A Book of Sketches by Stephen Leacock: A, B and C, the Human Element in Mathematics

From the Project Gutenberg

This was the final chapter in a book called Literary Lapses, written in 1910 by the author Stephen Leacock. It was one of the short stories included in the Exploring English anthology, one of the three books we studied for the Intermediate Certificate. The other two books were devoted to prose and poetry.
In the story, Stephen humorously personifies A, B and C as they negotiate the algebra problems encountered in maths problems.
The book is in the public domain and copyright has elapsed, hence the reason I have copied a portion.

 

A, B, and C, The Human Element in Mathematics

The student of arithmetic who has mastered the first four rules of his art, and successfully striven with money sums and fractions, finds himself confronted by an unbroken expanse of questions known as problems. These are short stories of adventure and industry with the end omitted, and though betraying a strong family resemblance, are not without a certain element of romance.

The characters in the plot of a problem are three people called A, B, and C. The form of the question is generally of this sort:

“A, B, and C do a certain piece of work. A can do as much work in one hour as B in two, or C in four. Find how long they work at it.”

Or thus:

“A, B, and C are employed to dig a ditch. A can dig as much in one hour as B can dig in two, and B can dig twice as fast as C. Find how long, etc. etc.”

Or after this wise:

“A lays a wager that he can walk faster than B or C. A can walk half as fast again as B, and C is only an indifferent walker. Find how far, and so forth.”

The occupations of A, B, and C are many and varied. In the older arithmetics they contented themselves with doing “a certain piece of work.” This statement of the case however, was found too sly and mysterious, or possibly lacking in romantic charm. It became the fashion to define the job more clearly and to set them at walking matches, ditch-digging, regattas, and piling cord wood. At times, they became commercial and entered into partnership, having with their old mystery a “certain” capital. Above all they revel in motion. When they tire of walking-matches—A rides on horseback, or borrows a bicycle and competes with his weaker-minded associates on foot. Now they race on locomotives; now they row; or again they become historical and engage stage-coaches; or at times they are aquatic and swim. If their occupation is actual work they prefer to pump water into cisterns, two of which leak through holes in the bottom and one of which is water-tight. A, of course, has the good one; he also takes the bicycle, and the best locomotive, and the right of swimming with the current. Whatever they do they put money on it, being all three sports. A always wins.

In the early chapters of the arithmetic, their identity is concealed under the names John, William, and Henry, and they wrangle over the division of marbles. In algebra they are often called X, Y, Z. But these are only their Christian names, and they are really the same people.

Now to one who has followed the history of these men through countless pages of problems, watched them in their leisure hours dallying with cord wood, and seen their panting sides heave in the full frenzy of filling a cistern with a leak in it, they become something more than mere symbols. They appear as creatures of flesh and blood, living men with their own passions, ambitions, and aspirations like the rest of us. Let us view them in turn. A is a full-blooded blustering fellow, of energetic temperament, hot-headed and strong-willed. It is he who proposes everything, challenges B to work, makes the bets, and bends the others to his will. He is a man of great physical strength and phenomenal endurance. He has been known to walk forty-eight hours at a stretch, and to pump ninety-six. His life is arduous and full of peril. A mistake in the working of a sum may keep him digging a fortnight without sleep. A repeating decimal in the answer might kill him.

B is a quiet, easy-going fellow, afraid of A and bullied by him, but very gentle and brotherly to little C, the weakling. He is quite in A's power, having lost all his money in bets.

Poor C is an undersized, frail man, with a plaintive face. Constant walking, digging, and pumping has broken his health and ruined his nervous system. His joyless life has driven him to drink and smoke more than is good for him, and his hand often shakes as he digs ditches. He has not the strength to work as the others can, in fact, as Hamlin Smith has said, “A can do more work in one hour than C in four.”

The first time that ever I saw these men was one evening after a regatta. They had all been rowing in it, and it had transpired that A could row as much in one hour as B in two, or C in four. B and C had come in dead fagged and C was coughing badly. “Never mind, old fellow,” I heard B say, “I'll fix you up on the sofa and get you some hot tea.” Just then A came blustering in and shouted, “I say, you fellows, Hamlin Smith has shown me three cisterns in his garden and he says we can pump them until to-morrow night. I bet I can beat you both. Come on. You can pump in your rowing things, you know. Your cistern leaks a little, I think, C.” I heard B growl that it was a dirty shame and that C was used up now, but they went, and presently I could tell from the sound of the water that A was pumping four times as fast as C.

For years after that I used to see them constantly about town and always busy. I never heard of any of them eating or sleeping. Then owing to a long absence from home, I lost sight of them. On my return I was surprised to no longer find A, B, and C at their accustomed tasks; on inquiry I heard that work in this line was now done by N, M, and O, and that some people were employing for algebraica jobs four foreigners called Alpha, Beta, Gamma, and Delta.

Now it chanced one day that I stumbled upon old D, in the little garden in front of his cottage, hoeing in the sun. D is an aged labouring man who used occasionally to be called in to help A, B, and C. “Did I know 'em, sir?” he answered, “why, I knowed 'em ever since they was little fellows in brackets. Master A, he were a fine lad, sir, though I always said, give me Master B for kind-heartedness-like. Many's the job as we've been on together, sir, though I never did no racing nor aught of that, but just the plain labour, as you might say. I'm getting a bit too old and stiff for it nowadays, sir—just scratch about in the garden here and grow a bit of a logarithm, or raise a common denominator or two. But Mr. Euclid he use me still for them propositions, he do.”

From the garrulous old man I learned the melancholy end of my former acquaintances. Soon after I left town, he told me, C had been taken ill. It seems that A and B had been rowing on the river for a wager, and C had been running on the bank and then sat in a draught. Of course the bank had refused the draught and C was taken ill. A and B came home and found C lying helpless in bed. A shook him roughly and said, “Get up, C, we're going to pile wood.” C looked so worn and pitiful that B said, “Look here, A, I won't stand this, he isn't fit to pile wood to-night.” C smiled feebly and said, “Perhaps I might pile a little if I sat up in bed.” Then B, thoroughly alarmed, said, “See here, A, I'm going to fetch a doctor; he's dying.” A flared up and answered, “You've no money to fetch a doctor.” “I'll reduce him to his lowest terms,” B said firmly, “that'll fetch him.” C's life might even then have been saved but they made a mistake about the medicine. It stood at the head of the bed on a bracket, and the nurse accidentally removed it from the bracket without changing the sign. After the fatal blunder C seems to have sunk rapidly. On the evening of the next day, as the shadows deepened in the little room, it was clear to all that the end was near. I think that even A was affected at the last as he stood with bowed head, aimlessly offering to bet with the doctor on C's laboured breathing. “A,” whispered C, “I think I'm going fast.” “How fast do you think you'll go, old man?” murmured A. “I don't know,” said C, “but I'm going at any rate.”—The end came soon after that. C rallied for a moment and asked for a certain piece of work that he had left downstairs. A put it in his arms and he expired. As his soul sped heavenward A watched its flight with melancholy admiration. B burst into a passionate flood of tears and sobbed, “Put away his little cistern and the rowing clothes he used to wear, I feel as if I could hardly ever dig again.”—The funeral was plain and unostentatious. It differed in nothing from the ordinary, except that out of deference to sporting men and mathematicians, A engaged two hearses. Both vehicles started at the same time, B driving the one which bore the sable parallelopiped containing the last remains of his ill-fated friend. A on the box of the empty hearse generously consented to a handicap of a hundred yards, but arrived first at the cemetery by driving four times as fast as B. (Find the distance to the cemetery.) As the sarcophagus was lowered, the grave was surrounded by the broken figures of the first book of Euclid.—It was noticed that after the death of C, A became a changed man. He lost interest in racing with B, and dug but languidly. He finally gave up his work and settled down to live on the interest of his bets.—B never recovered from the shock of C's death; his grief preyed upon his intellect and it became deranged. He grew moody and spoke only in monosyllables. His disease became rapidly aggravated, and he presently spoke only in words whose spelling was regular and which presented no difficulty to the beginner. Realizing his precarious condition he voluntarily submitted to be incarcerated in an asylum, where he abjured mathematics and devoted himself to writing the History of the Swiss Family Robinson in words of one syllable.

END

Simple Machines: How Does a Lever Work?

© Eugene Brennan

What Is a Lever?

A lever is a simple device that works by changing the input force needed to do work and move something. Some levers do this by magnifying the force applied to them. In this tutorial, we'll explore how levers are used in our everyday lives and then learn the simple physics of how they work.

Examples of levers. © Eugene Brennan

The Lever: One of the Six Classical Simple Machines

The lever is one of the six simple machines which were defined by Renaissance scientists hundreds of years ago.

Six Classical Simple Machines

• Lever
• Wheel
• Inclined plane
• Screw
• Wedge
• Pulley

You've used a lever in some shape or form without actually realizing it. So for instance scissors, nutcrackers, pliers, hedge shears, bolt cutters and lopping shears all use levers in their design. A prybar or crowbar is a lever also, and when you prise open the lid of a tin with the handle of a spoon, you're using "the law of the lever" to create a greater force. A long handle on a wrench provides more "leverage". A claw hammer also acts as a lever when pulling out nails. A see-saw and wheelbarrow are also levers.

What Is a Force?

To understand how a lever works, we first need to learn about forces. A force can be thought of as a "push" or "pull". A force is required for example to lift a weight or slide it on a surface.

Examples of Forces

• A forklift lifting a load
• Tension in a spring when you pull on it
• A magnet pulling a piece of iron
• Air in a balloon, football or tire, pushing outwards on its walls
• The force of gravity keeping things on the ground
• Air or water resisting the movement of a car, aircraft or ship. This is called drag.

An active force results in a reactive force, so for instance when you pull on a spring, this is the active force. The tension in the spring is the reactive force pulling back.

Different types of forces. © Eugene Brennan

What Does Mechanical Advantage Mean?

A simple machine can magnify a force. The degree to which the force is magnified is called the mechanical advantage. Levers are great because they can increase mechanical advantage and generate much larger forces. For example a hammer or crowbar can easily produce a ton of force for pulling out nails, lifting a rock or prising up boards.

What Are the Parts of a Lever?

In general, levers can be actual tools or components in a machine, but they also appear in our bodies and in nature. A lever is made up of several parts:

• Beam: The physical lever itself is made of materials such as wood, metal or plastic, bone in humans and animals etc., which can pivot or move on the fulcrum.
• Effort: The force that is exerted on the lever
• Fulcrum: The point at which a lever pivots or hinges
• Load: The object that is acted on by the lever.

Levers can increase a force. I.e they give a mechanical advantage. © Eugene Brennan

Using the handle of a spoon to open a tin. The spoon acts as a lever, creating a larger force to lift the lid. The fulcrum is the rim of the tin. © Eugene Brennan

Examples of Levers in Everyday Life

• Crowbars and prybars
• Pliers
• Scissors
• Bottle openers
• Bolt cutters
• Nut crackers
• Claw hammer
• Wheel barrow
• Parts of machines such as engines and production machines in factories
• Bones and joints in your body

From "The World of Wonder" a children's science periodical from the 1930s.

Three Classes of Levers

The class of a lever depends on the position of the effort, fulcrum and load.

First Class Lever

The effort is on one side of the lever and the load is on the other side. The fulcrum is in the middle. Moving the fulcrum closer to the load increases the mechanical advantage and increases the force on the load.

Examples: Scissors, pliers, hammer

Second Class Lever

The effort is on one side of the lever and the fulcrum is on the other side with the load between the effort and fulcrum. Keeping the effort in the same position and moving the load closer to the fulcrum, increases the force on the load.

Examples: nutcracker, wheelbarrow

Third Class Lever

The fulcrum is on one end of the lever, the load is on the other side and the effort is between the load and fulcrum. A third class lever has less of a mechanical advantage than the other two types because the distance from the load to the fulcrum is greater than the distance from the effort to the fulcrum.

Examples: human arm, broom, sporting equipment (e.g. baseball bat)

Three classes of levers. © Eugene Brennan

Examples of Levers

Bolt cutters. Annawaldl, public domain image via Pixabay.com

Using a crowbar as a lever to lift a heavy piece of stone.  Public domain image via Pixabay.com

Pliers and side cutters. © Eugene Brennan

An excavator (digger) has several connected levers on its boom. Hydraulic cylinders produce the force required to move the levers. Didgeman, public domain image via Pixabay.com

The Physics of How Levers Work

What Is the Moment of a Force?

To understand how levers work, we first need to understand the concept of moment of a force. The moment of a force about a point is the magnitude of the force multiplied by the perpendicular distance from the point, to the line of direction of the force. So in the diagram below, if the magnitude of the force is F and the distance is d, the the moment = Fd

Moment of a force. © Eugene Brennan

In the 2nd diagram below, two forces act on a lever. This is a schematic or diagram, but it symbolically represents any of the real life levers mentioned above.

The lever pivots at a point called a fulcrum represented by the black triangle (in real life, this could be the screw holding the two blades of a scissors together). A lever is said to be balanced when the lever doesn't rotate and everything is in equilibrium (e.g. two people of equal weight sitting on a see-saw, at equal distances from the pivot point).

Forces on a lever. © Eugene Brennan

In the diagram, a force F1 acts downward on the lever at a distance d1 from the fulcrum.

Another force F2 at distance d2 from the fulcrum acts downwards on the lever. When the lever is balanced, F2 balances the effects of F1 and the lever is stationary, i.e. there is no net turning force.

When balanced:

"The sum of the clockwise moments equals the sum of the counter-clockwise moments"

So for F1, the clockwise moment is F1d1

and for F2, the counter-clockwise moment is F2d2

So the clockwise moments = the anticlockwise moments

and

F1d1 = F2d2

Imagine if F1 is the active force and is known. F2 is unknown but must push down on the lever to balance it.

Dividing both sides of the equation by d2 and switching the left and right gives:

F2 = F1(d1/d2)

So F2 must have this value to balance the force F1 acting down on the right-hand side.

Since the lever is balanced, we can think of there being an equivalent force equal to F2 (and due to F1), shown in orange in the diagram below, pushing upwards on the left side of the lever.

If the distance d2 is a lot smaller than d1 (which would be the case with a crowbar or pliers), the term (d1/d2) in the equation above is greater than unity and F2 becomes greater than F1. (a long-handled crowbar can easily produce a ton of force).

This is intuitively correct since we know how a long crowbar can create a lot of force for lifting or prying things, or if you put your fingers between the jaws of a pliers and squeeze, you know all about it!

If F2 is removed and the lever becomes unbalanced, the upwards force due to the force F1 on the right is still F1(d1/d2). This force magnifying effect or mechanical advantage of a lever is one of the features that makes it so useful.

When the lever is balanced, the force F1 produces an equivalent force of magnitude F2 (shown in orange). This balances F2 (shown in blue) acting downwards. © Eugene Brennan

The Law of the Lever

We can summarise the above reasoning into a simple equation known as the law of the lever:

Mechanical advantage = F2/F1 = d1/d2

d1 is called the effort arm and d2 the load arm. If F1 is the effort and F2 is the load, then:

Interesting Fact: We Have Levers in Our Body!

Many of the bones in your body act as third class levers. For instance in your arm, the elbow is the pivot, the biceps muscle creates the effort acting on the forearm and the load is held by a hand. The small bones in the ear also form a lever system. These bones are the hammer, anvil and stirrup and act as levers to magnify sound coming from the eardrum.

The bones in our arms and other part of the body are third class levers. Original image without text, OpenStax College, CC BY SA 3.0 unported via Wikimedia Commons

What Is a Counterbalance Used For?

A counterbalance is a weight added to one end of a lever or other pivoting structure so that it becomes balanced (the turning moments clockwise and anti-clockwise are equalised). The weight of the counterbalance and its position relative to the pivot are set so that the lever can stay at any angle without turning. The advantage of a counterbalance is that a lever only has to be displaced and doesn't have to be physically lifted. So for instance a heavy road barrier could be raised by a human if it moves freely on its pivot. If there was no counterbalance, they would have to push down a lot harder on the barrier to lift the other end. Counterbalances are also used on tower cranes to balance the boom so that the crane doesn't topple over. Swing bridges use counterbalances to balance the weight of the swing section. Sometimes the counterbalancing force is provided by a spring instead of a weight. For instance springs are sometimes used on the deck of a lawn mower so a person doesn't have to lift the deck when adjusting the height. Also springs might be used on the lid of a home appliance such as a chest freezer to stop the lid falling down when raised.

A counterbalance used to balance a lever. These are often seen on road barriers where one end of the lever is much shorter than the other end. © Eugene Brennan

A tower crane. The counterbalance consists of a collection of concrete slabs mounted near the end of the boom. Conquip, public domain image via Pixabay.com

Counterbalance on a similar crane. User:HighContrast, CC 3.0 via Wikimedia Commons

Not All Levers Magnify Force: Levers Increasing Range of Motion

We discovered that many levers have a mechanical advantage and increase the force on a load when an input force is applied to the lever. This is really useful in a tool such as a wire snips, bolt cutter or gardening lopping shears for creating large forces that can shear through materials. However another function of a lever is to increase range of motion. In this case, the force on the load is less than the input force to the lever, but the lever produces a greater range of motion. An example is the biceps and forearm. The biceps muscle typically can move the hand times eight times further than the displacement of the point where the muscle attaches to the arm. The increased range of motion is achieved by positioning the effort closer to the fulcrum than the load.

The biceps and forearm form a 3rd class lever system that increase range of motion. Niwadare, CC BY-SA 4.0 international via Wikimedia Commons

References

Curley, R. (2017, June 26). Simple machines. Encyclopaedia Britannica.

Hannah, J. and Hillerr, M. J., (1971) Applied Mechanics (First metric ed. 1971) Pitman Books Ltd., London, England.

Disclaimer

This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.

© 2018 Eugene Brennan

Another Faulty LED Street Lamp

From the DeadSure app for reporting faulty street lighting.

Too early to make a judgement, but something tells me these new LED street lights are less reliable than the older sodium ones. Which makes sense, considering sodium lamps are simpler and purely electrical/electromechanical, whereas LED lamps have more working parts— not moving parts, but semiconductors—that can fail. I reported this one that I noticed tonight because the upper path in the Valley Park seemed dark. There may be a couple more not lighting, further out the road closer to St. Brigid's Well.

LED lighting theoretically has a lifespan in the tens of thousands of hours (30,000 to 40,000 hours). However, in reality it can be less than this because the driver electronics fails.

57 Modes and Nothing Wrong

Bicycle rear light. © Eugene Brennan

Mystery light.

I've bashed it almost to the point of cracking the lens, put it in the freezer for half an hour while turned on and left a strong magnet in its vicinity and it still won't turn off. Yet when I'm cycling, after 100 m or so, it manages to turn itself off. And that keeps happening. The only thing I can think of is that deceleration in a certain direction is bending a spring in a clicky button. Or maybe it's condensation on the PCB. Or possibly constant vibration rather than large shocks.

Anyway, I hate the way they don't put proper switches on these things instead of buttons for changing modes that inevitably cause them to switch off on bad roads. It's difficult to get a decent backlight anywhere. I've tried eBay, Amazon and AliExpress.

 

Archimedes' Principle, Buoyancy Experiments and Flotation Force

Archimedes' principle. © Eugene Brennan

What is Buoyancy?

We've all experienced buoyancy. If you throw a football into water, it floats on the surface. In a swimming pool, you can float on your back and large ships float in the sea. Even things that sink such as stones, experience a buoyant force, but the force is insufficient to keep them afloat. Buoyancy is a force that pushes up on the underside of an object that's placed in a fluid. The fluid doesn't have to be water, it can be a gas such as air and helium balloons experience buoyancy, causing them to rise upwards.

Who Was Archimedes?

Archimedes of Syracuse was a Greek astronomer, scientist and mathematician who was born circa 287 BC. Amongst his many works as a great scientist of the classical period was laying the ground work for modern calculus as well as proving geometric theorems, working out approximations for pi and calculating the surface area and volumes of 3D solids.

What is Archimedes' Principle?

Archimedes' principle, also called the principle of Archimedes or the Archimedean principle, states that the upthrust or buoyant force on an object in a fluid is equal to the weight of the displaced fluid. Displaced means pushed out of the way, so for instance when you drop stones into a container of water, you displace the water and it rises in the container. A force can be though of as a push or pull. The fluid doesn't have to be water, it can be any other liquid or gas, e.g. air. For more detailed information on forces, see my physics tutorial:

Examples of Forces in Everyday Life and How They Affect Things 

Negative, Positive and Neutral Buoyancy

An object placed in a fluid such as water can do three things:

1. It can sink. We call this negative buoyancy.
2. It can float. We call this positive buoyancy. If we push the object below the surface of the water and let go, the positive buoyancy force pushes it back up again above the surface.
3. It can stay submerged below the surface and neither sink to the bottom nor float back to the surface. If it's moved to a different depth below the surface, it stays in that position. This is called neutral buoyancy.

Experiments to Understand the Archimedean Principle

Let's do some experiments to investigate and understand the principle of Archimedes.

Step 1. Weigh the Object

Imagine we have an object suspended from a weighing scales. For instance it could be an iron weight like the one in the diagram below. We're going to lower it into a tank of water filled to the brim, level with the overflow outlet. The weight may float or it may become submerged, but it doesn't matter and doesn't affect our experiment. Before we lower it into the tank, the weighing scales tells us that its weight is 6 kg.

Experiment to investigate the principle of Archimedes. © Eugene Brennan

Step 2. Weigh the Displaced Water

As the weight is lowered, water is displaced and overflows into the pan on the second scales. When the weight is fully submerged we find that the water we collected weighs 2 kg.

Demonstrating the Archimedes' principle. Weight submerged in water. Displaced water is weighed. © Eugene Brennan

Step 3. Check the Weight on the First Scales

We now check the weight on the first scales again.

We find that the weight indicated is only 4 kg this time.

Step 4. Do Some Calculations

We discover that when we subtract the new measurement of the iron weight indicated on the scales from the previous measurement, it tallies with the weight we measured on the second scales.

So 6 kg - 4 kg = 2 kg

Principle of Archimedes

We've just discovered Archimedes' principle!

"The upthrust on a body submerged or floating in a fluid equals the weight of the fluid displaced"

How come the weight indicated on the first scales is now less than it was before?

It's because of the upthrust or buoyancy force.

This accounts for the difference and the object appearing lighter.

The 6 kg weight acts downwards, but it's as though 2 kg is pushing upwards acting as support and lessening the weight of the iron. So the scales indicates a smaller net weight of 4 kg. This upthrust equals the weight of the displaced water we collected in the pan of the second scales.

However, the mass of the object is still the same = 6 kg. The mass of objects stays the same, assuming they don't lose any material. Weight however changes. Weight is the force on an object due to gravity. On the Moon, where the force is smaller, objects weigh less.

The principle of Archimedes. Buoyant force equals the weight of the displaced liquid. © Eugene Brennan

Negative Buoyancy and Sinking Bodies

In the first experiment we did earlier, the iron weight sank below the water as it was lowered. The 6 kg iron weight we used displaces water. However the weight of the water displaced is only 2 kg. So according to the principle of Archimedes, the buoyant force is 2 kg acting upwards on the iron weight. Since this is less than 6 kg, it isn't enough to support the weight in the water. We call this negative buoyancy. If the weight was detached from the hook of the weighing scales, it would sink.

Negative buoyancy. Buoyant force is less than the weight of the submerged body. © Eugene Brennan

What are Examples of Things That Need Negative Buoyancy?

• Anchors need to have negative buoyancy so they can sink to the ocean floor.
• Fishing net sinkers to keep nets open

An anchor on a ship. Analogicus via Pixabay.com

Experiment 2. Investigating Positive Buoyancy

This time we lower a hollow steel ball onto the surface.

Positive Buoyancy and Floating Objects

What happens if a weight floats and doesn't sink? In the diagram below we lower a hollow steel ball into the tank. This time we know the weight is 3 kg. The chain goes slack if we move the scales closer to the water surface, because the weight floats and doesn't pull down on it. (In reality the chain has weight and will pull down on the scales, but for the sake of the experiment, let's imagine it's weightless.) The scale indicates 0 kg. The water displaced weighs the same as the weight this time.

So what happens in this scenario is that the ball displaces water and settles lower and lower in the water until the upthrust equals its weight. The force of gravity on the object acting downwards, i.e. its weight, is balanced by a buoyant force or upthrust acting upwards. Since the two are the same, the object floats.

In this second scenario, the object doesn't become fully submerged.

If we push the ball below the surface, it will displace more water, increasing the buoyant force. This force will be greater than the weight of the ball and the positive buoyancy will cause it to rise up out of the water and just displace enough water until the buoyant force and weight are equal again.

Positive buoyancy. The buoyant force and weight of the hollow steel ball are equal. © Eugene Brennan

 

What are Examples of Things That Need Positive Buoyancy?

• Lifebelts (lifebuoys)
• Marking and meteorological buoys
• Ships
• Swimmers
• Life jackets
• Floats on fishing lines
• Floats in toilet cisterns and float switches
• Flotation tanks/bags for recovering lost cargo/archaeological artefacts/submerged vessels
• Floating oil rigs and wind turbines

Things that need to have positive buoyancy. Clockwise from the top: A life belt, marking buoy, swimmer, ship.

Experiment 3. Investigating Neutral Buoyancy

In this experiment, the object we use has neutral buoyancy and can stay suspended under the water surface without sinking down or being pushed back up by the buoyant force of the water.

Neutral buoyancy occurs when the average density of an object is the same as the density of the fluid it is immersed in. When the object is below the surface, it neither sinks nor floats. It can be positioned at any depth below the surface and will stay there until another force moves it to a new location.

Neutral Buoyancy. Body can be positioned anywhere under the surface. Buoyancy force and weight of ball are equal. © Eugene Brennan

What are Examples of Things That Need Neutral Buoyancy?

• Diver
• Submarine

Submarines need to be able to control their buoyancy. So when there is a requirement to dive, large tanks are filled with water, producing negative buoyancy enabling them to sink. Once they reach the required depth, buoyancy is stabilised so that it becomes neutral. The sub can then cruise at a constant depth. When the sub needs to rise again, water is pumped out of the ballast tanks and replaced by air from compression tanks. This gives the submarine positive buoyancy, allowing it to float to the surface.

Humans naturally float in a vertical position with their noses just under the water if they relax their muscles. Scuba divers keep their buoyancy neutral by using belts with lead weights attached. This allows them to stay underwater at a desired depth without having to continually swim downwards.

A scuba diver needs to have neutral buoyancy. A submarine needs to have neutral, positive and negative buoyancy. Skeeze and Joakant. Public domain images via Pixabay.com

Negative, neutral and positive buoyancy. 

Formula for the Buoyant force

 

Where

ρ is the density of the displaced fluid

V is the volume of the displaced fluid

and g is the acceleration due to gravity

If we choose the the weight acting downwards to be a positive force, the negative sign in the equation for buoyant force is because it is a vector and acts in the opposite direction.

The weight of an object is

F_g = mg

where m is the mass of the object

Why do Ships Float?

Ships weigh thousands of tons, so how come they can float? If a stone or a coin is dropped into water it will sink straight to the bottom.

The reason ships float is because they displace lots of water. Think of all the space inside a ship. When a ship is launched into water, it pushes all the water out of the way and the massive upthrust balances the downwards weight of the ship, allowing it to float.

Why do Ships Sink?

Positive buoyancy keeps a ship afloat because the weight of the ship and buoyant force are balanced. However if too much heavy cargo is taken on by a ship, its total weight could exceed the buoyant force and it could sink. If the hull of a ship is holed, water will run into the hold. As water rises in the ship, it weighs down on the inside of the hull, causing the total weight to be greater than the buoyant force, making the ship sink.

A ship would also sink if we could magically crush all the steel structures and hull into a block. Because the block would take up a small fraction of the original volume of the ship, it wouldn't have the same displacement and therefore negative buoyancy.

Ships float because they displace a huge amount of water and the buoyant force can support the weight of the ship. Susannp4, public domain image via Pixabay.com

How Does Density of a Liquid Affect Buoyancy?

The density of the fluid an object is placed in affects buoyancy, however Archimedes' principle still applies.

Average density of object

If m is the mass of an object and V is its volume, then the average density ρ of the object is:

ρ = m / V

An object may not be homogenous. This means that the density could vary throughout the volume of the object. For instance if we have a large, hollow steel ball, the density of the steel shell would be about 8000 times the density of the air inside it. The ball could weigh tons, however when we work out the average density using the equation above, if the diameter is large, the average density is much less than the density of a solid steel ball because the mass is a lot less. If the density is less than that of water, the ball will float when placed in water.

Buoyancy and average density

• If the average density of an object is > density of the fluid, it will have negative buoyancy
• If the average density of an object is < density of the fluid, it will have positive buoyancy
• If the average density of an object = density of the fluid, it will have neutral buoyancy

Remember for an object to float, its average density must be lower than the density of the fluid it is placed in. So for instance if the density is less than water but greater than that of kerosene, it will float in water, but not in kerosene. 

A coin floats in mercury because the density of the metal that the coin is made from is less than that of mercury. Alby, CC BY-SA 3.0 via Wikimedia Commons

How Do Helium Balloons Float?

The principle of Archimedes works for objects not just in a liquid like water, but other fluids also, like air. Just like an aeroplane, a balloon needs a force called lift to make it rise in the air. Balloons don't have wings to provide lift and instead use the buoyant force of displaced air.

Hot air and helium balloons rely on buoyancy to give them lift and keep them aloft. © Eugene Brennan

What gives a balloon lift to rise in the surrounding air?

Remember the Archimedes principle states that the upthrust or buoyant force is equal to the weight of the displaced fluid. In the case of a balloon, the displaced fluid is air.

First let's imagine a scenario where we have a large balloon and just fill it with air. The weight acting downwards consists of the weight of the balloon plus the weight of the air inside. However the buoyancy force is the weight of the displaced air (which is approximately the same as the weight of the air inside the balloon, because the displaced air has the same volume, neglecting the volume of the balloon material).

So the force acting downwards = weight of balloon + weight of air inside balloon

From Archimedes' Principle, the force acting upwards = weight of displaced air ≈ weight of air inside balloon

Net force acting downwards = (weight of balloon + weight of air inside balloon) - weight of air inside balloon = weight of balloon

Therefore the balloon will sink.

Weight of balloon and air inside (and also the basket and people, ropes etc) is greater than the buoyant force which is the weight of displaced air, so it sinks. © Eugene Brennan

Now imagine we make the balloon large so that it has a lot of space inside.

Let's make it a sphere 10 metres in diameter and fill it with helium. Helium has a density less than that of air.

The volume is approximately 524 cubic metres.

This much helium weighs about 94 kilos.

The balloon displaces 524 cubic metres of air, however air is nearly six times denser than helium, so that air weighs about 642 kg.

So from Archimedes principle, we know that the upthrust equals this weight. The upthrust of 642 kg acting upwards on the balloon is greater than the weight of the helium inside the balloon and this gives it lift.

In practice the weight acting downwards would be greater because of the weight of the skin of the balloon, basket, people etc.

Weight of balloon and helium inside it is less than weight of displaced air, so the buoyant force gives enough lift to make it rise. © Eugene Brennan

Why Do Hot Air Balloons Float ?

Helium balloons float because they're filled with helium which is less dense than air. Hot air balloons have tanks of propane and burners on board in the basket. Propane is the gas used for camping stoves and outdoor cooking grills. When the gas is burned, it heats the air. This rises upwards and fills the balloon, displacing the air inside. Because the air inside the balloon is hotter than the ambient temperature of the air outside, it's less dense and weighs less. So the air displaced by the balloon is heavier than the air inside it. Since the upthrust force equals the weight of the displaced air, this exceeds the weight of the balloon and the less dense hot air inside it and this lift force causes the balloon to rise.

A hot air balloon. Stux, public domain image via Pixabay.com

The weight of displaced air (which produces the buoyant force) is greater than the weight of the balloon's skin, basket, burners and less dense hot air inside it and this gives it enough lift to rise. © Eugene Brennan

Worked Examples on Buoyancy

Question 1:

A hollow steel ball weighing 10 kg and diameter 30 cm is pushed below the surface of water in a pool.
Calculate the net force pushing the ball back to the surface.

Calculate the buoyant force on a steel ball submerged in water. © Eugene Brennan

Answer:

We need to calculate the volume of water displaced. Then knowing the density of water, we can work out the weight of water and thus the buoyant force.

Volume of a sphere V = 4/3 π r³

r is the radius of of the sphere

π = 3.1416 approx

We know the diameter of the sphere is 30 cm = 30 x 10^-2 m

so r = 15 x 10^-2 m

Substituting for r and π gives us

V = 4/3 x 3.1416 x (15 x 10^-2)³

Now work out the mass of water displaced by this volume.

ρ = m / V

where ρ is the density of a material, m is its mass and V is the volume.

Rearranging

m = ρV

for pure water ρ = 1000 kg / m³

Substituting for ρ and V calculated previously gives us the mass m

m = ρV = 1000 x 4/3 x 3.1416 x (15 x 10^-2)³

= 14.137 kg approx

So the ball weighs 10 kg, but the displaced water weighs 14.137 kg. This results in a buoyant force of 14.137 kg acting upwards.

The net force pushing the ball to the surface is 14.137 - 10 = 4.137 kg

The ball has positive buoyancy, so it will rise to the surface and float, stabilizing with enough of its volume submerged to displace 10kg of water to balance its own 10kg weight.

Note: Strictly speaking force is measured in Newtons. If m is mass and g is the acceleration due to gravity = 9.81 approx, then force F = mg. A weight of 1 kg is equivalent to approximately 1 x g = 1 x 9.81 = 9.81 N
See my guide on mechanics for more info:

Newton's Laws of Motion and Understanding Force, Mass, Acceleration, Velocity, Friction, Power and Vectors

Question 2:

How many balloons would it take to lift a person if the balloons were filled with helium? Each balloon is spherical, 30 cm diameter and weighs 10g (including string). The person weighs 90 kg.
Assume that the thickness of the skin of the balloon is small compared to the diameter, so the volume of helium internally = volume of air displaced.

How many helium balloons does it take to lift a person? © Eugene Brennan

 

Answer:

First assign variables and convert to standard SI units.

Let d be the diameter of a balloon = 30 cm = 30 x 10^-2 m

so r the radius of a balloon = 15 x 10^-2 m

Let N be the number of balloons.

Let ρ_h be the density of helium = 0.1786 g / L = 1.786 x 10^-1 kg/m³

Let ρ_a be the density of air = 1.225 kg/m³

Let V be the volume of a balloon

Let w be the weight of each balloon = 10 g = 10 x 10^-3 kg

Let W_p be the weight of the person = 90 kg

The volume of each balloon is V = 4/3πr³

Total volume of all the balloons = NV

Total weight of helium = volume of balloons x density of helium = NVρ_h

Total weight of air displaced = buoyancy force = volume of balloons x density of air = NVρ_a

Total weight of balloons = Nw

Buoyancy force = weight of air displaced and this must be greater than the weight of the helium + the weight of balloons + person's weight.

Volume of a balloon V = 4/3πr³ = 4/3 x 3.1416 x (15 x 10^-2)³ = 1.414 x 10^-2 m³

So write the equation:

NVρ_a > NVρ_h + Nw + W_p

Rearrange giving N(Vρ_a - Vρ_h - w) > W_p

So N > W_p / (Vρ_a - Vρ_h - w)

Plug in the numbers:

N > 90 / (1.414 x 10^-2 x 1.225 - 1.414 x 10^-2 x 1.786 x 10^-1 - 10 x 10^-3) = 18,765

That's a lot of balloons and the problem is that as the number of the balloons increases, the helium also has to lift the weight of the material they're made of. A better solution is to use bigger balloons. If we increase the diameter, volume increases with the cube of the radius while surface area only increases with the square of the radius. So there's more volume of helium per weight of balloon and less balloons are needed. Imagine we increase the size of the balloons by a factor of ten from 30 cm to 3 m.

Surface area = 4πr²

If r is now 10 times greater, the r squared factor results in a 100 times increase in surface area (assuming the material is stretched by the same amount on inflation.) So weight increases 100 times and a balloon now weighs 100 x 10 g = 1 kg.

However volume is proportional to r³ so if r increases ten times, volume increases 1000 times, but balloon weight has only increased 100 times. The bottom line is that a bigger balloon has greater net lifting force, not just because of the bigger dimensions, but because balloon weight/volume becomes smaller.

The volume of a balloon is now:

V = 4/3πr³ = 4/3 x 3.1416 x (1.5 )³ = 14.137 m³

Plug in the numbers again:

N > 90 / (14.137 x 1.225 - 14.137x 1.786 x 10^-1 - 1) = 6.5

That's a bit better, so only 7 balloons needed this time.

This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional. 

Related Reading

Newton's 3 Laws of Motion: Force, Mass and Acceleration

© 2019 Eugene Brennan

How to Understand Electricity: Volts, Amps and Watts Explained on Appliances

© Eugene Brennan

This comprehensive guide for consumers and students explains everything about volts, amps and watts and how they apply to home appliances and circuits. The equations are really quite simple, and you'll find some examples on how to apply them to home appliances.

Electricity 101: Understanding the Basics

In this tutorial, you'll learn about:

• Volts, watts, amps
• Power consumption of appliances and kilowatt hours (kWh)
• Ohm's law and resistance
• Resistivity and how it affects the resistance of a material
• Fuses and how they protect wiring and appliances
• How electricity is produced
• Devices used to measure voltage, current and resistance
• The effects of electric and magnetic fields
• Conductors, insulators and superconductors
• The basics of AC and DC
• Arcs and sparks
• Power supplies and voltage regulation
• Tracking electricity usage in the home

What Is Electricity and an Electric Current?

All matter is made from basic building blocks called atoms. A simplistic model of an atom, known as the Rutherford–Bohr model, Bohr model or Bohr diagram has a central nucleus made up of particles called protons and neutrons. The nucleus is surrounded by orbitals containing electrons. In some materials such as metals, electrons are bound loosely to the nucleus so they can detach and move when a voltage is applied. These materials are known as conductors and can conduct electricity. The flow of electrons is called an electric current.

Conceptual image of atom with protons and neutrons in the central nucleus and electrons in outer orbitals. Geralt, public domain image via Pixabay.com

Electrons with a negative charge flowing through a conductor. © Eugene Brennan

In reality, electrons don't actually move from one end of a conductor to another. Instead they transfer energy between each other just like the steel balls in this Newton's cradle, while moving negligibly themselves. DemondeLuxe, Creative Commons Image 3.0 unported

 

How Does Current Flow in a Circuit?

In the photo below, an AA cell, which is an example of a voltage source, powers a torch bulb. Electric current first flows out the top of the battery, through the wire and bulb and then returns via the bottom wire. So it always flows in a loop and two wires are needed to connect the voltage source to the load. 

In this example of a simple circuit, an AA cell forces current through the wires and lights up a bulb. © Eugene Brennan

Schematic or Circuit Diagram 

We can represent this circuit in a simple manner using a schematic or circuit diagram. Looking at the schematic below, a voltage source V will force a current I around the circuit through the load (the bulb in this case) whose resistance is R.

The load R could be an electrical appliance such as a heater, bulb, LED, motor or component in an electronic circuit. The lines joining the source to the resistance would be the connecting wires inside an appliance or power flex, or tracks on a printed circuit board.

A schematic of of a simple circuit. The voltage source V causes current I to flow around in a loop through the resistance or load R. © Eugene Brennan

 

Current Measured in Amps

The current I measured in amps is given by the equation:

I = V/R

So current is simply equal to voltage divided by resistance. This is called Ohm's Law and we'll examine it in more detail later and see some examples.

 

Voltage (measured in volts) forces current (measured in amps) through the resistance of a circuit (measured in ohms).

Current Direction in a Circuit

Conventionally we think of current flowing out the positive terminal of a source such as a battery. However current is a flow of sub-atomic particles called electrons which are negatively charged, so current actually flows the other way, from the negative terminal of the battery.

What Are Examples of Voltage Sources?

• Battery
• Mains voltage at a socket outlet
• Alternator or DC generator (dynamo)
• Solar cell
• Thermopile
• Laboratory power supply

The Water Pipe Analogy for Explaining Electricity

Voltage and current are like water pressure and water flow rate respectively, and reference is often made to pumps and water pipes as an analogy to explain electrical circuitry. A pump forces water through pipes and when it pumps harder, more water is forced through. The walls of pipes cause resistance to the flow of water. Similarly a voltage source "pumps" electric current through conductors and the current depends on the "pressure" of the source.

What's the Difference Between Watts, Amps and Volts?

Like any discipline, electrical engineering has jargon or specialised terminology.
 

What Are Volts?

Voltage is the pressure in a circuit and measured in volts. Think of a pump in a water pipe. The greater the pressure and the force which the pump exerts, the greater will be the flow of water through the pipe. Similarly a voltage source is like a pump and pushes electrons around the circuit. The higher the voltage applied to a circuit, the greater the current which will be forced through it.

What Are Amps?

An electric current is due to the movement of electrons through a conductor and load and is measured in amps. High current means lots of electrons flowing through the circuit. The water analogy is water flow rate in gallons per minute or litres per second

What Is a Load?

This is the device connected to a voltage source. It could be a motor, bulb, heater, LED, or an electronic resistor.

What Are Watts?

Power is the rate at which energy is consumed by a load and is measured in watts. A kilowatt is 1000 watts, also abbreviated to kW. Low powers are measured in milliwatts (mW) or thousandths of a watt.

What Are KWh or Kilowatt Hours?

Kwh are a measure of energy consumption. KWh are sometimes called units and are what you pay for on your electricity bill. A 1 kilowatt (1000 watt) appliance uses 1 kilowatt hour of electricity (1 kWh) in one hour. Similarly a 500 watt device uses 1 kWh of electricity in 2 hours.

What Are Some Commonly Used Voltages?

Commonly used voltages of devices and systems

How to Convert Between Volts, Amps and Watts

Now let's examine the quantities which are usually of interest when dealing with appliances, such as volts, amps and watts and how to convert between them. If you look at the casing of an appliance (see photo below) you can usually find a specification label or panel which indicates the voltage supply, frequency, wattage and possibly current. On some appliances e.g. TVs and washing machines, this panel may be mounted at the back of the device.

This is the equation we need to remember. Once we know it, we can rearrange to find the other two.

Examples of Working Out Watts, Amps and Volts for Appliances

How to Work Out Watts

Watts = Volts x Amps

e.g. A 120 volt appliance takes 2 amps, what is the power?

Power in watts = 120 x 2 = 240 watts

How to Work Out Amps

Amps = Watts / Volts

e.g. A 240 volt appliance consumes 480 watts of power, How much current does it draw?

Current in amps = 480 / 240 = 2 amps

How to Work Out Volts

Volts = Watts / Amps

e.g. A 720 watt appliance draws 3 amps, What voltage is it running on?

Voltage in volts = 720 / 3 = 240 volts

So it's really that simple. Notice I have chosen values in the examples so that everything works out nicely. You only really need to remember the first equation and if you know basic algebra you can rearrange to give the other two equations. However as you can see, you always need to know two of the quantities before you can work out the third quantity. From looking at the Google Analytics statistics and the questions which land people on this webpage, I often see questions asked such as "how many watts are in 480 volts?", which obviously makes no sense!

For high powered appliances, power is often specified in kilowatts ( abbreviated to kW)

1 kilowatt = 1000 watts

What Is a kWh? How to Calculate Energy Consumption of Appliances

Power is the rate at which a device uses energy. So for instance an air conditioning unit, shower or kitchen range/cooker uses electrical energy much faster than a light bulb. Power is normally written on a label or embossed into the plastic casing of an appliance.

Energy Used = Power x Time

So to figure out the energy usage of an appliance, you multiply its power rating by the time period for which it is running. The standard unit of energy is the joule or calorie, but generally energy used in the home by appliances is measured in kWh, also known as "units". To work out the number of kwh, you divide the power in watts by 1000 to convert to kilowatt (kW) and then multiply by time in hours to give kWh. So:

Energy in kWh = Watts / 1000 x Time in Hours

Kilowatt hours, kWh or units are what you pay for on your bill. Your electricity meter counts and displays the number of units used by all the appliances and lighting in your home.

e.g. A 2500 watt drier runs for 3 hours a day, how many kWh does it consume and if electricity costs 24c per unit, what is the cost of running it?

kWh = watts/1000 x time = 2500 / 1000 x 3 = 7.5 kWh or units

Cost = 7.5 x 24c = 180 cent

Some appliances don't run continuously. Examples are devices controlled by a thermostat such as refrigerators, freezers, ovens in cookers and air conditioning systems. The time for which the appliance is powered on and consuming power is called the duty cycle and it is often quoted as a percentage. So for instance a fridge which stays on half of the time has a duty cycle of 50%.

Labels on Appliances Showing Volts, Current and Power Rating

Typical electrical appliance labels/panels indicating voltage and current rating, power rating and frequency in hertz. © Eugene Brennan

How to Convert Horsepower to Watts

Horsepower is a measure of ... you guessed it! ... power!

Just as an engine's mechanical output can be measured in horsepower, so can the power of an electric motor.

1 Horsepower = 746 Watts

E.g. A fractional horsepower motor in a washing machine is rated at 1/2 horsepower.

So the power output of the motor = 746 watts x 0.5 = 373 watts

A motor is not 100% efficient, in other words not all the electrical power input is converted into mechanical power at the output shaft, some being wasted as heat in the windings.

Kilowatt hour electricity meter for measuring energy used in a home. © Eugene Brennan

Recommended Books

Introductory Circuit Analysis by Robert L Boylestad covers the basics of electricity and also more advanced topics such as AC theory, magnetic circuits, electrostatics and electronic simulation using SPICE based software. It's well illustrated and suitable for high school students and also first and second year electric or electronic engineering students.

Introductory Circuit Analysis. Source image, Amazon

 

Ohms's Law and Electrical Resistance

A simple circuit with a voltage source and load. The load has a resistance measured in ohms. © Eugene Brennan 

In the circuit above, a voltage V pushes a current I around the circuit and through the load. As you may remember, this could be a device such as a bulb, electrical heater, motor, LED or other electrical appliance. The load resists the flow of current and the magnitude of its resistance is R ohms.

So

I = V / R

or

R = V / I

This is known as Ohm's law and basically says that the current is proportional to the voltage and inversely proportional to the resistance (As resistance increases, current decreases and vice versa) Remember the resistance measured in ohms is just a measure of how the load or appliance in the circuit "resists" the flow of current. In electronic circuits and some electrical appliances, components called resistors have precise values of resistance so that they can be used to control the value of current flowing in a circuit.

Every electrical device or load has resistance. Resistance is like a restriction to the flow of electrons and electricity is dissipated as heat energy in a resistance. For a fixed voltage applied to a load, the higher the resistance, the lower the current. Going back to the water analogy, when you stand on a hose, you increase the resistance and restrict the flow. The only way to restore the flow is by getting the pump to pump harder, and force water through the restriction, i.e. the pump needs to have a higher pressure. Alternatively if you take your foot off the hose, you increase the diameter and lower the resistance and more water can be forced through. In an electrical circuit, if the voltage of the source is increased, more current is forced through the resistance. If the resistance is lowered, more current will flow even if the voltage of the source doesn't change. Even connecting wires in a circuit have resistance, so when high currents need to be carried by a cable, the copper or aluminium cores need to have a sufficiently large cross sectional area (CSA) to avoid overheating.

Calculating the Current Flowing Through a Resistance

The resistance in a circuit is 100 ohms, a voltage of 120 volts is applied, what is the current?

Current = 120 / 100 = 1.2 amps

A load could be an electronic resistor like this one, or an electrical appliance. © Eugene Brennan

How to Work Out Power in a Circuit Knowing the Resistance and Voltage or Current

Remember watts = volts x amps? An alternative way to work out power is from the resistance in ohms:

So if I is the current in amps, V is the voltage, R is the resistance in ohms and P is the power in watts,

Then:

I = V / R from Ohm's law

But also power in watts = volts x amps, i.e. P = VI

So substituting the expression I =V/R into P = VI gives:

P = VI = V(V/R) = V²/ R

similarly

P = VI = (IR)I = I²R

It's unlikely when dealing with appliances in the home to need to use the last two equations. However here is an example.

A 240 volt supply is connected to a load of 100 ohms. What is the power consumption of the load?

Power = V²/ R = (240)² / 100 = 576 watts

Summary of Equations for an Electric Circuit

A summary of equations for an electric circuit

 

What Are Electrical Conductors?

A conductor is a physical medium which carries an electric current. This could be a power cable, prongs on a plug, a liquid such as water, battery acid or ionised gas in a discharge lamp (e.g. fluorescent or sodium lamp). Pure deionised water doesn't conduct electricity, but most water contains ions, hence why it's a hazard in contact with electrical appliances. Many solutions of water and other substances conduct electricity, e.g. water and common table salt.

In the case of a solid conductor such as copper wire, the electrical resistance is proportional to the length of the conductor and inversely proportional to its cross-sectional area. In effect this means that the longer a piece of wire, the higher its resistance. Similarly the greater the diameter of the wire, the lower its resistance. This has implications for conductors used in appliances and power transmission. For example, the gauge of wire used in an extension lead is important, if the wire is too thin, the resistance will be high and the cable can overheat. If a power cable is very long, its resistance may be too high if not properly rated, resulting in an unacceptable voltage drop at the end of the cable (because of the resistance).

What Is Resistivity and How Does It Affect Resistance?

For a conductor with cross sectional area A and length l, the resistance R can be calculated using the equation:

R = ρl / A

ρ (Greek letter "rho") is a constant known as the resistivity and is a measure of how good the material is at conducting electricity. The lower the resistivity of a material, the lower will be the resistance of the conductor.

Copper has the lowest resistivity of most common materials and this is why it is widely used in the manufacture of cables. Silver has a lower resistivity than copper, but it is much more expensive. Aluminium is generally used for overhead cables and although it has a higher resistivity than copper, it is lighter. Gold has a resistivity about 1.5 times that of copper, however it is unreactive and doesn't oxidise (tarnish). A tarnish coating on a conductor increases contact resistance, so this is why gold is often used as a coating on audio / video connectors. Gold is also used for the miniature connecting wires in integrated circuits.
Insulators are conductors with very high and for all practical purposes infinite resistance.

Resistivities of Various Materials

Materials with increasing resistivity.

What Is an Insulator?

An electrical insulator is a material which has a very high resistance because there are no free electrons to carry current. For all practical purposes an insulator can be considered to have infinite resistance. Because resistance is infinite (infinity is represented by the symbol ∞), then current through an insulator is:

Current = Voltage / resistance = voltage / ∞ = 0

Insulators are used to prevent current flow between two electrical points with differing voltage e.g. insulation on the individual cores of a power cable, the plastic of a power plug or glass/ceramic insulators on power lines. They also prevent high voltage from causing electric shock.

What Are Insulators Made Of?

Typical insulating material used for electrical purposes are:

• Plastic
• Ceramic
• Glass
• Glass epoxy (used for PCBs)
• Bakelite (an older style thermosetting plastic)
• Mica

Detail of the insulator string (the vertical string of discs) on a 275,000 volt electricity pylon near Thornbury, South Gloucestershire, England. Adrian Pingstone, Public domain image via Wikimedia Commons

PVC insulation on the cores of a power flex. © Eugene Brennan

 

The insulating black plastic shrouds on the pins of this plug prevent contact with the pins during insertion/removal. © Eugene Brennan

How Is Electricity Made?

Since electricity is a flow of electrons, it isn't really made. Instead it is produced or generated when these electrons are moved.

Electricity is produced from:

• Batteries
• DC generators or AC alternators
• Solar cells
• Thermopiles

Early 20th century (1909) alternators in a hydroelectric power station. Sergei Prokudin-Gorskii - Public domain image via Wikipedia

What Does a Power Plant Do?

A power station, also known as a power plant generates electricity using alternators or solar cells. There are several types of power plants, thermal, hydroelectric, wind, wave, tide and solar.

Power Stations That Use Alternators to Generate Electricity

• Hydroelectric Power Station: In a hydroelectric power station, water flowing through pipes from a dammed lake turns the blades of a turbine attached to the shaft of an alternator. The alternator then generates electricity.

• Thermal Power Station: Fossil fuels such as coal, oil, gas and peat or renewable energy crops like willow are burned and the heat is used to boil water and generate steam at high pressure. The steam passes through pipes to a steam turbine and turns it at high speed. Again the steam turbine is connected to the shaft of an alternator, turning it and generating electricity. Nuclear power stations are also thermal using the heat of nuclear fission to boil water and turn it into steam.

• Wind Farm: A wind farm uses windmills to generate electricity. Wind turns the blades of the windmill which are connected to a metal shaft. This shaft turns an alternator and this generates electricity. Wind farms can have several hundred windmills spread over hundreds of acres. 

• Wave and Tide Generation: Wave energy generators use the motion of waves to operate an electric generator. Tidal generators are like undersea windmills and use the flow of water during rising and ebbing tides to turn giant under water "propellers". Like a windmill, the propeller is connected to an alternator that generates electricity.

A wind farm. BKnight97, CC by SA 4.0 via Wikimedia Commons

 

Power Stations That Don't Use Alternators to Produce Electricity

• Solar Farm: Solar panels are large flat panels made of special semiconductor material. When sunshine lands on the panels, they produce an electric current. The larger the area of the panel, the greater is the electricity produced. Just like wind farms, solar-generating farms can be spread over a large area and consist of hundreds of panels. However people can also have solar panels fixed on their roof to generate some of their electricity requirements. Solar panels are becoming more efficient, which means that they can produce useful amounts of electricity even on cloudy days.

Solar panels don't have any moving parts unlike an alternator.  RosiePosie, public domain image via Pixabay.com

What Is a Thermopile?

A thermopile is an array of thermocouples connected together, usually in series. A thermocouple works on the principle of the Seebeck Effect to produce electricity. Thermopiles aren't really used to produce energy commercially, but are the only method of generating the electricity required by probes in deep space, far from the Sun. Solar power isn't an option and batteries wouldn't have the capacity to last the mission. So radioisotope thermoelectric generators (RTG) are used consisting of a nuclear source to produce heat and a thermopile to produce electricity.

What Is the Voltage Supplied to Our Homes?

In general, the voltage supply to your home is nominally 230 or 120 volts. Voltage in the USA is 120 volts, but two "hots" are supplied to homes so that a 240 volt supply is also available between the hots. The higher voltage is used for high powered appliances such as washers, driers, kitchen ranges (cookers) and air conditioning. 120 volts is used for lower power and portable devices. It is also safer because in the event of an electrical shock, less current flows through the body so there is a lesser risk of electrocution.

In countries where 230 volts is standard, generators or step down isolating transformers are used to provide a 110 volt supply for power tools. This is normally mandatory on construction sites. Again the idea of the lower voltage is to lessen the danger of electrocution, if for example a power flex is inadvertently cut, or a tool gets wet.

Utility voltage by country. SomnusDe, Public domain, via Wikimedia Commons

What Are Fuses For?

As we will see later, electrical cables, appliances, wires inside appliances, components etc all have resistance. This resistance produces heat when current flows through it. Any electrical conductor can get excessively hot if too much current flows and in the case of wires, this can cause the plastic insulation covering the cable to melt or even catch fire. So fuses are used in series with a cable or appliance to limit current flow and make everything safe. Fuses are like a "weak link" in a chain and blow before damage can occur. They have a specified rating and this is not the current they blow at, but the current they will carry without blowing. Once current exceeds the rating of the fuse, the fuse will blow. The length of time it takes for the fuse to blow is proportional to the current. So minor overloads can result in a fuse blowing in minutes, but if there is a large current or short circuit scenario, the fuse will blow in seconds or milliseconds.

Breaking Capacity of Fuses

Fuses have a max current they can carry without the encapsulation of the fuse rupturing. So fuses on the secondary of domestic power supplies in TVs, battery chargers and other electronic appliances are often glass types because the supply will only source a relatively small amount of energy if there is a fault. Ceramic types are used to resist the heat and shock that occurs when the inrush current can be perhaps hundred or thousands of amps, potentially feeding a huge amount of instantaneous power. If a short circuit occurs in an appliance, it's quite possible that the utility transformer in your street can feed current of this magnitude into the short. So for example the BS1362 fuse in a UK style plug has a ceramic body. Blown fuses should always be replaced by the same type, ceramic if necessary, to avoid a fire occurring.

Fuse Types

In general, fuses are fast blow (F) and time lag (T). Time lag types are often used for power supplies in electronic equipment because the capacitors take a surge of current as they charge up, which would blow a fast acting fuse.

How Many Amps Are Supplied to Our Homes?

Typically for a 230 volt supply to a home, the main fuse rating is 80 to 100 amps at the consumer unit. So this is the maximum current that will flow before the fuse blows. At 80 A and 230 volts, this allows a power draw of 230 x 80 = 18.4 kW.

A High Breaking Capacity (HBC) BS1362 fuse, used as standard in a UK style plug. © Eugene Brennan

Testing and Measuring Voltages

How Do You Check Voltage With a Multimeter?

A multimeter is an instrument that can measure voltage, current, resistance and possibly additional parameters. You can also use it to check continuity of cables and check fuses. If you don't know how to use one, read my guide How to Use a Digital Multimeter (DMM) to Measure Voltage, Current, and Resistance. Multimeters normally have a continuity range also, and this comes in useful for checking breaks in cables, fuses and loose connections.

 

Fluke digital multimeter. © Eugene Brennan

How to Test a Live Wire

For this it's best to stay safe and use a non-contact volt tester or phase tester screwdriver. These will indicate if voltage is e.g > 100 volts. A multimeter can only measure the voltage between live and neutral or live and earth (ground) if these conductors/terminals are accessible, which may not always be the case.

A non-contact detector is a standard tool in any electrician's tool kit, but useful for homeowners also. I use one of these for identifying which conductor is live whenever I'm doing any home maintenance. Unlike a neon screwdriver (phase tester), you can use one of these in situations when live parts/wires are shrouded or covered with insulation and you can't make contact with wires. It also comes in useful for checking whether there's a break in a power flex and where the break occurs.

Note: It's always a good idea to use a neon tester to double check that power is definitely off when doing any electrical maintenance.

How Can I Measure Electricity Usage in My Home?

An electricity usage monitor or tracker tells you everything you want to know about your appliance behaviour. The parameters are displayed on an LCD and include voltage, current, power consumption, kWh used, cost of running and run time of appliance. The latter is useful for troubleshooting fridges, freezers, air conditioners etc which are controlled by a thermostat and switch on and off. A failed thermostat or waterlogged insulation can cause an appliance to run constantly, so this problem can be identified.

Power consumption monitoring adapter. © Eugene Brennan

What Exactly is Meant By "Electrical Energy Consumption"?

What happens when an appliance is powered from electricity? Scientists tell us that energy cannot be destroyed, it just changes from one form to another. This process happens all the time - on Earth and throughout the Universe. For instance a rock on the edge of a cliff has potential energy, because of its altitude above the ground. If it falls over the edge of the cliff, it starts to pick up velocity, i.e. gains kinetic energy (motion energy) while losing potential energy. When it hits the ground, this energy is dissipated as heat (think of the heat produced by an asteroid impact). Similarly when an appliance is plugged in, the electricity doesn't get wasted or "consumed", in the sense of being destroyed, it simply changes form. So in the case of a lamp, it ends up as light energy or as heat energy when a heater is used. Electrical energy can also be converted to sound in a loudspeaker or electromagnetic radiation (microwave oven or radio transmitter), all forms of energy. Electrical energy can also be converted to kinetic energy in an electric motor or to potential energy when an elevator is raised in a building.

Power is a measure of the rate at which energy is used. So for instance a 1000 watt heater or high powered hvac air conditioning system uses energy at a higher rate than a 60 watt light bulb.

How is Electricity Converted to Other Forms of Energy?

Converting energy from one form to another

What Is a Superconductor?

When certain materials are subjected to very low temperatures, their resistance falls to zero.

Since V = IR, if R is zero, then V becomes 0 even if I is non zero.

The consequences of this are that a current can flow even if the voltage source is removed. Because resistance is zero, and no heat is dissipated, huge currents can be carried by thin cables. Superconductors are used for example in MRI machines to carry the high currents required by powerful magnets.

 

Superconducting cables. RRama, CC BY-SA 2.0 FR , via Wikimedia Co

What Are AC, DC and Three-Phase?

The current produced by a power source can take one of two forms, AC or DC. The power source could be a battery, electrical generator, power transmitted along service cables to your home or the output of a signal generator, a device used in laboratories or by test personnel when testing or designing electronic systems.

DC Explained

This stands for direct current so the current provided by the source only flows one way. A DC source will have a nominal value voltage level and this voltage will fall as the source is loaded and outputs more current. This drop is due to inherent internal resistance within the source. The resistance is not due to an actual resistor, but can be modelled as such, and is composed of actual resistance of conductors, electronic components, electrolyte in batteries etc.

Examples of DC sources are batteries, DC generators known as dynamos, solar cells and thermocouples.

AC Explained

This stands for "alternating current" and means that the current "alternates" or changes direction. So current flows one way, reaches a peak, falls to zero, changes direction, reaches a peak and then falls back to zero again before the whole cycle is repeated. The number of times this cycle happens per second is called the frequency. In the U.S. the frequency is 60 Hertz (Hz) or cycles per second. In other countries it is 50 Hz. The electricity supply in your home is AC.

The advantage of AC is the ease by which it can be transformed from one voltage level to another by a device known as a transformer.

AC sources include the electrical supply to your home, generators in power stations, transformers, DC to AC inverters (allowing you to power appliances from the cigarette lighter in your car), signal generators and variable frequency drives for controlling the speed of motors. The alternator in a vehicle generates electricity as AC before it is rectified and converted to DC. New generation brushless, cordless tools convert the DC voltage of the battery to AC for driving the motor.

Reducing Costs of Transmitting Electricity Over the Grid

Because AC can so easily be transformed from one voltage to another, it is more advantageous for power transmission over the electricity grid. Generators in power stations output a relatively low voltage, typically 10,000 volts. Transformers can then step this up to a higher voltage, 200,000, 400,000 volts or higher for transmission through the country. A step up transformer, converts the input power to a higher voltage, lower current output. Now this decrease in current is the desired effect for two reasons. Firstly, voltage drop is reduced in the transmission lines because of the lower current flowing through the resistance of cables (since V = IR). Secondly, reducing current reduces power loss as current flows through the resistance of the distribution cables (remember power = I²R in the equations above?). Power is wasted as heat in transmission cables, which obviously isn't wanted. If current is halved, power loss becomes a quarter of what it was previously (because of the squared term in the equation for power), If current is made 10 times smaller, power loss is 1% of what it was, and so on.

The AC waveform of the the domestic supply to our homes is sinusoidal. © Eugene Brennan

AC voltage is sinusoidal. © Eugene Brennan

Transformer in an electrical sub-station. The function of a transformer is to either increase or decrease voltage. Image: Rainer Knäpper, Free Art License via Wikimedia Commons

What Is Three-Phase Voltage?

Very long distance transmission lines may use DC to reduce losses, however power is normally distributed nationwide using a 3 phase system. Phase voltages have a sinusoidal waveform and each of the phases is separated by 120 degrees. So in the graph below, phase 1 is a sine wave, phase 2 lags by 120 degrees and phase 3 lags by 240 degrees (or leads by 120 degrees). Only 3 wires are needed to transmit power because it turns out that no current flows in the neutral (for a balanced load). The transformer supplying your home, has 3 phase lines as input and the output is a star source so it provides 3 phase lines plus neutral. In countries such as the UK, homes are fed by one of the phases plus a neutral. In the US, one of the phases is split to provide the two 'hot' legs of the supply.

Why Is 3 Phase Used?

• More power can be transmitted using just 1.5 times the number of wires
• Motors powered by 3 phase are smaller than a similar single phase motor of the same power
• Evening of output torque smooths operation and results in less vibration of motors powered by 3 phase
• Neutral conductor can be reduced in size because of lower current flow
• Neutral is unnecessary for transmitting power between substations and transformers

Note: A "phase" is a single winding in an alternator, winding in a transformer, coil in a motor or resistive load. The phase voltage is the voltage across a phase.

3 Phase Formulas

If Vₚ is the phase voltage

and VL is the line voltage between each phase

Delta load:

Vₚ = VL
Iₚ = IL / √3

Star load:

Vₚ = VL / √3

Iₚ = IL

 
For both star and delta balanced loads with a power factor of 1:

Power = √3VLIL

Delta Star Transformer

A Delta-star (also known as delta - wye or delta Y) transformer is often used for producing a 3 phase, or single phase and neutral supply to homes and industry. The incoming supply is typically 11kv and output phase voltage is 230 volts (in countries which use this voltage).

3 Phase voltages. Each phase is sinusoidal with a phase difference of 120 degrees. J JMesserly modification of original svg by User:SiriusA, Public Domain image via Wikimedia Commons

Delta-Star(Wye) transformer which can supply single or 3-phase supply. © Eugene Brennan

 

Three-phase power lines. Each overhead line is a single phase. Wing-Chi Poon, Creative Commons Attribution-Share Alike 2.5 Generic via Wikimedia Commons

 

What Are Other Effects When a Current Flows?

As mentioned above, when current flows through the resistance of a load, it gets hot. This is sometimes the desired effect, e.g. an electrical heater. However it is an unwanted effect in lamps, because the desired function of the device is to convert electricity to light, and not produce heat as a byproduct. Excessive current in power cables during an overload can potentially cause a fire if protective devices such as fuses or MCBs (Miniature Circuit Breakers) aren't included in line with the cable.

What Does Current Flow Produce?

Magnetic fields. This phenomenon is used in a device called a solenoid or electromagnet which is basically like a spool or coil of wire through which a current flows. Electromagnets are used in the old style, non-electronic, door and phone bells, water inlet valves on washing machines, relays (a switch operated by an electromagnet), starter motors on vehicles and in salvage for lifting iron and steel. Magnetic fields are also the principle upon which all motors work.

Electric fields. Current flowing through a conductor also produces an electric field. An extreme example of this is the high intensity field produced under a high voltage power line which is sufficient to illuminate a fluorescent tube held in the hand.

Magnetic field lines around a conductor. © Eugene Brennan

The electric field under a high voltage power line is sufficient to produce an electric discharge in a fluorescent tube. Image BaronAlaric GNU_Free_Documentation_License version 1.2 via Wikimedia Commons

How Do Switches Work and What Are Sparks?

As you've discovered, if resistance is increased in a circuit, current decreases. If you just break the conductor in a circuit and create an air gap, the magnitude of the resistance for all practical purposes is infinite because air is a good insulator and no current will flow. I.e.

Current = Voltage / Resistance = Voltage / ∞ = 0

So this is how a switch works. Two contacts, usually made of brass in a domestic switch, are pressed together when the switch is on and closed. When the switch is turned off, the contacts rapidly separate and interrupt current.

What Are Sparks?

Imagine two electrodes or points in a circuit separated by an air gap (e.g. the gap in an automotive spark plug). If voltage is high enough, the air between the two points becomes so stressed by the electric field that it becomes ionized, i.e. atoms have their electrons ripped off. These electrons are then able to traverse the gap, attracted by the positive electrode and in doing so, collide with other gas molecules and release more electrons. Eventually an avalanche of electrons occurs (all of this happening in a split second) and the result is called a spark or spark discharge A spark produces a flash of visible light, heat, UV radiation and sound and it's temperature can be about 5000 deg C, hotter than the surface of the sun. The voltage required to produce a spark is about 3000 volts per mm between rounded electrodes in air. Sparks can be small, e.g. automotive spark plug or gas lighter, or much larger.

An example of a large spark is lightning. When clouds get charged up, voltage becomes so high that a spark jumps from cloud to cloud or cloud to ground. The sound we call thunder is caused by the explosive heating and expansion of air by the electrical discharge.

Sparks occur in an air gap when voltage exceeds the breakdown voltage of the gap. When two electrodes are separated, current tends to continue to flow and heating of the metal electrodes causes material to vaporise and also ionise the air. This results is a continuous spark discharge called an arc which is similar to a spark. If the electrodes are separated sufficiently, the arc won't be sustained and will stop abruptly. Arc welding makes use of an arc between two electrodes to melt metal. Switches must also be designed so that their contacts separate sufficiently apart and quickly enough so that arcs are rapidly quenched and reduce damage to the contacts. In substations, large air gaps or oil filled circuit breakers are necessary to quench the high current arcs which occur when high voltage is switched.

References

Boylestad, Robert L. (1968). Introductory Circuit Analysis. (6th ed. 1990) .Merrill Publishing Company, London, England.

This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualised advice from a qualified professional.

© 2012 Eugene Brennan