Work, energy and power for NDA

The NDA syllabus asks for "elementary ideas of work, power and energy", and that is exactly the level of the questions. You will not see long derivations. You will see a force at an angle, a body dropped from a height, a motor lifting water, or a statement such as "a porter walking on a level road does no work against gravity" that you must judge true or false.

Two things make this chapter score well. The first is knowing precisely when work is zero, positive or negative, because statement questions hinge on it. The second is fluency with the ratios: what happens to kinetic energy when speed doubles, or when momentum doubles. Those questions need no arithmetic at all once the relationships are clear.

Typical question types are:

• whether work is done in a given situation, and its sign;
• one-step numericals with W = Fs cos θ, KE = ½mv² or PE = mgh;
• how kinetic energy changes with speed, mass or momentum;
• energy conversions in devices (motor, dynamo, solar cell, loudspeaker);
• power, horsepower and the kilowatt-hour;
• efficiency, and the three classes of lever.

Work: the exact definition

Work is done when a force moves its point of application. For a constant force F making an angle θ with the displacement s:

W = F s cos θ

The SI unit is the joule: 1 J is the work done when a force of 1 N moves its point of application 1 m along its own direction. Work is a scalar, even though force and displacement are vectors.

Work is also zero when there is no displacement. A person pushing hard against a wall, or holding a heavy bag still, feels tired because muscles use energy internally, but in physics no work is done on the wall or the bag.

Energy: the capacity to do work

Energy has the same unit as work, the joule. NDA mainly asks about mechanical energy and about conversions between forms.

Kinetic energy

A body of mass m moving at speed v has KE = ½mv². Two consequences come up again and again:

• KE depends on the square of speed. Double the speed and KE becomes 4 times; triple it and KE becomes 9 times.
• KE and momentum are linked: KE = p²/(2m), or p = √(2m × KE). If momentum doubles (at the same mass), KE becomes 4 times. If two bodies have equal KE, the heavier one has more momentum. If they have equal momentum, the lighter one has more KE.

The work–energy theorem says the net work done on a body equals the change in its kinetic energy. A braking force F that stops a car over a distance d does work Fd = ½mv². So at a fixed braking force, stopping distance is proportional to v²: a car at twice the speed needs four times the distance to stop. This is why speed limits matter far more than they seem to.

Potential energy

Energy stored because of position or shape.

• Gravitational: PE = mgh near the Earth's surface, with h measured from a chosen reference level.
• Elastic: a stretched or compressed spring stores ½kx², where k is the spring constant and x the extension. A wound watch spring, a drawn bow and a stretched catapult are examples.

Conservation of energy

Energy can neither be created nor destroyed; it only changes form. The total energy of an isolated system stays constant.

For mechanical energy, if only gravity does work (no air resistance or friction), KE + PE stays constant:

• A body dropped from height h loses PE and gains the same amount of KE. It lands at speed √(2gh), whatever its mass.
• A swinging pendulum has maximum PE and zero KE at the extreme positions, and maximum KE at the lowest point.
• A ball thrown upward trades KE for PE until it stops at the top.

When friction acts, mechanical energy is not conserved, but total energy still is: the "lost" mechanical energy becomes heat and sound.

Power

Power is the rate of doing work: P = W/t. Its SI unit is the watt (1 W = 1 J/s). Because W = Fs, power can also be written P = Fv for a force F acting on a body moving at speed v along the force.

• 1 kilowatt = 1,000 W. 1 horsepower ≈ 746 W.
• The kilowatt-hour is a unit of energy: 1 kWh = 1,000 W × 3,600 s = 3.6 × 10⁶ J. It is the "unit" on an electricity bill.

Two people climbing the same staircase do the same work if they weigh the same, but the one who climbs faster develops more power.

Efficiency and simple machines

No real machine gives out all the energy it takes in; some is lost as heat to friction.

Efficiency = (useful output ÷ input) × 100%, always less than 100% for a real machine.

A machine does not reduce the work needed; it lets a smaller effort do the job over a longer distance. Mechanical advantage (MA) = load ÷ effort.

A single fixed pulley has an ideal MA of 1; it only changes the direction of the effort, which makes lifting easier because you can pull down with your body weight. A single movable pulley has an ideal MA of 2.

Worked NDA-style MCQs

Q1. A force of 10 N moves a body 5 m, acting at 60° to the displacement. The work done is:
(a) 25 J (b) 50 J (c) 43.3 J (d) 0

W = Fs cos θ = 10 × 5 × cos 60° = 50 × ½ = 25 J. Answer: (a).

Q2. A 2 kg stone is dropped from a height of 20 m (g = 10 m/s², no air resistance). Its kinetic energy just before hitting the ground is:
(a) 200 J (b) 40 J (c) 800 J (d) 400 J

All the PE becomes KE: mgh = 2 × 10 × 20 = 400 J. Check: v = √(2gh) = √400 = 20 m/s, and ½ × 2 × 20² = 400 J. Answer: (d).

Q3. A 50 kg student runs up a flight of stairs 10 m high in 20 s (g = 10 m/s²). The power developed is:
(a) 25 W (b) 250 W (c) 500 W (d) 2,500 W

Work = mgh = 50 × 10 × 10 = 5,000 J. P = 5,000/20 = 250 W. Answer: (b).

Q4. Two bodies of mass 1 kg and 4 kg have equal kinetic energy. The ratio of their momenta is:
(a) 1 : 4 (b) 4 : 1 (c) 1 : 2 (d) 1 : 16

p = √(2mK). With K equal, p is proportional to √m: √1 : √4 = 1 : 2. Answer: (c).

Q5. A pump lifts 600 kg of water to a height of 10 m in 1 minute (g = 10 m/s²). Its useful power output is:
(a) 100 W (b) 1 kW (c) 6 kW (d) 10 kW

Work = 600 × 10 × 10 = 60,000 J in 60 s, so P = 1,000 W = 1 kW. Answer: (b).

Q6. A car's speed is doubled. With the same braking force, its minimum stopping distance becomes:
(a) the same (b) double (c) four times (d) half

Braking work Fd equals ½mv², so d is proportional to v². Answer: (c).

Common mistakes

• Saying work is done whenever effort is felt. Pushing a wall or holding a load still does no work on it.
• Forgetting cos θ. Only the component of force along the displacement does work.
• Doubling KE when speed doubles. KE goes as v², so it becomes four times.
• Calling the kilowatt-hour a unit of power. It is energy; the kilowatt is power.
• Thinking a machine saves work. It saves effort; the work in is at least the work out.

Practice set

1. The work done by the centripetal force on a body in uniform circular motion is: (a) positive (b) negative (c) maximum (d) zero
2. 1 horsepower is about: (a) 746 W (b) 546 W (c) 1,000 W (d) 100 W
3. A 5 kg body is 4 m above the ground (g = 10 m/s²). Its potential energy is: (a) 20 J (b) 50 J (c) 200 J (d) 2,000 J
4. If the momentum of a body is doubled, its kinetic energy becomes: (a) double (b) half (c) unchanged (d) four times
5. A 1,000 W heater runs for 2 hours. The energy used is: (a) 2 × 10³ J (b) 3.6 × 10⁶ J (c) 7.2 × 10⁶ J (d) 7.2 × 10³ J
6. A nutcracker is a lever of: (a) class I (b) class II (c) class III (d) no class
7. A machine takes in 500 J and gives out 400 J of useful work. Its efficiency is: (a) 80% (b) 125% (c) 20% (d) 100%
8. A dynamo converts: (a) electrical energy into mechanical (b) mechanical energy into electrical (c) chemical energy into electrical (d) heat into electrical

Answers:

1. (d). The force points to the centre, at 90° to the motion.
2. (a). Worth memorising along with 1 kWh = 3.6 × 10⁶ J.
3. (c). mgh = 5 × 10 × 4.
4. (d). KE = p²/(2m), so doubling p multiplies KE by 4.
5. (c). 1 kW × 2 h = 2 kWh = 2 × 3.6 × 10⁶ J.
6. (b). The load (the nut) lies between the hinge and your hand.
7. (a). 400/500 × 100.
8. (b). A motor does the reverse.

What to do next

• Draw the angle table from memory and add one everyday example of your own to each row.
• Practise the ratio questions (speed doubled, momentum doubled, equal KE, equal momentum) until you can answer each in under 15 seconds.
• Solve 20 numericals on work, PE, KE and power with g = 10, writing the unit every time.
• Revise laws of motion and momentum if the momentum link felt shaky, then move on to gravitation.