Work Done and Energy Transfer

In physics, "work" has a precise meaning that is narrower than in everyday language.

Work is done when a force makes an object move through a distance in the direction of that force.

Holding a heavy shopping bag still might feel like hard work, but in physics terms no work is done on the bag, because it is not moving. Work requires both a force and movement in the direction of that force.

When work is done, energy is transferred from one store to another. The amount of energy transferred is exactly equal to the work done.

Work done = energy transferred.

For example, lifting a box transfers energy from your chemical (food) store to the box's gravitational potential store. Pushing a crate along the floor against friction transfers energy to a thermal store. The bigger the force and the further the object moves, the more work is done and the more energy is transferred.

Work done depends on the size of the force and the distance moved along the line of that force.

$W=Fs$

where $W$ is work done in joules (J), $F$ is the force in newtons (N) and $s$ is the distance moved along the direction of the force in metres (m).

You must recall and apply this equation.

Worked example — a person pushes a box with a steady force of 25 N and it moves 4.0 m along the floor in the direction of the push. Calculate the work done.

$W=Fs=25\times 4.0=100\text{J}$

The work done is 100 J, and 100 J of energy is transferred in the process.

Worked example — a crane lifts a load, applying an upward force of 800 N to raise it 3.0 m. Calculate the work done.

$W=Fs=800\times 3.0=2400\text{J}$

The work done is 2400 J (or 2.4 kJ), transferred to the load's gravitational potential energy store.

Work done is measured in joules (J), the same unit as energy, because work done is energy transferred.

One joule is the work done when a force of one newton moves an object a distance of one metre: $1\text{J}=1\text{N⋅m}$.

This gives a handy conversion. A value in newton-metres is numerically the same as a value in joules, so no arithmetic is needed to switch between them.

Worked example — a force does 350 N·m of work on a trolley. Express this in joules.

Because 1 J = 1 N·m, the work done is simply 350 J.

Worked example — a machine transfers 6.0 kJ of energy. Express this work done in newton-metres.

First convert kilojoules to joules: 6.0 kJ = 6000 J. Then, since 1 J = 1 N·m, the work done is 6000 N·m.

Being fluent in this conversion saves time and avoids errors when a question mixes the two units.

If you know the work done and one of the other quantities, rearrange $W=Fs$ to find the third.

To find the force:

$F=\frac{W}{s}$

To find the distance moved:

$s=\frac{W}{F}$

Worked example — a cyclist does 4500 J of work travelling 300 m along a flat road at steady speed. Calculate the average resistive force the cyclist works against.

$F=\frac{W}{s}=\frac{4500}{300}=15\text{N}$

The average force is 15 N.

Worked example — a motor does 2000 J of work pulling a cart with a constant force of 250 N. Calculate the distance the cart moves.

$s=\frac{W}{F}=\frac{2000}{250}=8.0\text{m}$

The cart moves 8.0 m. Check that force is in newtons and distance is in metres before substituting.

When an object is dragged across a surface, the force does work against friction, a contact force opposing the motion.

Work done against friction is transferred to the thermal energy store of the object and its surroundings, causing a rise in temperature.

The energy does not disappear. It is transferred to thermal stores, so the sliding surfaces get warmer. This is why:

• Rubbing your hands together quickly makes them feel warm.
• Car brakes and brake discs get hot after heavy braking.
• A rope pulled fast through your hands can burn them.
• A drill bit becomes hot as it cuts through metal.

In each case, useful movement energy is being transferred to a thermal store by the work done against friction. Because this thermal energy spreads out to the surroundings, it is usually wasted rather than useful energy.

1. Thinking a force always does work

Work is only done if the object moves in the direction of the force. Pushing hard on a wall that does not move means no distance is covered, so no work is done, however great the force.

2. Using a distance in the wrong direction

The distance $s$ in $W=Fs$ must be measured along the direction of the force. For a horizontal push, use the horizontal distance moved, not a diagonal or vertical distance.

3. Forgetting to convert units

Force must be in newtons and distance in metres. A distance given in centimetres must be converted to metres (divide by 100) before using $W=Fs$.

4. Mixing up joules and kilojoules

1 kJ = 1000 J. When a question gives energy in kilojoules, convert to joules before combining it with values in newtons and metres.

5. Saying energy is lost to friction

Energy is never destroyed. Work done against friction transfers energy to thermal stores; the energy is dissipated to the surroundings, not lost.