The Physics
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Opus in profectus

Power

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Problems

practice

  1. 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).
  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 [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).
    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            
    3. 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 and West Chester University in Pennsylvania 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 two 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 30 km/h
    max. force 1700 N 290 N 290 N 290 N
    power
  10. The English scientist Thomas Young (1773–1829) was the first person to define work as the force–displacement product. In the passage below, he almost defined power also.

    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 Nineteenth 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?
    2. What is the "daily work of a labouring man" in joules?

investigative

  1. 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.
    1. Fill the following table with the data you will need to solve the remaining problems.
      ringworld
      parameter
      terrestrial
      equivalent
      calculated
      value
       
      unit
      radius earth-sun distance m
      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
      n/a radius of earth m
    2. Determine the surface area of…
      1. Ringworld
      2. the earth
      3. a sphere with radius equal to the Earth-sun distance
    3. How fast does Ringworld spin to provide its inhabitants with the sensation of normal earth gravity? State your answer in…
      1. meters per second
      2. earth days per rotation
      3. rotations per earth year
    4. Determine the…
      1. kinetic energy of Ringworld due to its rotation and
      2. 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.