Work and power

Work

• Work is done when a force moves an object in the direction of the force. Doing work is how energy is transferred.

$$W = Fd$$

• $W$ — work done, measured in joules (J)

• $F$ — the force applied, in the direction of motion (N)

• $d$ — the distance moved in the direction of the force (m)

• One joule is the work done when a force of $1$ N moves an object $1$ m.

• Work done on an object increases its energy; work done by an object decreases it.

When no work is done

Work needs both a force and a displacement along that force. So no work is done when:

• there is no movement — holding a heavy bag still, or pushing on a wall that does not move. You get tired, but no work is done on the object in the physics sense.

• the force is perpendicular to the motion — carrying a bag horizontally does no work against gravity, because the weight acts downward while the motion is sideways.

• there is no force — an object coasting in space.

• For the same reason, the centripetal force does no work on an object in circular motion: it is always at right angles to the velocity, which is why the speed stays constant.

Work done against gravity and friction

• Lifting an object at steady speed means applying a force equal to its weight over the height raised:

$$W = mgh$$

• This work becomes gravitational potential energy.
• Dragging an object at steady speed against friction does work equal to the friction force times the distance:

$$W = fd$$

• This work becomes heat in the surfaces, not stored energy — which is why friction is called dissipative.

Power

• Power is the rate of doing work, or the rate at which energy is transferred:

$$P = \frac{W}{t}$$

• $P$ — power, measured in watts (W)

• $W$ — work done or energy transferred (J)

• $t$ — time taken (s)

• One watt is one joule per second.

• A more powerful machine does the same work in less time, or more work in the same time. It does not do more work overall.

• A useful alternative form, for a force moving at a steady speed:

$$P = \frac{Fd}{t} = Fv$$

• $v$ — the steady speed at which the force acts (m s⁻¹)
• This is why a car's engine must produce more power to maintain a higher speed against the same drag.

Efficiency in words

• No real machine transfers all its input energy usefully — some always becomes heat and sound.
• The useful output is always less than the input, but the total energy is unchanged; the difference has simply been transformed into forms that are not wanted.