Work, energy and power

📚The physics

Work is a transfer, not a thing. A body does not contain work; work is what happens when a force moves its point of application.

where \(\theta\) is the angle between the force and the displacement. Three consequences fall straight out of that cosine and they are worth more marks than the formula itself. A force perpendicular to the motion does no work — which is why the tension in a string does no work on a ball whirled in a circle, and why the normal force does no work on a sliding block. A force opposing the motion does negative work, as friction does. And if nothing moves, no work is done however hard you push.

When the force varies, the formula stops working and you use the graph instead: the work done is the area under the force–distance graph. For a spring obeying Hooke’s law that area is a triangle, which is exactly where the \( \tfrac{1}{2}kx^{2} \) comes from.

The three stores you need.

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

A useful identity: since \( W = \Delta E_k \), the resultant force multiplied by distance gives the change in kinetic energy. That is often a one-line route to an answer that would take three suvat steps.

Conservation of energy is the most powerful tool in this theme, because it skips the middle of a problem entirely. A ball rolling down a curved track has a horrible force analysis and a trivial energy analysis: \(mgh\) 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 gone to internal energy, and you can calculate exactly how much.

Power is the rate of energy transfer, \( P = W/t \). The version that catches people out is \( P = Fv \), which follows from \( W = Fs \) divided by \(t\). It is the form to reach for whenever a question mentions a vehicle moving at constant speed: at constant speed the driving force equals the resistive force, so the engine power equals resistance × speed.

Efficiency is useful output divided by total input, as a fraction or a percentage, and it is never greater than 1. A Sankey diagram shows the same information as a picture: the width of each arrow is proportional to the energy per second, the arrows always sum to the input, and the wasted branches point downwards. If your arrows do not add up, the diagram is wrong.