1The idea in one paragraph

Energy is a quantity that is never created or destroyed; it only moves from one store to another. The way a force moves energy is by doing work: push something along and the energy you transfer is the force times the distance moved in the direction of the force. When nothing resists the motion, a moving body simply trades kinetic energy for potential energy and back, and the total, its mechanical energy, stays fixed. When friction or drag acts, the mechanical energy falls by exactly the work they do, and that energy ends up as internal energy of the surroundings. Power is how fast the transfer happens, efficiency is what fraction of it was useful, and energy density says how much a kilogram of fuel can supply.

2Conservation of energy, and work as a transfer

Principle of conservation of energy: energy cannot be created or destroyed. The total energy of an isolated system is constant; energy can only be transferred from one store to another.

A system is the set of bodies you choose to look at. An isolated system exchanges no energy with anything outside it. Every energy calculation in this subtopic is the same sum: energy in the stores at the start, plus energy put in, equals energy in the stores at the end, plus energy that left.

Energy is moved from one store to another in a few ways, and in mechanics the way is work. When a force moves a body, the force does work on it, and the work done equals the energy transferred. A crane lifting a crate does work on the crate, and the crane's electrical energy becomes the crate's gravitational potential energy. Friction slowing a sliding book does (negative) work on it, and the book's kinetic energy becomes internal energy of the book and the table, which get very slightly warmer.

A Sankey diagram draws those transfers to scale. The input is an arrow whose width stands for the energy supplied; it splits into arrows for each output, the useful one carried straight on, the wasted ones bent away, and the widths always add back up to the input. Figure 1 is an electric motor lifting a load.

Of the 500 J supplied, 320 J becomes gravitational potential energy of the load. The rest is dissipated: 150 J heats the motor's windings and bearings and the air around them, and 30 J leaves as sound. Energy that has spread out into the surroundings as internal energy is still there; it is just no longer useful to anyone. Check any Sankey diagram you draw by adding: 320 + 150 + 30 = 500.

3Work done by a constant force

When a constant force F moves a body through a displacement s, only the part of the force along the displacement does work. Figure 2 shows a suitcase pulled by its handle.

The force splits into F cos θ along the ground and F sin θ upwards. The upward part does not move the case upwards, so it does no work. Only F cos θ, acting through s, transfers energy:

where θ is the angle between the force and the displacement. Three cases are worth knowing by heart.

• θ = 0°, force along the motion: cos θ = 1 and W = Fs, the largest possible.
• θ = 90°, force at right angles to the motion: cos θ = 0 and no work is done. The normal force on a car driving along a level road does no work, and neither does the centripetal force on a body moving in a circle, which is why a body in uniform circular motion keeps the same kinetic energy.
• θ = 180°, force against the motion: cos θ = −1 and the work done is negative. Friction and drag do negative work: they take energy out of the moving body.

Worked example 1. A traveller pulls a suitcase 40 m along a level platform with a force of 60 N along the handle, at 35° to the horizontal. How much work does she do on the case?

Work is a scalar. It has a sign (positive when the force helps the motion, negative when it opposes it) but no direction.

Work as an area. When the force changes as the body moves, W = Fs cos θ cannot be used directly. Plot the force along the displacement against the displacement; the work done is the area under the graph, as Figure 3 shows.

For a constant force, panel (a), the area is a rectangle, Fs. For a spring, panel (b), the force grows in proportion to the extension, F = kx, so the area is a triangle, ½ × x × kx = ½kx². That work is stored in the stretched spring as elastic potential energy. Stretching a spring of spring constant 200 N m⁻¹ by 0.15 m stores ½ × 200 × 0.15² = 2.3 J (2.25 J).

4The three stores of mechanical energy

Mechanical energy is the sum of three stores: kinetic energy, gravitational potential energy and elastic potential energy. The data booklet gives each.

Kinetic energy is the energy of translational motion:

The second form uses momentum p = mv, and it is the one to reach for when a question gives you momentum, as in explosions in A.2. Note the square: double the speed and the kinetic energy is four times as large.

Gravitational potential energy changes when a body moves up or down near the Earth's surface:

Only the change in height matters, so you can put the zero of height wherever is convenient, usually the lowest point. The equation assumes g is constant, which is true only close to the surface; far from the Earth, D.1 replaces it.

Elastic potential energy is stored in a stretched or compressed body that obeys Hooke's law:

where Δx is the extension or compression and k the spring constant, exactly the area in Figure 3(b).

5Work done by the resultant force

The work done by the resultant force on a body equals the change in its energy. For a body on a level surface, where only its kinetic energy changes, this is the most useful tool in mechanics after F = ma, and it follows straight from it: the resultant force F gives an acceleration a = F ÷ m, and over a displacement s, v² = u² + 2as. Multiply through by ½m and

The work done by the resultant force is the change in kinetic energy.

Worked example 2 · braking distance. A 1500 kg car travelling at 20 m s⁻¹ brakes with a constant resultant force of 6000 N. How far does it travel before stopping? How far if it was going twice as fast?

Twice the speed, four times the kinetic energy, four times the braking distance with the same brakes. That square is the physics behind every speed limit near a school.

6Conservation of mechanical energy, and when it fails

If no friction, drag or other resistive force acts, the total mechanical energy of a system is conserved. Energy moves between kinetic, gravitational and elastic stores, and the work done by gravity or by a spring is exactly the amount transferred from one mechanical store to another. Figure 4(a) shows a ball falling: the gravitational potential energy it loses is the kinetic energy it gains, and the sum is a flat line.

This is how energy solves kinematics problems. Take Worked example 2 of A.1 again: a ball is thrown at 12.0 m s⁻¹ from a roof 20.0 m high. Mechanical energy is conserved, so

The mass cancels, and so does the direction of the throw. Thrown up, down or sideways at 12.0 m s⁻¹, the ball lands at 23.2 m s⁻¹, as the A.1 calculation found the long way. When a question asks only for a speed, and gives heights, an energy method is shorter and safer than the equations of motion. It will not give you a time or a direction; for those you still need A.1.

Worked example 3 · a spring launcher. A toy launcher has a spring of spring constant 400 N m⁻¹, compressed by 5.0 cm. It fires a 12 g dart horizontally. Find the launch speed, assuming all the elastic energy becomes kinetic energy.

The same idea gave you the speed at the bottom of the vertical circle in A.2: 4.0 m s⁻¹ at the top of a circle of radius 0.80 m becomes √(4.0² + 2 × 9.8 × 1.6) = 6.9 m s⁻¹ after falling 1.6 m.

When resistive forces act. Friction and drag are non-conservative forces: the work they do does not come back when the motion reverses. The energy leaves the mechanical stores and becomes internal energy of the body and its surroundings. So the rule becomes a bookkeeping rule.

Change in total mechanical energy = work done on the system by non-conservative forces (negative for friction and drag).

Figure 4(b) is the falling ball with air resistance. The total falls as the ball falls, slowly at first, while the ball is slow and the drag is small, and faster lower down.

Worked example 4 · a skateboarder. A 55 kg skateboarder starts from rest at the top of a ramp 4.0 m high. The ramp is 12 m long. Without friction she would reach the bottom at 8.9 m s⁻¹; she actually reaches it at 7.5 m s⁻¹. Find the average resistive force on her.

Figure 5 shows where the energy went.

The 609 J is not destroyed. The wheels, bearings, ramp and air are warmer by that much in total, which is too spread out to notice or to use.

7Power

Power is the rate of doing work, or the rate of transferring energy:

It is measured in watts, W, where 1 W = 1 J s⁻¹. (Careful: W in italics is work, W upright is the watt.) The second form follows from the first: work Fs done in time t gives power Fs ÷ t, and s ÷ t is the speed. Use P = Fv when a force keeps something moving at constant velocity.

Worked example 5 · a lift. A motor raises an 800 kg lift at a steady 2.0 m s⁻¹. At constant velocity the resultant force is zero, so the cable tension equals the weight.

The same logic applies to a car on a level road. At a steady 30 m s⁻¹, the driving force equals the total resistive force, say 600 N, so the engine delivers P = 600 × 30 = 18 kW to the wheels. The power goes into pushing air aside and flexing tyres, not into speeding the car up. The steady speed needs power even though the kinetic energy is not changing.

8Efficiency

Efficiency is the fraction of the energy supplied that ends up where you wanted it:

It is a ratio of like quantities, so it has no unit, and it is written as a decimal (0.64) or a percentage (64%). It is never greater than 1: a device cannot put out more energy than it takes in. An answer above 1 means the ratio is upside down.

The motor in Figure 1 has η = 320 ÷ 500 = 0.64. If the lift motor of Worked example 5 draws 25 kW of electrical power from the supply, its efficiency is 15.7 ÷ 25 = 0.63, and 9.3 kW is dissipated in the motor, the cables and the pulleys. Using powers and using energies give the same answer, because both measure the same transfer over the same time.

9Energy density of fuel sources

The energy density of a fuel is the energy it releases per kilogram, in J kg⁻¹ (in practice MJ kg⁻¹, where 1 MJ = 10⁶ J). It decides how heavy a vehicle's fuel or battery must be for a given journey, and it is one reason the world's energy supply looks the way it does. Figure 6 compares typical, rounded values.

Some tables quote energy density per cubic metre instead, in J m⁻³, which matters for a fuel tank's size rather than its mass. The unit tells you which is meant; read it before you divide.

Worked example 6 · fuel for a journey. A car needs 18 kW at the wheels for one hour. Its petrol engine is 25% efficient and petrol has an energy density of 46 MJ kg⁻¹. What mass of petrol does it burn?

The step that goes wrong is the efficiency. The fuel must supply more than the useful energy, so you divide by 0.25; multiplying would make the engine a source of energy.

Two things in Figure 6 are worth noticing. Fossil fuels pack roughly 30 to 55 MJ into each kilogram, while a lithium-ion battery holds around 0.6 MJ, less than a seventieth of petrol, which is why electric cars carry batteries weighing hundreds of kilograms, a disadvantage partly offset by the fact that electric motors are far more efficient than petrol engines. And nuclear fuel, with an energy density about a million times that of any chemical fuel, sits off the chart altogether. E.4 explains where that energy comes from.