Work Done in Physics

work done example - cable car

Work done is a boring topic. It is the very definition of a boring topic, even the name of this physics concept is dull. The word “work” doesn’t exactly make you jump up with excitement in the same way as “explosion” or “play” might, does it?

The definition of work done also seems very arbitrary, force x distance. Why? Why should we care about something defined as force x distance? Then you are expected to rote learn that work done is measured in joules, the unit of energy, but it’s not an energy store because it’s work done. Got that? This is just confusing to the average student.

Often work done is taught along with a demonstration of something being lifted up like the ski lift in the picture. Maybe the class sets out to find the work done by a person climbing stairs by firstly weighing the person then finding the height of the staircase. You are informed that motion in a horizontal direction doesn’t get included, no work done walking across the room. This contradicts the common experience of getting out of breath when walking quickly. Surely I’m doing work? The term work done also crops up in electrical circuits too where there are no stairs. None of it ties together very well. Ask for help from a teacher and the reply is usually, “that is the definition of work”!

There is an way to think about work done which makes a whole lot more sense! Imagine for a moment that we are stepping into a chapter of the brilliant series Horrible Histories, I would like to take you back to the Industrial Revolution.

 

Pit Ponies and Barges.

Back in ye olden days when there were no motors or even steam engines, heavy things were moved by teams of people or horses. Horses were used in mines to haul carts full of ore and coal, and to pull things up from the mine to the surface. Barge horses were used to pull canal barges laden with goods along canal ways.

winching children into a mine shaft

Two children are lowered down the mine-shaft to the coal- face by a woman working a winch Date: 1842 Source: ‘The Condition & Treatment of the Children employed in the Mines & Collieries of the United Kingdom’, page 7

And yes just like in Horrible Histories, young children were hand cranked down dark mine shafts to chip away at coal and mineral ores in polluted air with nothing more than a candle for company. Makes double maths look slightly more appealing.

 

winding horse whim

A sepia photograph of a winding-horse whim from Wheal Geevor Tin Mine copyright of Geevor Tin Mine and Pendeen Community Heritage, permission given for education use.

 

canal barge horse

A working canal barge with horse pulling the barge along, approx 1900

Then came the invention of steam engines and machinery to automate this process. It was important for the new industrial engineers to be able to explain their machines in a language that the mine owners could understand. They used the term “horse power”. This is the machine equivalent of the work that could be done by an average horse. Horses being more useful than small children.

Do you notice how the term work done slipped into that sentence? What work was being done here? In each example a heavy object is being pulled some distance by a horse. Pulling is an example of a force. The depth of the mine shaft, the length of the tow path, these are distances.

Now the definition of the work being done by the horse makes a bit more sense. The horse has to do some physical work to move an object a distance along or up. The horse has to pull, so there is definitely a force involved. In the mine the pull is upwards to overcome the downwards pull of gravity. On the towpath the pull is forwards, but what is the horse pulling against? Not gravity this time. The horse pulls against the water as it drags the canal barge forwards.

So the work being done by the horse is the result of the force it has to use to move the object and the distance it has to be moved.

A machine like a steam engine would need to be as good as, or better than, the work that could be done by a horse. That would make the steam engine attractive to potential buyers.

 

Defining Work Done

The term work was first adopted in the 1820s. There is a reference by the French mathematician GaspardGustave Coriolis in “Calculation of the Effect of Machines, or Considerations on the Use of Engines and their Evaluation“. He used it in the sense of a “weight lifted through a height”, from  the use of early steam engines to lift buckets of water out of flooded ore mines.

A “weight lifted through a height”, was also the way Sadi Carnot defined work in his famous paper Reflections on the Motive Power of Fire in 1824. Carnot said:

We use here motive power (work) to express the useful effect that a motor is capable of producing. This effect can always be likened to the elevation of a weight to a certain height. It has, as we know, as a measure, the product of the weight multiplied by the height to which it is raised.

Now hopefully you can see how work done = force x distance was a really useful quantity to a Victorian engineer.

A horse hauls 500kg up a 20m mine shaft. What is the work done by the horse on the load? (take g = 10 m/s^2)

 

A barge horse is pulling a barge with an effective mass of 8000 lb along a river for 1km. 1kg = 2.2lbs. Covert the weight to kgs and calculate the work done by the horse on the barge.

 

A steam engine can do the work of 4 horses. One horse can move 600kg a distance of 3km in 1 hour. How much work can each horse do? How much work does the engine do in 1 hour? What is the power (joules per second) of the steam engine?

When something moves it has kinetic energy. By doing work the horse is getting some heavy object to move. No movement and no work has been done. If the thing doesn’t budge, even though the horse might be straining all its muscles, in the strict physics sense no work has been done.

Giving these heavy barges and piles of coal some kinetic energy is what the horse is for. The horse’s store of chemical energy in its muscles slowly runs down while the heavy barge’s kinetic energy store slowly fills up. There has been a transfer of energy from one to the other through the action of the force. Work done is an energy transfer process. It is not a store of energy but a way of showing energy in flux. It is an amount of energy going from something that exerts a force to make something else move. This is why it is measured in Joules. It is the amount of energy being reduced in the horse to make the kinetic energy of the barge increase.

Amount of energy reduced in horse = amount of energy gained by barge = work done by pulling.

The work-energy principle basically sums up what this red equation says. An increase in the kinetic energy of an object is caused by an equal amount of work done on the object by the force acting on the object.

Work done fits nicely into our ideas about the conservation of energy. It allows us to explain why energy isn’t lost or destroyed by the horse when we get something moving. The changes in energy are a measure of the work being done to move the thing.

 

Generalising the Concept of Work Done

Physicists are practical people. If they come up with a new idea then they are going to look for different ways to use it. This makes life easier as you then have fewer laws of physics to learn. In fact some physicists devote their lives to finding the fewest number of rules you can have to describe the interactions of all things.

Theory of everything cartoon

Calamities of Nature cartoon Strip from 2012

Work done gets taught to school students because it is one of those concepts which has lots of practical applications. It can be used in many different areas of physics. Any time a force is acting on something and it moves you can say work is being done.

Hopefully you can see how work done by a horse depends on a force (due to the horse pulling on a weight) and a distance. To make this more general we can remove the horse altogether. Imagine a force acting on something that moves that thing some distance. We don’t need to worry about the actual object doing the pull (or push). We only need to think about the force involved and the distance moved. When the force only pulls in one direction we only include movements that are in that direction in our calculation of work done.

How about a tractor beam from the Death Star pulling on the Millennium Falcon? Or a magnet pulling the cover of a bag shut. Or a strong wind propelling a windsurfer. Or the negative end of a battery pushing away negative electrons in a wire?

A force of 50 newtons acts to move an object 2.5m. What is the work done by the force on the object?

From this more general way of thinking we can jump across to other areas of physics. The only thing that changes is how we come up with the value of the force. In the case of ponies and mines we use weight, a force equal to the mass x the pull of Earth’s gravity. In a different situation the force could be due to electromagnetism, gas pressure, radiation pressure, pretty much anything that can exert a force and cause a movement.

 

Electrical Work Done

Electrical work done, or electrical work for short, is the work done by an electric field on a charged particle which causes the particle to move.

The electrical field in a circuit is produced by the battery. Inside the battery charges separate out towards a positive terminal and a negative terminal. This separation of charge results in an electric field between the positive and negative charges, just like the magnetic field between the north and south poles of a magnet. The electrical field fills the space between the two terminals. To make use of this electric field hold it in a loop of conductive wire. This is a circuit.

This electrical field does work on the free electrons in the conductive metal wire. The electrons move as a result and this average drift of the electrons along the wire is what we call current.

Electrical work done = force from electric field x distance charge moves

In this way chemical energy that was stored inside the battery is reduced and the kinetic energy of the electrons in the wire is increased as work is done on them through the action of the field. This is how batteries run out. Eventually the battery exhausts the chemicals inside it and not enough charges can be separated out to maintain a strong enough electrical field to exert a force on electrons in the wire.

An electric field due to a battery exerts a force of 6×10^-18 newtons on an electron in a 30cm wire. There are 2.5×10^28 free electrons in the wire, what is the total work done to move all these electrons through the wire?

 

Thermodynamic Work Done

There was a famous experiment done by the British scientist James Joule which links the work done by a weight moving through a height, to the heat energy gained by some water. It is regarded as one of the starting points of a whole branch of physics, thermodynamics. Thermo for heat energy and dynamics for how heat is exchanged between systems and their surroundings.

James Joule heat engine

The apparatus used by Joule in his famous experiment

The apparatus is ingenious. By rotating the paddle and transferring energy to the water through friction, Joule was planning to measure the heat energy gained by the water. I always find this astonishing. When I first learnt physics, I had no idea agitating water would heat it up! I always thought of stirring things to cool them down but that is a different process (evaporation) altogether.

Joule figured out that linking the rotating axis of the paddle to a string and pulley system  meant he could lift a weight up at the same time as rotating the paddle. He slowly wound the weight up high, let the system settle and then released the weight. As it dropped, it rapidly turned the paddle and the thermometer reading went up. He was relating an increase in temperature to the work done moving a mass.

Work done to lift weight = energy needed to rotate paddle = energy gained by water

Joule used this to define the mechanical equivalent of heat. Now any change in temperature of a system could be given a number relating it to work done. Work could be done to a system or by a system. This was fantastically useful in the time of steam engines, with their hot boiler tanks, pressurised gases pushing mechanical parts around causing motion.

When we use the word “system” in physics we mean whatever object or objects are being affected. It could be a boiler filled with water turning into steam. The entire thing, container, liquid and gas would be the system. Energy would be supplied to the system through a fire heating the boiler. It would be transferred out of the system through heat loss and the steam escaping to push a turbine or piston.

Work done in thermodynamics is the energy transferred by a hot system to its surroundings. The amount of work done by this system is found from the effect it has on its surroundings. So if a system of hot gas expands and pushes on a piston the work done is defined by the movement of the piston caused by an expansion of the gas.

heat transferred into a system = work done on surroundings + heat left in the system 

Concluding Work Done

Work is such a useful and big concept. We have barely scratched the surface here. The main points to take away are

  • Work done is a way of transferring energy from one store to another.
  • As a sort of energy in flux, it is also measured in Joules.
  • Work always gets things moving, its results in an increase in kinetic energy of something.
  • It doesn’t matter what sort of force is acting, so long as it makes something move.
  • The movement can be as general as the pressure of a gas pushing on its surroundings.
  • All calculations of work done reduce to force acting x distance moved.

 

Answers to Questions

A horse hauls 500kg up a 20m mine shaft. What is the work done by the horse on the load? (take g = 10 m/s^2)

Work done = force x distance, force in this case is a weight of 500x10N = 5000N using the relationship weight = mass x g. 

Thus work done = 5000N x 20m = 100,000J

A barge horse is pulling barge with an effective mass of 8000 lb along a river for 1km. 1kg = 2.2lbs. Covert the weight to kgs and calculate the work done by the horse on the barge.

The effective mass is how heavy it feels to the horse to pull barge probably weighing tonnes along in water. The horse isn’t actually lifting the barge up so this isn’t the actual mass of the barge.

Force of barge in N = effective mass in kg x g = 8000 x 2.2 x 10 = 176,000

Work done = force (N) x distance (m) = 8000x 2.2 x 10 x 1000 = 1.76 x 10^8J

A steam engine can do the work of 4 horses. One horse can move 600kg a distance of 3km in 1 hour. How much work can each horse do? How much work does the engine do in 1 hour? What is the power (joules per second) of the steam engine?

One horse can do 600 x 10 x 3000 = 18,000,000J of work.

The engine does 4 x work of one horse = 4 x 18,000,000J = 72,000,000J in one hour

power = energy (J) / time (s) = 72,000,000 / (60×60) = 2000W

A force of 50 newtons acts to move an object 2.5m. What is the work done by the force on the object?

work done = force x distance = 50N x 2.5m = 125J

An electric field due to a battery exerts a force of 6×10^-18 newtons on an electron in a 30cm wire. There are 2.5×10^28 free electrons in the wire, what is the total work done to move all these electrons through the wire?

work done = force on one electron x length of wire x number of electrons

work done = 6×10^-18 x 0.3 x 2.5x 10^28 = 4.5 x 10^10 J

 

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