The Physics
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Opus in profectus

# Power

## Problems

### practice

1. A typical adult in the United States consumes something like 2000 calories of food per day. Determine the average power generated by such an adult (assuming he or she is not gaining or losing weight).
2. 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.
3. A supertanker doesn't come with brakes. Using engines alone, it takes a loaded supertanker 13 km (8 miles) to stop. A typical vessel of this class has a gross mass of about 150 million kilograms and a cruising speed of 50 kph (30 mph). Determine…
1. the average stopping force applied to the ship and
2. the average power dissipated while stopping it.
4. The athlete in this video clip 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
• the video advances at 25 frames per second (with 71 frames total).
Determine the following quantities for the barbell this athlete is lifting as functions of time…
1. height
2. velocity
3. acceleration
4. applied force
5. work
6. power
Most of this question is a review of mechanical concepts discussed in previous sections in this book. Only the last part deals with power. To begin this problem, you will need some sort of screen measuring tool. Many basic image editing applications have this function built into them.

### numerical

1. A 64 kg student travels from the first floor to the fourth floor of a school (a height of 15 m).
1. What total work did she do climbing the stairs?
2. How long would this trip last if the student produced 240 W of power?
2. A motorized winch is rated at 10.0 kW. At what maximum constant velocity can this winch raise a mass of 27,500 kg?
3. 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.
4. 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?
5. The graph below shows the power output vs. time for an elevator motor in operation.

1. What does the area under this curve represent?
2. Calculate its cumulative value at 2 s intervals. Compile your results in a table like the one below.
interval ending at 0 s 2 s 4 s 6 s 8 s 10 s
interval area
cumulative area
1. Sketch a graph of this quantity with respect to time.

6. The world's most powerful laser in 1996 was the Petawatt — a special prototype laser built at the Lawrence Livermore National Laboratory (LLNL) in California. This laser produced a peak power of 1.25 petawatts, 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. How long could an ordinary 60 W light bulb run on the energy delivered in one pulse of the Petawatt?
7. A problem for Americans and other children of the former British Empire. James Watt defined the horsepower as being sufficient to raise 33,000 pounds 1 foot every 1 minute (often stated in reduced form as 550 foot pounds per second). Show that one horsepower is approximately equal to…
1. 476.75 watts
2. one pound of thrust at 375 mph
8. A document based question. Read the following excerpt from the New York Times.

Lance Armstrong's strength and endurance sometimes seem too extraordinary to be believed.

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….

New York Times, 2005

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?
9. In 2008, a team of engineers at Rensselaer Polytechnic Institute in New York developed a technique for measuring the forces generated while swimming. They first tested olympic athletes and then moved on to dolphins. The table below combines measurements taken by the researchers with the Olympic results for some of the swimmers tested. Complete the table and determine the power output of humans and dolphins.
dolphins humans
Primo & Puka Ariana Kukors Megan Jendrick Beth Botsford
venue Santa Cruz 2008 Seattle 2008 Sydney 2000 Atlanta 1996
style dolphin kick freestyle breaststroke backstroke
distance n/a 100 m 100 m 100 m
time n/a 49.62 s 67.05 s 61.19
speed 9.0 m/s
typical force 1600 N 290 N 290 N 290 N
power
10. The English scientist Thomas Young (1773–1829) was the first person to use the word energy in its modern sense. In the passage below he almost defined a new unit for power — something along the lines of the horsepower, but using people.

The daily work of a labouring man, of middle age, and in good health, will serve as a convenient unit for the comparison of moving powers of all kinds. It may be most easily remembered in this form: a man can raise a weight of 10 pounds to the height of 10 feet in a second, and can continue this labour for 10 hours a day.

Thomas Young, 1807

Assume that British men in the early 19th century actually had the stamina to do as much work as Young says.
1. What is the power of a "labouring man, of middle age, and in good health" in watts? In horsepower?
2. What is the "daily work of a labouring man" in joules? In kilocalories? In Btu?
11. A 940 W motor is used to lift 200 kg of supplies 11 m above street level to the roof of a building.
1. If the motor ran for 24 s how much work did it do?
2. What is the final potential energy of the supplies relative to street level?
3. How much work was done against friction?
4. What was the average force of friction on the cable?
12. Read the following passage from Wired describing an extreme cycling competition called the UCI Hour Record.

The hour is widely considered to be cycling's purest record, albeit an unusual one: Instead of requiring them to traverse a set distance, this event allots cyclists a set time of 60 minutes to pedal as many laps as they can around a velodrome. And whereas other competitive pursuits typically pit multiple athletes against one another, the hour is a solo affair. The race, if you can call it that, is against the clock….

Despite her preparation, [Olympic cyclist Evelyn] Stevens' hour attempt nearly broke her. During minutes 50 through 55, "I was physically in the most painful place I had ever been," she says. She remembers sounds fading away, her vision going dark, and her thoughts turning to all the wrong things. "You want oxygen, you want water, your body is screaming: Stop, stop, stop." Around the 55th minute, the idea of letting down her coach brought her back. "I just thought, oh gosh, he'd be so disappointed. His family, who sacrificed so much of their time with him so he could coach me, would be so disappointed!" The guilt returned her attention to her breath, to her mantra, to the state of mind she needed to traverse a then-unprecedented 47.980 km — 29.81 miles — in 60 minutes flat….

To stand a shot at the hour record, an athlete must maintain for 60 minutes a power output that most people would struggle to hold for 60 seconds. Stevens averaged just over 300 watts for the duration of her attempt. British cyclist Bradley Wiggins, who, in 2015, pedaled 54.526 kilometers (33.881 miles) to set the current men's record, is estimated to have averaged 440 watts. If you've ever paid attention to your numbers during spin class, those figures will no doubt astound you. If spinning's not your thing, imagine this: 440 watts is the energy it takes a 150-pound person to climb a flight of stairs in about 5.5 seconds. Now imagine climbing 655 flights at that pace. That's an hour.

Robbie Gonzalez, 2019

Using the quantities stated in this passage, complete a table like the one below.

The hour record * The hypothetical 150 pound person is climbing a flight of stairs in 5.5 seconds. The real-life athletes are cycling for one hour.
quantity Evelyn
Stevens
Wiggins
150 pound
person*
time
(s)
distance
(m)
power
(W)
speed
(m/s)
applied force
(N)
total work
(J)
13. The Petawatt at Lawrence Livermore National Laboratory in California was the world's most powerful laser from 1996 to 1999, when it was decommissioned. Since 2015, when it was last upgraded, the world's most powerful has been the Laser for Fast Ignition Experiment (LFEX) at Osaka University in Japan. Both have beams of about 1015 watts — a petawatt. Read the following passages and compute a more exact value for the power of the Petawatt and the LFEX. (The passage on the left is from a publication meant for a more general audience. The passage on the right is from a scientific journal.) How do the two lasers compare?

Livermore's Petawatt laser operated for three years, until its last shot was fired on May 27, 1999. At full energy of about 680 joules, the shots delivered more than a quadrillion watts (or petawatt, which equals 1015 watts) of power, exceeding the entire electrical generating capacity of the U.S. by more than 1,200 times. But the Petawatt's shots lasted for just a fleeting moment — less than a trillionth of a second, or 440 femtoseconds to be precise.

M. Perry, 2000

In this work we show a method to reduce the laser pulse pedestal level, by implementing a plasma mirror (PM) device on an LFEX laser at the Institute of Laser Engineering, Osaka University. The LFEX laser is a kJ, PetaWatt class laser capable of delivering 1.6 kJ of laser light on target in 1.5 ps, with an intensity of ~2 × 1019 W cm−2. The PM is constituted by a transparent dielectric (usually BK7 glass) with an antireflective (AR) coating….

A. Morace, et al. 2017

14. The most prominent structure in the ZARM research center at the University of Bremen is the 120 meter tall drop tower. Sealed experiment "capsules" weighing up to 500 kilograms are launched vertically upwards inside a cylindrical steel vacuum tube to a maximum height of 110 meters. Free fall conditions inside the capsule simulate weightlessness that lasts almost 10 seconds. (For comparison, the American ZERO-G and French AirZeroG airplane flights can simulate weightlessness for up to 30 seconds.) Capsules are launched using a hydraulically-driven piston system called the "catapult" located in the basement several meters below the base of the tower. A capsule in the catapult goes from zero to launch speed in a space of 7 meters and a time of 0.3 seconds. Immediately after launch, a "deceleration unit" filled with polystyrene pellets slides into the base of the tower to safely capture the returning capsule. The capsule is then removed so the next experiment can begin. Determine the following quantities for a capsule with a maximum mass and maximum flight time (disregarding any height differences between the top of the catapult, the top of the deceleration unit, and the base of the tower)…
1. its maximum gravitational potential energy relative to the base of the tower
2. its launch speed
3. its total flight time from when it left the catapult to when it impacted the deceleration unit
4. the average power delivered to the capsule by the piston in the catapult
5. its average speed inside the catapult
6. the average force applied to the capsule by the piston in the catapult

### statistical

1. zarm-power.txt
The data in the accompanying tab delimited text file give the net force on a 500 kg projectile sitting atop a vertically mounted piston as a function of time and displacement. Use this data set and your favorite application for analyzing data to solve the following problems.
1. Using the force-time data given, determine…
1. the maximum net force and
2. the average net force on the projectile while it was accelerating
2. Determine the work done on the projectile as a function of time and use this to determine…
1. the total work done on the projectile during while it was being launched
3. Determine the power delivered to the projectile as a function of time and use this to determine…
1. the maximum power and
2. the average power used to launch the projectile
(Two equally valid methods can be used to solve this last part, but they will not yield exactly the same results. Welcome to the real world.)

Data adapted from Kampen, Kaczmarczik, and Rath; 2006.

### investigative

1. Ringworld (paid link) 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.
1. Fill the following table with the data you will need to solve the remaining problems.
ringworld
parameter
terrestrial
equivalent
calculated
value

unit
width diameter of Sun m
mass of Jupiter kg
mass of Saturn kg
mass of Uranus kg
mass of Neptune kg
mass total mass kg
available power luminosity of Sun W