Work, energy and power

Work and energy are the same quantity seen from two sides: work is the transfer, energy is the amount transferred, and both are measured in joules. That single sentence is why this topic is so powerful — it lets you answer questions about the start and the end of a process without knowing anything about the messy middle.

🔨When work is done

Work is done when a force moves its own point of application in the direction of that force. If the object does not move, or moves at right angles to the force, no work has been done — however much effort it took.

The amount of energy transferred is equal to the work done, so the units must match: work is measured in joules, and 1 J = 1 N m. Work is a scalar, even though it is built from two vectors.

where \(\theta\) is the angle between the force and the displacement. Only the component of the force along the displacement, \( F\cos\theta \), does any work at all.

The middle case is the one that earns marks: a force perpendicular to the motion transfers no energy. That is why the normal force does no work on a sliding block, why the tension does no work on a ball whirled in a circle, and why carrying a suitcase along a level corridor does no work on it, however tired your arm gets.

Two routes to the same place make the point about machines. Lifting a crate straight onto a truck needs a large force over a short distance; pushing it up a ramp needs a smaller force over a longer one.

Work from a graph

When the force is not constant the formula stops working, and you use the graph instead: the work done is the area under the force–displacement graph.

⚡Energy and its stores

Energy is the capacity to do work. It comes in many forms — kinetic, gravitational potential, elastic potential, thermal, chemical, nuclear, electrical, radiant — but three have equations you must be able to use:

Note the square on the speed: doubling the speed quadruples the kinetic energy, which is the whole argument behind speed limits and braking distances. The gravitational form applies near the Earth’s surface, where \( \Delta h \) is the vertical height change — the path taken is irrelevant. In the elastic form, \( \Delta x \) is the extension from the natural length, not the total length.

Mechanical energy is the sum of the kinetic, gravitational potential and elastic potential energies. In the absence of frictional and resistive forces, the total mechanical energy of a system is conserved.

Conservation of energy

In any situation you must be able to account for the changes in energy. If energy is “lost” by one object it must be gained by another. Three ways of saying the same principle:

• the total energy of any closed system is constant;
• energy is never created or destroyed, only changed in form;
• there is no change in the total energy of the Universe.

Conservation is powerful because it skips the middle of a problem. A ball rolling down a curved track has a horrible force analysis and a trivial energy one: \( mgh \) at the top becomes \( \tfrac{1}{2}mv^{2} \) at the bottom, so \( v = \sqrt{2gh} \) and the shape of the track never enters the calculation. Add friction and the same statement still works — the missing energy has become internal energy, and you can say exactly how much.

🔋Power and efficiency

Power is the rate at which energy is transferred, which is the same as the rate at which work is done:

There is a second form worth having ready. If something moves at constant velocity \(v\) against a constant resistive force \(F\), then in each second it travels \(v\) metres and does \(Fv\) joules of work:

Reach for that one whenever a question mentions a vehicle at steady speed: at constant speed the driving force equals the resistive force, so the engine’s useful power output is simply resistance × speed.

Efficiency is the ratio of useful energy out to total energy in. Because it is a ratio, it has no units, and it can never exceed 1:

Whether energy counts as “useful” depends on what you wanted. In a filament lamp the light is useful and the thermal energy is waste; in an electric heater the thermal energy is the whole point, and such a heater is essentially 100% efficient.

📈Sankey diagrams

Almost all of the world’s electrical power is generated the same way: a fuel releases thermal energy, the thermal energy boils water, the steam turns a turbine, and the turbine turns a generator.

Any cyclical process must dump some energy into the surroundings where it can no longer do useful work. That is the second law of thermodynamics talking, and it is why no power station is anywhere near 100% efficient. A Sankey diagram shows exactly where the energy went: an arrow drawn left to right, with the width of every branch proportional to the energy or power it carries, and losses branching off up or down.

⛰Energy density and specific energy

Two similar-sounding quantities, and questions do test the difference:

• Specific energy is the energy liberated per unit mass of fuel consumed, in J kg\(^{-1}\). It is what matters when you have to carry the fuel — aircraft, rockets, your own body.
• Energy density is the energy liberated per unit volume of fuel consumed, in J m\(^{-3}\). It is what matters when you have to store the fuel in a tank of fixed size.

Comparing energy sources

No source is best on every measure, and the IB expects you to be able to argue the trade-offs rather than recite a list.

• Fossil fuels — coal, oil, gas. High specific energy and energy density, cheap, easy to transport, and a station can be built almost anywhere. Against: they are non-renewable, their combustion products include greenhouse gases and the pollutants that cause acid rain, and extracting them damages the environment.
• Solar, wind, wave. No chemical by-products, renewable, and the energy itself is free. Against: the supply is unreliable and depends on weather, location and time of day, the initial cost is high, and a very large area is needed for a significant output.
• Hydroelectric and tidal. Clean and renewable, and hydro is reliable in a way that wind is not. Against: they need particular geography, and building a dam floods land. Pumped storage belongs here but is not a source at all — it is one of the few large-scale ways of storing energy, filling in when demand peaks.
• Geothermal. Clean, renewable and steady, but only usable where the geology allows it.
• Nuclear fission. An enormous energy density and no carbon dioxide during operation. Against: the waste stays radioactive for a very long time, decommissioning is expensive, and public acceptance is hard to win. Fusion is a different prospect again — effectively unlimited fuel and far less radioactive waste — but the engineering of confining a plasma at scale is still unsolved.