Work done and energy transferred
Definition
Work done
Work done is the energy transferred when a force moves an object through a distance in the direction of the force.
Definition
Joule
The joule is the unit of energy and of work done, equal to the work done when a force of one newton moves an object one metre in the direction of the force.
- Work done is the energy a force transfers when it moves an object through a distance in the direction of that force.
- Work done and energy transferred are the same quantity, so the work done by a force equals the energy it transfers.
- Both are measured in joules, J\text{J}J, while force is measured in newtons, N\text{N}N, and distance in metres, m\text{m}m.
- One joule is the work done when a force of 1 N1\ \text{N}1 N moves an object 1 m1\ \text{m}1 m in the direction of the force, so 1 J=1 N m1\ \text{J} = 1\ \text{N}\ \text{m}1 J=1 N m.
- A force that acts without producing any movement does no work, so it transfers no energy.
- Only the part of the movement lying along the force counts, so a bag carried horizontally gains nothing from the upward force holding it.
The work done equation
- Work done is calculated from E=F×dE = F \times dE=F×d.
- In this equation EEE is the work done in joules, FFF is the force in newtons and ddd is the distance moved in the direction of the force in metres.
- Rearranging gives F=EdF = \dfrac{E}{d}F=dE when the force is wanted and d=EFd = \dfrac{E}{F}d=FE when the distance is wanted.
- Convert every distance into metres before substituting, so 60 cm=0.60 m60\ \text{cm} = 0.60\ \text{m}60 cm=0.60 m and 1.5 km=1500 m1.5\ \text{km} = 1500\ \text{m}1.5 km=1500 m.
- For a lift straight upwards at a steady speed the force needed equals the weight, so the work done is the weight in newtons multiplied by the vertical distance in metres.
- Doubling the force doubles the work done over the same distance, and doubling the distance doubles the work done for the same force.
- A force at right angles to the movement contributes no distance in its own direction, so it does no work.
Example
Dragging a crate across a floor
- A stagehand drags a crate 4.0 m4.0\ \text{m}4.0 m along a flat floor using a horizontal force of 25 N25\ \text{N}25 N.
- The equation is E=F×dE = F \times dE=F×d.
- Substituting gives E=25×4.0E = 25 \times 4.0E=25×4.0.
- The work done is E=100 JE = 100\ \text{J}E=100 J, so the pulling force transfers 100 J100\ \text{J}100 J of energy.
Example
Finding a force from the work done
- A winch transfers 4500 J4500\ \text{J}4500 J while pulling a sledge 30 m30\ \text{m}30 m along level ground.
- Rearranging E=F×dE = F \times dE=F×d gives F=EdF = \dfrac{E}{d}F=dE.
- Substituting gives F=450030F = \dfrac{4500}{30}F=304500.
- The force is F=150 NF = 150\ \text{N}F=150 N, acting along the direction of the movement.
Practical
Measuring the work done by a force
- Aim: to measure the work done by a known force as it drags a wooden block along a bench.
- Apparatus: wooden block, newton meter reading to 0.1 N0.1\ \text{N}0.1 N, metre rule, bench or runway, masking tape, slotted masses, balance and clamp.
- Variables: the distance moved in the direction of the force is the independent variable, the work done is the dependent variable, and the block, the mass on top of it, the bench surface and the pulling direction are controlled.
- Method, set-up:
- Weigh the block, then tape a start line and a finish line to the bench and measure the distance between them with the metre rule.
- Hook the newton meter to the block and hold it horizontally, in line with the direction of travel, so the reading is the pulling force.
- Zero the newton meter while it is held in the pulling direction with nothing attached.
- Method, measurements:
- Pull the block from the start line to the finish line at a slow, steady speed so the reading stays as constant as possible.
- Read the newton meter while the block is moving rather than before it starts, because a larger force is needed to break it free.
- Record the steady force in newtons and the distance in metres.
- Repeat each run three times and calculate the mean force.
- Repeat for at least five distances by moving the finish line further along the bench.
- Results: the work done rises in proportion to the distance moved while the pulling force is kept the same.
- Maths: calculate the work done for each distance from E=F×dE = F \times dE=F×d, then plot EEE on the vertical axis against ddd on the horizontal axis; the graph is a straight line through the origin and its gradient equals the pulling force FFF in newtons.
- Watch out: a newton meter held at an angle reads more than the horizontal pulling force, a block that accelerates is not being pulled by a steady force, and a dusty or worn patch of bench changes the friction part way along.
- Safety: clamp the runway or keep the block well away from the bench edge, keep fingers clear of falling masses, and use modest masses so the block stays under control.
Energy changes when work is done
- Work done by a force always changes a store, so every E=F×dE = F \times dE=F×d calculation can also be described as an energy transfer.
- Lifting an object at a steady speed puts the work done into the gravitational potential energy store of the object and Earth.
- A resultant force that speeds an object up puts the work done into the object's kinetic energy store.
- Stretching a spring puts the work done into the elastic potential energy store of the spring.
- Work done against friction or air resistance puts the energy into the thermal energy stores of the surfaces and the surroundings, so their temperature rises.
- A car braking from 30 m/s30\ \text{m/s}30 m/s has the work done by the braking force taken out of its kinetic energy store and put into the thermal energy stores of the brakes, tyres and road.
- To describe any of these changes, name the force, state that the object moves in the direction of that force, state that work is done, then name the store that increases and the store that decreases.
Example
Work done against friction while braking
- A cyclist and bicycle of total mass 85 kg85\ \text{kg}85 kg are brought to rest by a braking force of 340 N340\ \text{N}340 N acting over 12 m12\ \text{m}12 m.
- The work done by the braking force is E=F×d=340×12E = F \times d = 340 \times 12E=F×d=340×12.
- This gives E=4080 JE = 4080\ \text{J}E=4080 J.
- The cyclist's kinetic energy store therefore falls by 4080 J4080\ \text{J}4080 J as the bicycle stops.
- Those joules are transferred into the thermal energy stores of the brake blocks, the wheel rims and the surrounding air, which is why the rims feel warm.
Exam technique
Setting out a work done answer
- Write E=F×dE = F \times dE=F×d before any numbers, because the equation itself often carries a mark.
- Check that the distance used is measured along the direction of the force and not the whole path length.
- Convert centimetres and kilometres into metres, and never substitute a mass where a force is needed.
- Give the answer in joules, and use kJ\text{kJ}kJ only when the question asks for it.
- For a describe question, join the ideas with so, such as the force moves the crate, so work is done and energy is transferred to the thermal store of the floor.
Common Mistake
- Do not give a force in joules, because joules measure work done and energy while newtons measure force.
- Do not multiply by the whole path length when only part of the movement lies along the force.
- Do not claim work is done while a bag is simply held still, because there is no movement.
- Do not confuse mass in kilograms with weight in newtons when the work done in a lift is calculated.
Self review
- Define work done and state its unit.
- Write the equation linking work done, force and distance.
- Explain why 1 J1\ \text{J}1 J is the same as 1 N m1\ \text{N}\ \text{m}1 N m.
- Describe how to measure the work done by a force dragging a block along a bench.
- Name the store that increases when work is done against friction.
