Work done, gravitational potential energy and kinetic energy

Here, $E$ is work done or energy transferred in joules, $F$ is force in newtons (N), and $d$ is distance moved in the direction of the force in metres (m). A larger force or a greater distance means more energy is transferred. A force of 1 N acting through 1 m transfers 1 J.

For example, pushing a shopping trolley with a constant force of 250 N through 20 m in the direction of the push transfers:

$$E = 250 \times 20 = 5000\text{ J}$$

This tells us how much energy the push transfers; it does not mean that all of that energy must become kinetic energy. Some may be transferred by work against friction.

Measuring work done

To measure work, you need to measure both the force and the distance over which it acts. A newton meter measures force; a ruler or tape measure measures distance.

Attach a newton meter to an object and pull it along a horizontal surface. Keep the meter horizontal and pull steadily so that its reading remains approximately constant. Record the force in newtons. Mark the object's starting and finishing positions and measure the distance between them in metres, along the direction of the pull. Multiply the measured force by this distance to find the work done by the pulling force.

Moving the object at a steady speed does not mean that no work is done. The pulling force still transfers energy, while friction transfers energy to thermal stores rather than allowing the object to gain kinetic energy.

Lifting increases gravitational potential energy

Gravitational potential energy is energy stored because of an object's position in a gravitational field. Raising an object transfers energy to its gravitational potential store; lowering it reduces this store.

The change is calculated using:

$$\Delta GPE = m \times g \times \Delta h$$

The symbol $\Delta$ means ‘change in’. Here, $m$ is mass in kilograms (kg), $g$ is gravitational field strength in newtons per kilogram (N/kg), and $\Delta h$ is the change in vertical height in metres. The energy change is in joules.

This equation connects directly to work done. An object's weight is $mg$. When it is lifted at a steady speed, the upward lifting force equals its weight. Multiplying that force by the vertical distance gives $mg\Delta h$: the work done against gravity equals the gain in gravitational potential energy.

A bucket containing water has a total mass of 12 kg and is raised vertically by 15 m. With $g = 10\text{ N/kg}$:

$$\Delta GPE = 12 \times 10 \times 15 = 1800\text{ J}$$

The bucket gains 1800 J in its gravitational potential store. The lifting force does 1800 J of work if the bucket's speed does not change and other transfers can be neglected. For a fixed mass and gravitational field strength, doubling the height gained doubles the energy gained.

Movement stores kinetic energy

Kinetic energy is the energy an object has because it is moving. Its value depends on both mass and speed:

$$KE = \frac{1}{2} \times m \times v^2$$

Here, $KE$ is in joules, $m$ is mass in kilograms, and $v$ is speed in metres per second (m/s).

At the same speed, doubling an object's mass doubles its kinetic energy. Speed has a stronger effect because it is squared: doubling the speed gives four times the kinetic energy, while tripling it gives nine times the kinetic energy.

Connecting work to changes in energy stores

A force can increase or decrease an object's kinetic energy. A forward force can make an object speed up; braking forces reduce its kinetic energy and transfer energy to thermal stores of the brakes and surroundings.

For a car stopped by a braking force of 500 N over 23 m, the amount of work done by the brakes is:

$$E = 500 \times 23 = 11\,500\text{ J}$$

If braking accounts for the whole decrease, the car loses 11 500 J from its kinetic store. The energy is transferred, not destroyed.

When an object falls, gravity does work and energy is transferred from its gravitational potential store to its kinetic store. If it starts from rest and air resistance is negligible, the kinetic energy gained equals the gravitational potential energy lost:

$$\frac{1}{2}mv^2 = mg\Delta h$$

The same connection applies to a rollercoaster descending a track when friction and air resistance are ignored. If resistive forces matter, some energy is transferred to thermal stores, so the kinetic energy gained is smaller than the gravitational potential energy lost.

• Work done = energy transferred, in J.
• $F$: force in N; $d$: distance in the direction of the force, in m.
• Measure force with a newton meter and distance with a ruler or tape measure; multiply them for a constant force.
• No movement means no work done on the object.

Gravitational potential energy