Energy and Work Calculations

A person lifting a box does work against its weight. Energy is transferred to the box’s gravitational potential energy store as it rises. A force that speeds up a trolley transfers energy to its kinetic energy store. In both cases, the force causes movement in its direction.

Calculate the energy transferred

For a force acting in the direction of movement,

$E=F\times d$

Here $E$ is the work done—and therefore the energy transferred—in joules (J), $F$ is the force in newtons (N), and $d$ is the distance moved in the direction of that force in metres (m). Thus $1\,\text{N m}=1\,\text{J}$.

Suppose a box is lifted vertically through $2\,\text{m}$ by an upward force of $100\,\text{N}$. The work done by that force is $100\times2=200\,\text{J}$. If no significant energy is transferred elsewhere, the box’s gravitational potential energy store increases by $200\,\text{J}$. The relevant distance is its vertical rise, not the length of a sloping route used to carry it.

When more than one force does work

Forces can transfer energy in different directions at the same time. A push can increase an object’s kinetic energy store, while friction opposes its motion and transfers energy to thermal stores of the object and its surroundings.

Imagine a crate pushed $4\,\text{m}$ along a floor by a $60\,\text{N}$ force, with a $20\,\text{N}$ frictional force opposing its motion. The push does $60\times4=240\,\text{J}$ of work on the crate. Work done against friction is $20\times4=80\,\text{J}$. If there are no other energy transfers, the crate’s kinetic energy store increases by $240-80=160\,\text{J}$. The energy has been redistributed, not lost.

A braking force provides the reverse kind of example: it acts against a moving vehicle, so the vehicle’s kinetic energy store decreases. The work done against the braking force gives the amount of energy transferred away from that store. It is useful to calculate this as a positive amount and describe separately which store decreases.