm |
5. Power — rate of doing work
$$P=\frac{W}{t}=\frac{F\times s}{t}=F v\quad(v=\text{average speed})$$ Power depends on (i) the work done and (ii) the time taken. A coolie taking 1 min spends twice the power of one taking 2 min for the same work.| System | Unit | Relation |
|---|---|---|
| SI | watt (W) = 1 J s⁻¹ | $1\,\text{kW}=10^3$, $1\,\text{MW}=10^6$, $1\,\text{GW}=10^9$ W |
| CGS | erg s⁻¹ | $1\,\text{W}=10^{7}\,\text{erg s}^{-1}$ |
| Engineering | horse power (hp) | $1\,\text{hp}=746\,\text{W}=0.746\,\text{kW}$ |
6. Energy — capacity to do work
When a body does work its energy decreases; when work is done on it, its energy increases → work is a transfer of energy. Energy is a scalar; same units as work.| Unit | Value | Use |
|---|---|---|
| joule (J) / erg | $1\,\text{J}=10^{7}$ erg | SI / CGS |
| watt-hour (Wh) | $1\,\text{Wh}=3600\,\text{J}$ | electrical |
| kilowatt-hour (kWh) | $1\,\text{kWh}=3.6\times10^{6}\,\text{J}$ | "unit" of electricity |
| calorie (cal) | $1\,\text{cal}=4.18\,\text{J}$ | heat (1 g water through 1 °C) |
| electron-volt (eV) | $1\,\text{eV}=1.6\times10^{-19}\,\text{J}$ | atomic particles |
Energy vs Power: energy = total capacity to do work (unit J); power = rate at which energy is supplied/spent (unit W).
7. Mechanical Energy: Potential & Kinetic
Mechanical energy = potential energy (PE) + kinetic energy (KE).7.1 Potential Energy (U)
Energy possessed by a body by virtue of its position or configuration. Forms:- Gravitational PE: energy due to position above the ground. Derivation: to lift mass m through height h, least upward force $=mg$, so $U = mg\times h$:
- Elastic PE: energy stored in a deformed body (stretched rubber, compressed/wound spring, bent bow) = work done in deforming it.
7.2 Kinetic Energy (K)
Energy possessed by a body by virtue of its motion. Derivation: a body of mass m moving with speed v is stopped by a retarding force F in distance S; using $v^2=u^2+2aS$ with the work it can do: $$\boxed{K=\tfrac12 mv^2}$$- KE–momentum relation: $K=\dfrac{p^2}{2m}$, i.e. $p=\sqrt{2mK}$.
- Forms of KE: translational (car in a straight line), rotational (spinning top, Earth's spin), vibrational (a struck wire). A rolling body has both translational + rotational KE.
7.3 Work–Energy Theorem
The work done by a net force on a moving body equals the change in its kinetic energy: $$W = \tfrac12 mv^2 - \tfrac12 mu^2 = K_f - K_i$$PE vs KE: PE depends on position/configuration (not speed); KE depends on speed/motion. PE can change only into KE first; KE can change into any form.
8. Conversion of PE into KE
PE converts to KE whenever it is "put to use":- A stone at a height falls → PE → KE → drives a nail.
- A wound watch spring unwinds → elastic PE → KE of the hands.
- A stretched bow → elastic PE → KE of the arrow.
- A compressed spring released → elastic PE → KE of the ball.
9. Principle of Conservation of Energy
Energy can neither be created nor destroyed; it only changes from one form to another, the total remaining constant.
Free fall (verification): for a body of mass m dropped from height H ($v^2=2gx$ after falling x):
| Position | PE | KE | Total (PE+KE) |
|---|---|---|---|
| Top (x = 0) | $mgH$ | 0 | $mgH$ |
| Fallen H/2 | $mgH/2$ | $mgH/2$ | $mgH$ |
| Just before ground (x = H) | 0 | $mgH$ | $mgH$ |
Simple pendulum: PE is maximum at the extreme positions (KE = 0) and KE is maximum at the mean position (PE = minimum); energy oscillates between the two while the total stays constant (ignoring air resistance).
10. Different Forms of Energy & Transformations
Forms: solar, heat, light, chemical, hydro, electrical, nuclear, geothermal, wind, sound, magnetic, mechanical — all inter-convertible.| Device | Transformation |
|---|---|
| Electric bulb | electrical → light + heat |
| Electric motor / fan | electrical → mechanical |
| Loudspeaker | electrical → sound |
| Cell / battery | chemical → electrical |
| Microphone | sound → electrical |
| Solar cell | light → electrical |
| Dynamo / generator | mechanical → electrical |
| Burning fuel | chemical → heat + light |
| Green plant (photosynthesis) | light → chemical |
11. Worked Numerical Examples (ICSE pattern)
Q1. A crane lifts a 500 kg car to a height of 4 m. Work done? (g = 9.8) Solution: $W = mgh = 500\times9.8\times4 = \mathbf{19600\ J}$.Q2. A 10 N force displaces a body 2 m at 60° to the force. Work done? Solution: $W = Fs\cos\theta = 10\times2\times\cos60^\circ = 10\times2\times0.5 = \mathbf{10\ J}$.
Q3. A 40 kg boy climbs 30 steps each 20 cm high in 2 min; a 30 kg girl does the same in 1.5 min. Compare (i) work, (ii) power. (g = 10) Solution: $h = 30\times0.20 = 6$ m. $W_{boy}=40\times10\times6 = 2400$ J; $W_{girl}=30\times10\times6 = 1800$ J → work 4 : 3. $P_{boy}=2400/120 = 20$ W; $P_{girl}=1800/90 = 20$ W → power 1 : 1.
Q4. A 15 N force pulls a 2 kg body 5 m up a 30° incline. Find work by the force and work against gravity. (g = 9.8) Solution: Work by force $=15\times5 = \mathbf{75\ J}$. Height $=5\sin30^\circ = 2.5$ m, so work against gravity $=mgh = 2\times9.8\times2.5 = \mathbf{49\ J}$. The 26 J difference is work against friction.
Q5. Power of an engine to lift $10^5$ kg of coal per hour from a 360 m deep mine. (g = 10) Solution: $W = mgh = 10^5\times10\times360$; $P = W/t = \dfrac{3.6\times10^{8}}{3600} = \mathbf{100\ kW}$.
Q6. A man pulls a cart at constant 16 m s⁻¹ with 200 N force. Power? Solution: $P = Fv = 200\times16 = \mathbf{3200\ W}$.
Q7. Express 5 kWh in joules. Solution: $5\times3.6\times10^{6} = \mathbf{1.8\times10^{7}\ J}$.
12. Key Terms — Quick Glossary
| Term | One-line definition |
|---|---|
| Work | $W=Fs\cos\theta$; done only with displacement; unit J. |
| Energy | capacity to do work; unit J. |
| Power | rate of doing work, $P=W/t=Fv$; unit W. |
| Potential energy | energy due to position/configuration ($U=mgh$). |
| Kinetic energy | energy due to motion ($K=\tfrac12mv^2$). |
| Work–energy theorem | net work = change in KE. |
| Conservation of energy | total energy stays constant; only changes form. |
| 1 kWh | $3.6\times10^{6}$ J (commercial "unit"). |
| 1 hp | 746 W. |
13. Likely Exam Questions (with crisp answers)
- Define work; when is work done by a force? → $W=Fs\cos\theta$; done only when the body is displaced.
- When is work (a) positive, (b) negative, (c) zero? → θ < 90°; θ > 90°; θ = 90° or s = 0.
- A coolie carries a load on his head walking on level ground — work done? → Zero (force ⟂ displacement).
- Work done by a body in one full circular revolution? → Zero (net displacement = 0).
- Why is a satellite's orbital work zero? → Gravity (centripetal) is ⟂ to its displacement.
- Define 1 joule. → Work by a 1 N force moving a body 1 m in its direction.
- State the SI & CGS units of work and their relation. → joule and erg; $1\,\text{J}=10^{7}$ erg.
- Define power; SI unit. → Rate of doing work; watt.
- Differentiate work and power. → Work is force × displacement (no time); power is the rate of doing work (depends on time).
- Derive $K=\tfrac12mv^2$. → From $W=FS$, $F=ma$, $v^2=u^2+2aS$ (u = 0) ⟹ $K=\tfrac12mv^2$.
- State the work–energy theorem. → Net work done = change in kinetic energy.
- State the principle of conservation of energy. → Energy is neither created nor destroyed; total is constant, only the form changes.
- Show energy is conserved in free fall. → PE + KE = $mgH$ at every point.
- Define 1 kWh; relate to joule. → Energy spent by a 1 kW source in 1 h = $3.6\times10^{6}$ J.
- What does the electron-volt measure? → Energy; $1\,\text{eV}=1.6\times10^{-19}$ J.
14. The Twelve Forms of Energy (one-liners)
- Solar energy — radiated by the Sun; used via solar panels, furnaces, cells.
- Heat (thermal) energy — due to random motion of particles; from burning fuel, friction.
- Light energy — form of energy that produces the sensation of sight.
- Chemical (fuel) energy — stored in chemical bonds; released on burning/reaction (food, coal, cells).
- Hydro energy — energy of falling/flowing water; drives turbines for electricity.
- Electrical energy — energy of moving charges; most convenient, easily transmitted.
- Nuclear energy — released in fission/fusion of nuclei; huge energy from a tiny mass.
- Geothermal energy — heat from inside the Earth (hot springs, geysers).
- Wind energy — kinetic energy of moving air; drives windmills.
- Sound energy — produced by vibrating bodies; travels as a wave.
- Magnetic energy — energy stored in a magnetic field.
- Mechanical energy — sum of kinetic and potential energy.
15. More Worked Numericals (ICSE pattern)
Q8. A weight-lifter raises a 200 kgf load to 2.5 m in 5 s. Find (i) work, (ii) power. (g = 10 N kgf⁻¹) Solution: $W = 200\times10\times2.5 = \mathbf{5000\ J}$; $P = 5000/5 = \mathbf{1000\ W}$.Q9. A water pump raises 50 litre of water to 25 m in 5 s. Find its power. (g = 10, density = 1000 kg m⁻³) Solution: mass $= 50\times10^{-3}\times1000 = 50$ kg; $W = mgh = 50\times10\times25 = 12500$ J; $P = 12500/5 = \mathbf{2500\ W}$.
Q10. An electric heater of power 3 kW runs for 10 h. Energy consumed in (i) kWh, (ii) joule? Solution: (i) $E = 3\times10 = \mathbf{30\ kWh}$. (ii) $30\times3.6\times10^{6} = \mathbf{1.08\times10^{8}\ J}$.
Q11. A 100 W motor drives a stirrer; 50 % of the supplied energy stirs the water. Work done on the water in 1 minute? Solution: Useful power $= 50\%\times100 = 50$ W; $W = 50\times60 = \mathbf{3000\ J}$.
Q12. A heart does 1 J of work per beat and beats 72 times per minute. Find its power. Solution: beats per second $= 72/60 = 1.2$; $P = 1\times1.2 = \mathbf{1.2\ W}$.
Q13. The power of a motor is 40 kW. At what speed can it raise a load of 20 000 N? Solution: $v = P/F = 40000/20000 = \mathbf{2\ m\,s^{-1}}$.
Q14. The energy of an electron is $4.0\times10^{-19}$ J. Express it in eV. Solution: $\dfrac{4.0\times10^{-19}}{1.6\times10^{-19}} = \mathbf{2.5\ eV}$.
16. Common Mistakes to Avoid
- Forgetting cos θ in work (a force at an angle does less work than $Fs$).
- Saying a coolie carrying a load on his head "does work" — he does no work against gravity (displacement ⟂ weight).
- Confusing units of power (W, kW) with units of energy (Wh, kWh).
- Writing torque's unit (N m) as joule — joule is reserved for work/energy.
- Forgetting that $W=mgh$ is independent of the path taken.
- Mixing up $K=\tfrac12mv^2$ (quadratic in v) — doubling speed gives four times the KE.
17. More Exam Questions (with crisp answers)
- Why does a fielder lower his hands while catching a ball? → To increase the time/displacement of stopping, reducing the force (work–energy theorem).
- A body's speed is doubled — what happens to its KE? → It becomes four times ($K\propto v^2$).
- Name the energy change in a hydroelectric power station. → PE of water → KE → electrical energy.
- Name the energy change when a ball is dropped and bounces. → PE → KE → (sound + heat at impact) → KE → PE.
- State two factors on which the power of a source depends. → The work done and the time taken.
- Is energy a vector or scalar? → Scalar.
- Define 1 watt. → 1 joule of work done in 1 second.
- Why is the efficiency of a real machine/engine less than 100 %? → Some energy is always lost against friction and as heat/sound.