Work, Energy and Power

In physics, work is done when a force causes a displacement of an object in the direction of the force. If you push a box across a floor, you do work. If you push against a solid wall and nothing moves, no work is done — no matter how tired you feel.

Here F is the force (N), x is the displacement (m), and θ is the angle between the force and the displacement. The unit of work is the joule (J): 1 J = 1 N m.

When the force acts parallel to the displacement (θ = 0°), cos θ = 1 and W = Fx. When the force is perpendicular (θ = 90°), cos θ = 0 and no work is done. This is why a satellite in circular orbit experiences no work done by gravity — the force is always perpendicular to the velocity.

The principle of conservation of energy states that energy cannot be created or destroyed — it can only be transferred from one form to another. The total energy of a closed system remains constant. Energy exists in many forms: kinetic, gravitational potential, thermal, elastic, nuclear, chemical, electromagnetic. It can be transferred between these forms, but the total always stays the same.

This principle constrains what is physically possible: no machine can output more energy than is put in, and no process can simply make energy vanish.

Kinetic energy (E_k) is the energy an object possesses because of its motion. The faster it moves and the more massive it is, the more kinetic energy it has.

Gravitational potential energy (E_p) is the energy an object possesses because of its position in a gravitational field. Lifting an object vertically does work against gravity and stores GPE.

Here m is mass (kg), g is gravitational field strength (9.81 N kg⁻¹ near Earth's surface), and h is the change in height (m).

A classic example: a roller coaster at the top of a loop has maximum E_p and minimum E_k. As it descends, E_p is converted to E_k and the coaster speeds up. At the bottom, E_k is maximum and E_p is minimum (assuming the reference level is the lowest point).

Power is the rate at which work is done (or energy is transferred). A powerful machine doesn't necessarily do more work — it does the same work in less time.

The unit of power is the watt (W): 1 W = 1 J s⁻¹. A 60 W light bulb transfers 60 J of energy every second.

The equation P = Fv is particularly useful for vehicles. A car engine must produce enough power to overcome resistive forces (drag, friction) at a given speed. At higher speeds, more power is needed even though the force might stay the same, because v is larger. Conversely, when accelerating from rest, the engine delivers power at low v but high F.

Efficiency measures how much of the input energy is usefully transferred:

No real machine is 100% efficient — some energy is always dissipated, usually as thermal energy due to friction. A car engine might be 25–40% efficient; the rest of the chemical energy from the fuel is lost as heat. Improving efficiency is a major engineering challenge with real consequences for sustainability and cost.