Work, Energy & Power

In physics, work has a precise meaning — it is done only when a force acts on a body and the body moves in the direction of the force.

• W = F × s × cosθ — Work = Force × Displacement × cosine of the angle between them.
• Where W = work, F = force, s = displacement, and θ = angle between force and displacement.
• When force and displacement are in the same direction (θ = 0°), W = F × s since cos0° = 1.
• The SI unit of work is the joule (J).
• 1 joule = 1 newton × 1 metre (1 J = 1 N·m) — work done when a force of 1 N moves a body 1 m in its own direction.
• Work is a scalar quantity — it has magnitude but no direction.
• The CGS unit of work is the erg, where 1 joule = 10⁷ erg.

• A force must act on the body.
• The body must be displaced (move) in the direction of the force.
• If either is missing, no work is done.

The sign of work depends on the angle between the force and the displacement, so the same force can do positive, negative or zero work.

• Example: a body falling freely under gravity — gravity acts downward and the body moves downward.
• Example: a horse pulling a cart forward.

• Work is negative when force and displacement are in opposite directions (θ = 180°).
• Example: friction does negative work on a moving body, since it acts opposite to motion.
• Example: work done by gravity on a body thrown upward (force down, motion up).

• Work is zero when force acts at right angles to displacement (θ = 90°, cos90° = 0).
• Example: a coolie carrying a load on his head and walking horizontally — gravity acts downward, motion is horizontal, so work done against gravity is zero.
• Example: a body in uniform circular motion — the centripetal force is always ⟂ to the velocity, so work done is zero.
• Example: the Moon revolving around the Earth — gravitational pull is ⟂ to its circular path.
• Zero work also occurs when there is no displacement at all (e.g. pushing a wall that does not move).

Energy is the capacity of a body to do work, and a body that can do more work is said to possess more energy.

• Energy and work have the same SI unit — the joule (J).
• Energy is a scalar quantity.
• Larger practical units include the kilojoule (1 kJ = 10³ J) and the kilowatt-hour (kWh).
• Energy can neither be created nor destroyed, only changed from one form to another (law of conservation of energy).

Kinetic energy is the energy a body possesses by virtue of its motion — anything that moves has kinetic energy.

• KE = ½ m v² — where m = mass and v = velocity (speed) of the body.
• KE is directly proportional to the square of the velocity — doubling the speed makes the KE four times as large.
• KE is always positive and is a scalar quantity, measured in joule (J).
• Relation with momentum: KE = p² / 2m, where p = mv is the momentum.
• Examples: a moving car, flowing water, a flying bullet and blowing wind all possess kinetic energy.

Potential energy is the energy stored in a body because of its position or its state (such as being stretched or compressed).

• PE = m g h — Gravitational PE = Mass × Acceleration due to gravity × Height.
• Where m = mass, g = 9.8 m/s², and h = height above the ground.
• The higher a body is raised, the greater its potential energy.
• Examples: water stored in a dam, a stretched bow, a wound spring, and a stone held at a height all possess potential energy.
• The energy stored in a stretched or compressed spring is called elastic potential energy.

The work–energy theorem links the work done on a body directly to the change in its kinetic energy.

• W = ΔKE = ½mv² − ½mu² — where u = initial velocity and v = final velocity.
• If positive work is done, the body speeds up (kinetic energy increases).
• If negative work is done, the body slows down (kinetic energy decreases).
• It shows that work and energy are interchangeable, both measured in joule (J).

Energy exists in many forms, and one form can be converted into another. The chief forms tested in exams are listed below.

• Chemical energy is stored in food, coal, petrol and cells (batteries).
• Nuclear energy is released in the Sun by nuclear fusion and in reactors by nuclear fission.
• The Sun is the ultimate source of almost all energy on Earth.

Energy is constantly changing from one form to another, and most devices we use are simply converters of energy from one form into another.

• A microphone and a loudspeaker are opposites — one converts sound to electrical, the other electrical to sound.
• In every transformation, some energy is lost as heat — but no energy is destroyed.

The law of conservation of energy is one of the most fundamental laws of physics and governs every energy change.

• The total energy of an isolated system always remains constant.
• Total energy before a change = total energy after the change.

• For a falling body, KE + PE = constant (total mechanical energy) at every point.
• At the top: all the energy is potential (KE = 0).
• At the bottom: all the energy is kinetic (PE = 0).
• In between, PE keeps changing into KE, but their sum stays the same.

• At the extreme positions, energy is wholly potential.
• At the mean (lowest) position, energy is wholly kinetic.
• The total energy stays constant as it swings between potential and kinetic forms.

Power tells us not how much work is done, but how quickly it is done — it is the rate of doing work.

• P = W / t — Power = Work done ÷ Time taken.
• The SI unit of power is the watt (W).
• 1 watt = 1 joule per second (1 W = 1 J/s) — power of 1 watt means 1 joule of work is done each second.
• Power is a scalar quantity.
• Power can also be written as P = F × v (Force × velocity).
• The unit watt is named after the scientist James Watt.

• 1 kilowatt (kW) = 1000 watt.
• 1 megawatt (MW) = 10⁶ watt.

Horsepower is an older, practical unit of power still used to rate engines and motors.

• 1 horsepower = 746 watt (often rounded to 750 W).
• It was introduced by James Watt to compare the output of engines with that of horses.
• The power of car and pump motors is often rated in horsepower.

In homes and factories, electrical energy is sold in a much larger unit than the joule — the kilowatt-hour.

• 1 kilowatt-hour (kWh) = 1 unit of electricity (as printed on an electricity bill).
• 1 kWh = 3.6 × 10⁶ J (3.6 million joules) — the energy used by a 1 kW appliance running for 1 hour.
• Derivation: 1 kWh = 1000 W × 3600 s = 3.6 × 10⁶ J.
• kWh is a unit of energy, NOT of power (a common exam trap — the kilowatt is power, the kilowatt-hour is energy).

No machine gives back as much useful energy as it takes in; efficiency measures how much of the input energy a device turns into useful output.

• Efficiency (η) = (useful output ÷ total input) × 100%.
• Efficiency has no unit — it is a ratio, usually expressed as a percentage.
• Efficiency is always less than 100% because some energy is always lost (mostly as heat, sound or friction).
• No machine can be 100% efficient — a perfectly efficient machine is impossible.
• The lost energy is not destroyed — it is converted into other (often unusable) forms like heat and sound.