Power
The Physics Hypertextbook™
© 1998-2008 by Glenn Elert -- A Work in Progress
All Rights Reserved -- Fair Use Encouraged
Discussion
Intro
| 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) |
| 783,000 | most powerful truck (Terex TR100) |
| 468,000 | most powerful car (McLaren F1) |
| 10,000 | Watt's steam engine of 1778 |
| 746 | 1 horsepower |
| 100 | human, daily average |
| 0.293 | 1 btu/h |
| 18 × 10−6 | human, sounds produced during normal speech |
physiology
| Power of Various Human Activities | |
| power(W) | activity |
|---|---|
| 800 | playing basketball |
| 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 |
| 0.001 | sound waves produced by speaking |
| Source: Physics of the Body | |
bridge
| Power of Various Human Organs | ||||
| 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 |
| Source: Physics of the Body | ||||
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
Summary
- bullet
Problems
practice
- A typical adult in the United States consumes something like 2000 dietetic calories
of food per day. Determine the average power generated by such an adult (assuming
he or she is not gaining or losing weight).
Solution …
We have to assume here that all the food energy consumed goes into work on some level (mechanical or metabolic). This is true only so long as the person is not gaining or losing weight, which occurs when the energy consumed is greater than or less than the work done. Start the problem by converting the units -- convert the energy from calories to joules and convert the time from days to seconds.
P = W = 2000 kcal 4186 J 1 day = 97 W t 1 day kcal 24 × 60 × 60 s With something between 700 and 800 watts in a horsepower, this corresponds to a rate of energy conversion somewhere between one-seventh and one-eighth of a horsepower.
- Determine the cost of operating a 7000 Btu, room-sized air conditioner in
New York City for the duration of the summer. Assume that electricity costs 14¢
per kilowatt·hour and that the air conditioner will run about 10 hours
a day for 80 days.
Solution …
In the United States, the Btu is often confused with the Btu per hour (Btu/h). This is partially the fault of the heating, ventilation, and air conditioning industries (HVAC). The rate at which energy is transformed from one form to another or transferred from one place to another is called power. Power is stated in units of energy divided by time. Furnaces and air conditioners are rated according to their heating and cooling power; that is, how quickly they can add or subtract heat from a room or home. American appliance manufacturers often quote the power of their devices in "Btu" when they really mean "Btu/h". Most certainly, this is just a form of shorthand and does not reveal any malicious intent on the part of the industry. However …
Electric energy is sold by the kilowatt·hour while air conditioners are rated in Btu/h. This makes it extremely difficult to estimate the operating costs of these energy hungry appliances. For example, a 7000 Btu/h room air conditioner consumes energy at a rate of about …
7000 Btu 1055 J 1 h = 2050 W = 2.1 kW 1 h 1 Btu 3600 s while electricity in the New York City metropolitan area averages about 14 ¢ per kilowatt·hour. Thus, every ten hours of use costs …
2.1 kW 10 h 14 ¢ = 290 ¢ = $2.90 1 1 kWh Given that summer days in NYC range from semitropical to subtropical to absolutely tropical, it's quite reasonable to assume 80 days of air conditioner use in a typical season. This brings the cost of cooling one room to …
$2.90 80 AC days = $230 / AC season AC day AC season This cost may or may not be acceptable to an individual consumer for this particular use. That isn't the point. The point is that rating an air conditioner in a nonstandard unit adds one more step to the problem. Appliances should be rated in watts or kilowatts so that consumers would be able to make mental estimates of their operating costs more easily.
- Write something calculus based if possible.
- Answer it.
- The athlete in this video clip [mov]
is performing a weightlifting maneuver known as the snatch. In this maneuver,
the barbell must be lifted from the platform to a point above the head, with
the
arms and legs fully extended, in a single movement. The barbell must then be
held motionless until the referees give the signal and then returned to the
platform.
In this particular video …
- the mass of the barbell is 77.5 kg (for comparison, the mass of athlete is 58 kg),
- the disks on the barbell have a diameter of 450 mm, and
- the video advances at 25 frames per second (with 71 frames total).
- height
- velocity
- acceleration
- applied force
- work
- power
Solutions …
[magnify]- Height Determine the position of some easily identifiable
point on the barbell in any units that are convenient. For calibration
purposes, measure the diameter of the disk on the end of the barbell
in the same units. Set up a proportion to convert the position
measurements in arbitrary units to height in meters. Set up a similar
proportion to convert the frame number to elapsed time in seconds.
h = height on screen · 0.450 m t = frame number ⇒ diameter on screen 25
[magnify]- Velocity By definition, velocity is the rate of change
of position with time. Take the slope of the line tangent to the
height-time graph to produce the velocity-time graph or, in the
language of calculus, take the derivative of height with respect
to time.
v = dh ⇒ dt
[magnify]- Acceleration By definition, acceleration is the rate
of change of velocity with time. Take the slope of the line tangent
to the velocity-time graph to produce the acceleration-time graph
or, in the language of calculus, take the derivative of velocity
with respect to time.
a = dv ⇒ dt
[magnify]- Applied Force A net force causes an acceleration. The
net force in this situation is a combination of the force applied
by the athlete and the weight of the barbell. Since these forces
point in opposite directions, their vector sum is difference of
their magnitudes. Thus, the applied force of the athlete (the force
we care about) is the net force on the barbell plus its weight.
∑ F = ma = F − W = F − mg ⇒ F = m(a + g)
- Work (Three Steps)
[magnify]- Change the horizontal axis from time to height. Since the athlete lowered the barbell for a brief moment about halfway into the lift, the graph is a bit twisted in the middle.
[magnify]- Work is defined as the cumulative product of force
and displacement. Use the area under the force-height graph to
produce the work-height graph. In the language of calculus, the
work is the force-displacement integral. Like the force-height
graph, this graph is also bent out of shape.
W = ∫ F · dh ⇒
[magnify]- Change the horizontal axis back from height to time. Since time only marches forward, the graph is a classic, single-valued function again. Work can only have one value at any instant in time.
[magnify]- Power By definition, power is the rate at which work
is done. Take the derivative of the work-time graph to get the
power-time graph. Weightlifting is an exceptionally powerful sport.
This athlete had an average power of 390 W and a maximum power
of 1900 W (approximately ½ and 2½ horsepower,
respectively).
P = dW ⇒ dt
numerical
- A 64 kg student travels from the first floor to the fourth floor
of a school (a height of 15 m).
- What total work did she do climbing the stairs?
- How long would this trip last if the student produced 480 W of power?
- A motorized winch is rated at 10.0 kW. At what maximum speed can this winch raise a mass of 27,500 kg?
- A pedaling cyclist turns a 17.5 cm crank arm at 200 rpm. (The crank arm distance is measured from one pedal to the axle.) Calculate the average force exerted on the pedals if the cyclist does work at the rate of 600 W.
- How fast must a cyclist climb a 12° hill to maintain a power output of 190 W? Ignore friction and assume the mass of the cyclist plus bicycle is 85 kg?
- The graph below shows the power output vs. time for an elevator motor
in operation.
- What does the area under this curve represent?
- Calculate it.
- The world's most powerful laser (the Petawatt) went online at Lawrence Livermore National Laboratory (LLNL) in May of 1996. This laser produced a peak power of 1.25 petawatts (1.25 × 1015 W), ten times more power than the previous record holding laser (which was also built at LLNL) and 1200 times more powerful than the entire electrical generating capacity of the United States. Although it is incredibly powerful, the Petawatt is not particularly energetic. Pulses from the Petawatt typically last less than half a picosecond (10−12 s). How long could an ordinary 60 W light bulb run on the energy delivered in one pulse of the Petawatt?
- A problem for Americans only: Show that one horsepower is equivalent to one pound of thrust at 375 mph.
- A document based question. Read the following excerpt from the 14 June 2005 New
York Times.
Lance Armstrong's strength and endurance sometimes seem too extraordinary to be believed.
Lance Armstrong has a mass of about 70 kg and competes on a 7.5 kg bike. How many meters could he climb after 20 minutes of cycling?
Armstrong, a six-time winner of the Tour de France bicycle race who next month will try for his seventh straight victory, can cover 32 miles [51.5 km] in one hour of riding. In contrast, the average cyclist covers 16 miles [25.7 km]; a top marathon runner can cover 21 miles [33.8 km] on a bike.
Armstrong can ride up the mountains in France generating about 500 watts of power for 20 minutes, something a typical 25-year-old could do for only 30 seconds. A professional hockey player might last three minutes - and then throw up….
Armstrong showed subsequent improvements until his career was stopped short in 1996 with a diagnosis of testicular cancer. Eight months after his treatment ended, he was back in the Austin laboratory.
"He wanted to know if anything was permanently wrong," Dr. Coyle said.
They took measurements and found nothing to stop him, except his own willingness to compete.
Armstrong did compete. "In the next two years his heart got even better, his lactic acid dropped further and, amazingly, his efficiency increased to 23 percent," Dr. Coyle said.
investigative
- Ringworld
is the title of a classic science fiction novel written by Larry Niven in 1970. Set in the year 2850, it is the story of four adventurers (two human and two alien) who are chosen to explore an engineered world encircling a sun-like star. The Ringworld is an enormous cylindrical band with a radius roughly equal to that of the earth's orbit and a width about the same as the diameter of the sun. It was constructed by some unspecified form of matter transmutation using the planets and minor bodies that once orbited the Ringworld's sun as raw material. The flat, inner surface is covered with a natural-looking, earth-like terrain and it spins at a speed fast enough to provide its inhabitants with the sensation of earth gravity. Thousand mile high walls along the edges keep the Ringworld's atmosphere from spilling out into space. The Ringworld is the home of hundreds of hominid species, but they are mostly non-technological. The sufficiently advanced civilization that engineered the Ringworld collapsed centuries ago and the adventurers find only its remains.
- Fill the following table with the data you will need to solve the
remaining problems.
ringworld parameter terrestrial equivalent value unit radius earth-sun distance m width diameter of the sun m mass mass of jupiter kg " mass of saturn kg " mass of uranus kg " mass of neptune kg " total mass kg available power luminosity of the sun W n/a radius of the earth m - Determine the surface area of …
- Ringworld
- the earth
- a sphere with radius equal to the earth-sun distance
- How fast does Ringworld spin to provide its inhabitants with the
sensation of normal earth gravity? State your answer in …
- meters per second
- earth days per rotation
- rotations per earth year
- Determine the …
- kinetic energy of Ringworld due to its rotation and
- number of days it would take to accelerate a newly constructed Ringworld from rest up to its final rotational speed if all the solar energy that landed on its surface was converted to kinetic energy.
- Fill the following table with the data you will need to solve the
remaining problems.
Resources
- atheletes
- Blakeslee, Sandra. How Lance Armstrong Gets His Unusual Energy. New York Times, 14 June 2005.
- Ostroy, Alex. Training in NYC: Just how much better? you versus the pros. nyvelocity.com,
- lasers
- Petawatt, Lawrence Livermore National Laboratory. December 1996.
- machines
- Terex Construction Americas, Specifications on the most powerful trucks in production.
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