Power

Discussion

Introduction

definition

Power is the rate at which work is done or the rate at which energy is transfered from one place to another or transformed from one type to another. The symbol for power is an italicized, uppercase P.

The bar (or overline) above the symbol for power in this equation indicates that the value computed would be an average power. It is often omitted, even in formal writing, except when is being contrasted with a definition that uses calculus notation — like in the next sentence.

Power is also the first derivative of work done (or energy transferred or transformed) with respect to time.

When this equation is used, the value computed is an instantaneous power.

units

What is the unit of power? Watt is the unit of power.

The International System unit of power is the joule per second, which is given the special name watt (lowercase) with the symbol being W (uppercase, not italics).

James Watt (1736–1819) was a Scottish-English inventor, engineer, businessperson, scientist who is famous for his improvements to steam engines that powered the industrial revolution in England during the 19th century. The SI unit for power is named after him because he more or less invented the concept of power as a measurable quantity. The unit he invented along with the concept is the horsepower [hp], which is roughly the rate at which workhorses were made to do work grinding grain in England during the 19th century — about 746 watts. More info on the horsepower can be found further down the page.

[1 hp ≈ 746 W]

Now with calculus

The calculus applied to the definition we started with (power is the rate at which work is done) is an example of a derivative — a rate computed over a time so short it can never actually be computed. Instead we examine the limit of our time interval as it approaches, but never reaches, zero.

Condensed description, simply replace every Δ (delta) with a d (dee).

The value calculated this way is now an instantaneous value instead of an average value. You can now use the techniques of calculus when solving problems that involve power.

Once again with harder calculus

The formal definition of work is as a force-displacement path integral.

When written this way, any variance in the force is with respect to the distance traveled along the path.

What if instead I want to think about work as an integral with respect to time?

We begin by multiplying our integral by a special infinitesimal version of 1, dt/dt.

Rearrange the denominators as allowed by the commutative property of multiplication.

Replace the derivative of displacement with it's equivalent (velocity), stop writing the 1 in the denominator (it has no effect on anything here), and there you have it — another way to think about work, this time as a force-velocity path-in-space-and-time integral.

Combining the definition of power (the rate at which work is done)…

with the fundamental theorem of calculus (the integral is the anti-derivative)…

we can now safely say that instantaneous power is the force-velocity scalar product.

units in more detail

watt is the unit of power

From the basic definition…

any units of work (or energy) and time can be used to generate a unit of power. The International System uses joules [J] and seconds [s] for these, respectively.

A joule per second is called a watt [W] in honor of the Scottish mechanical engineer James Watt (1736–1819). Watt is most famous for inventing an improved steam engine in the years around 1770 and slightly less famous for inventing the concept of power shortly thereafter. Power was a new way to compare his engines to the machines they were designed to replace — horses. (More on that later.)

Watt wouldn't have thought about power they same way we do today. The concept of energy wasn't invented until after he died. For him, power was the product of force and velocity.

P = Fv

The units still work out the same way in the SI system, of course. Recall that the joule is the product of a newton and a meter.

But of course, Watt didn't use the SI system or even it's precursor, the metric system. There were no kilograms until 1795. The newton didn't become a unit until 1948. There was no joule in the world of units when Watt was alive because, essentially, there was no Joule in the world of people. (James Joule was eight months old when James Watt died.)

James Watt used pounds for force and a variety of English units for velocity — inches/second, feet/minute, miles/hour, etc.

the original horsepower

While the English engineer Thomas Newcomen (1664–1729) is generally regarded to have invented the steam engine around 1698, Watt's improved design patented in 1769 became the industry standard that powered the Industrial Revolution in Britain and elsewhere.

One of the earliest commercial engines Watt built was sold to a copper mine in Cornwall, a region of England where coal was expensive. Watt supervised the construction of purpose-built steam engines at the mines and then charged a licensing fee equal to a fraction of the money saved by switching to his improved design.

Newcomen and Watt engines are examples of reciprocating engines. So are the engines in most cars and trucks. Steam is pumped into a vertical cylinder, driving a piston up. The steam condenses and atmospheric pressure drives the piston down. In an engine with more than one cylinder, when one of the pistons is moving up, the other is moving down. The motion of one is reciprocated by the motion of the other. (Strangely, a piston driven engine with even one cylinder is still called a reciprocating engine.) Watt's pistons were originally attached to a rocking beam that was perfect for driving a lift pump. This is the classic, old-timey pump with a handle that everyone has probably seen — in photographs, at least, if not in person. Later mechanical additions allowed Watt to transform the reciprocating motion of the beam into the rotational motion of an axle. This opened the steam engine up to new applications.

The strongest competitor to the steam engine at the time of its invention was the horse. One of the more ingenious ways the power of the horse was harnessed was the horse mill (also known as the horse gin) — a large spoke and axle contraption like a wagon wheel without a rim that could be rotated horizontally. Horses were harnessed to the ends of the spokes (four to six at a time, for large applications) and compelled to walk in circles around the central drive shaft for hours at a time. The human powered equivalent of a horse mill is called a treadmill. 18th century treadmills weren't anything like the exercise equipment first sold in the 20th century. Horse mills and treadmills were preindustrial machines for doing work — not the abstract mathematical work of force times displacement, but genuine back breaking, hard labor.

In order for Watt to charge a licensing fee for his "rotative" steam engines, he needed an economic equivalent — something he could compare them to. Horses were the natural choice, but how much work does a horse do? Work isn't even the right concept. One horse can do so much work, but two horses will do it twice as fast. It's not the amount of work a horse does that matters, it's the rate at which it does it.

Watt identified the Whitbread Brewery in London as a potential customer. Large London breweries like Whitbread's are estimated to have employed an average of 20 horses for the mill at once. Whitbread's horses were strapped six to a mill and set to walk a 24 foot diameter circle 144 times per hour grinding malted barley into powder with a force of 180 pounds.

The resulting irrational number was rounded to two significant digits for convenience so that, by definition…

1 horsepower = 33,000 foot pounds per minute

or equivalently, since there are 60 seconds in a minute…

1 horsepower = 550 foot pounds per second

James Watt never published this definition himself. The first time it appeared in print was in a book review. The author, Olinthus Gregory, complained about Watt's "ridiculous" notion of the unit horsepower in his book on mechanics.

It follows, from what has been said, and from the consideration of the strengths of horses variously employed, such as waggon horses, dray horses, plough horses, heavy horses, light coach horses, &c. that what is called "horse power" is of so fluctuating and indefinite a nature, that it is perfectly ridiculous to assume it as a common measure, by which the force of steam engines and other machines should be appreciated.

Olinthus Gregory, 1806

An anonymous reviewer of Gregory's book defended Watt's notion of the unit horsepower and included the now standard definition.

Boulton and Watt, however, have not left the matter in a state that can be accounted incorrect in any case, but have given to it all the accuracy that can be required, when, from the result of experiments made with the strong horses employed by the brewers in London they have assumed, as the standard of a horse's power, a force able to raise 33,000 lib. one foot high in a minute; and this, no doubt, was meant to include an allowance of power sufficiently ample to cover the usual variations of the strength of horses, and of other circumstances that may affect the accuracy of the result.

Edinburgh Review, 1809

more horsepower, because one is never enough

There are several ways to interpret James Watt's definition of the unit horsepower in terms of the International System of Units (SI). The most direct is to convert the English units to their SI equivalents and use the standard value of gravity to convert mass into weight. This is the so called mechanical horsepower.

Electricians and electrical engineers decided to round this number to the nearest whole number. Nice and simple. This is the definition of the electrical horsepower.

1 electrical horsepower = 746 W

The electrical and mechanical definitions of the horsepower agree with each other to within 4 parts in 10,000.

James Watt's horsepower is also roughly the same as lifting a 75 kg mass at a speed of 1 m/s on Earth. This became the definition for the so called metric horsepower.

1 metric horsepower = (75 kg)(9.80665 m/s²)(1 m/s) = 735.499 W exactly

Using 76 kg for the mass would have brought the metric horsepower closer to the mechanical horsepower, but I supposed 76 kg didn't feel "metricky" enough. The whole exercise seems silly since neither 75 nor 76 is a multiple of 10. Standard gravity not equaling 10 doesn't help much either.

Comparing the mechanical horsepower to the metric horsepower gives a ratio that is exact to 8 decimal places.

The reciprocal ratio is a lot messier and is only approximate when stated with 8 decimal places.

Next up, the boiler horsepower. The Btu per hour (often erroneously shortened to Btu) is a unit of power used by the heating, ventilation, and cooling industry (HVAC). Or should the Btu/h be its own thing? COME BACK AND FIX THIS.

horsepower symbols

Since the horsepower was never a part of the International System of Units, there is no international standard symbol for it. In English, hp is used for horsepower and the reader is often just supposed to know what type it is from context. Other European nations use the symbol corresponding to the word for horsepower in their respective languages, and the fact that it is metric is assumed. (I assume.)

physiology