The Physics
Hypertextbook
Opus in profectus

# Power

## Discussion

Intro

 P̅ = ΔW Δt
 P = dW dt

= F̅v cos θ P = F · v
Power of some things
power (W) device, phenomena, process, event
gamma ray burster
3.6 × 1039 typical quasar
3.6 × 1026 the sun
1.25 × 1015 most powerful laser (Petawatt)
1.3 × 1013 total human consumption, global
3.2 × 1012 total human consumption, US
1.2 × 1010 space shuttle at launch
109 ~ 1010 large commercial power plant
4,700,000 most powerful locomotive (GE AC6000 CW)
2,610,000 most powerful truck (Komatsu 980E-4)
1,800,000 most powerful radio transmitter (VLF Cutler, Maine)
1,550,000 most powerful car (Arash AF10)
10,000 Watt's steam engine of 1778
746 1 horsepower
100 human, daily average
1 1 watt
0.293 1 Btu/h
10−5 human, sounds produced during normal speech

### units

From the basic definition…

 P̅ = ΔW Δt

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.

 ⎡⎣ W = J ⎤⎦ s

A joule per second is called a watt [W] in honor of the Scottish mechanical engineer James Watt. 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.

 ⎡⎣ W = J = N m = N m/s ⎤⎦ s s

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 men. (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 kilowatt-hour is a unit of energy used by electrical utilities.

The Btu per hour (often erroneously shortened to Btu) is a unit of power used by the heating, ventilation, and cooling industry (HVAC).

A horsepower is a unit of power sufficient to raise 33,000 pounds 1 foot every 1 minute (550 lbs, 1 ft, 1 sec) equivalent to roughly 745.70 W

### horse power and the horsepower

James Watt was a Scottish mechanical engineer most famous for his improvements in the design of steam engines. While Thomas Newcomen is generally regarded to have invented the steam engine in 1712, Watt's improved design became the industry standard that powered the Industrial Revolution in Britain and elsewhere.

The Newcomen steam engine had been used to pump water out of mines in England for decades, but it was inefficient. 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. Eighteenth Century treadmills weren't anything like the exercise equipment first sold in the Twentieth 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.

Large London breweries like Whitbread's are estimated to have employed an average of 20 horses for the mill at once

Watt identified the Whitbread Brewery in London as a potential customer.

for who used them to grind malt into powder.

Olinthus Gregory (1774–1841) England

Prior to Mr Watt's application of the steam engine to produce rotative motion, the great manufactories of the kingdom had their mill work set in motion by the agency of water, of wind, or of horses; and the latter had for many years, been almost exclusively employed in the breweries and distilleries of the metropolis. It was therefore natural for one who wished to substitute the power of steam for the power of horses, to state the number of the latter to which the new power, under given conditions, would be equivalent; and it is probable that Boulton and Watt felt that such a mode of comparison would be more intelligible to common apprehensions than a more accurate and scientific formula. It gave the power of an engine expressed in numbers, of which the ordinary strength of a horse is the unit….

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

physiology

Power of various human activities Source: Physics of the Body
power (W) activity
700 cycling (21 km/h)
685 climbing stairs (116 steps/min)
545 skating (15 km/h)
475 swimming (1.6 km/h)
440 playing tennis
400 cycling (15 km/h)
265 walking (5 km/h)
210 sitting with attention focused
125 standing at rest
120 sitting at rest
83 sleeping
Power of various human organs Source: Physics of the Body
organ
mass
(kg)
power
(W)
power density
(W/kg)
% of total
liver & spleen 23 27
brain 1.40 16 11 19
skeletal muscles 28.00 15 0.55 18
kidneys 0.30 9.1 30 10
heart 0.32 5.6 17 7
remainder 16 19
total 65 85 100