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Problems

   
     

5.1) Joe's Job

Joe the pony is used to help pull water from a well. His owner wants him to pull up large tubs of water, as much as 80 kg, in each load. Joe is a rather petite pony and can only exert .7 hp of power.

  1. How long does it take him to pull a full tub out of the 10 m-deep well?
  2. Joe’s owner then decides to rig up a fancy pulley system so that the weight of the water is held on four ropes instead of just one and Joe will have to use less force to pull each load up. Joe is excited, because he assumes he will now have to use less energy to pull up each tub. Explain to Joe why his assumption is wrong.

5.2) The Nuclear Alternative

A small town consumes about 10 MW of electricity. Currently the town is using a coal-fired power plant to provide this power, but they are concerned about greenhouse gases and global warming, and are wondering whether to close down their plant and begin to buy electricity from a nearby nuclear power plant. Both plants operate at about 34% efficiency. The coal-fired plant uses coal with an energy content of 29.3 J/kg, and every kg of coal (assumed to be pure C, with a molecular wt of 12 g) produces 3.67 kg of CO 2, while the nuclear plant uses uranium enriched to 4% U-235. A gram of U-235 produces 2.56 x 10 10 J.

  1. How much coal does the power plant use in a year? How much CO2 is added to the atmosphere?
  2. How much uranium is necessary for the town to operate for one year?
  3. What decision would you recommend? Why? Nuclear Power Plant
    Picture Newsletter

5.3) Fun in the Sun

Solar energy is one of the most important components of a renewable energy future. Your parents are considering installing photovoltaic cells, which convert the sun’s energy directly into electricity, on their roof, and have asked for your help. You do some research and learn that, on a sunny day in Colorado, sunlight at the earth’s surface provides about one kilowatt/m2. Solar panels have an area of 0.5 m2, operate at 10% efficiency, and cost $63. Your parents use electricity at a rate of 2 kW.

  1. If your parents want to rely on solar electricity exclusively, how many solar panels will they need to buy?
  2. If electricity costs $0.08 per kilowatt-hour, how long will it take your parents to recoup their initial investment?

5.4) Earthquake

The energy of earthquakes comes from the stresses which build up as the tectonic plates on the earth’s surface shift. An earthquake’s strength is typically given on the logarithmic Richter scale. In the past, scientists approximated the energy released by an earthquake from its magnitude, but modern technology allows us to measure the energy released, then determine the earthquake’s magnitude. The old formula for converting Richter magnitude to energy is

log E = 4.8 + 1.5Ms ,

where E is seismic energy in Joules and Ms is seismic magnitude.

  1. Plot a graph of energy released vs. magnitude for magnitudes 4 through eight, increasing by half-integers.
  2. How long could you run a 60 W light bulb with the energy from an earthquake of magnitude 2? 3? 7?

5.5) MagLev Trains

It takes about 560 kW to run a typical diesel locomotive at speeds around 110 mph. Magnetically levitated, or MagLev, trains, on the other hand, require substantially less power because the magnetic levitation eliminates energy loss to friction. The worst MagLev system requires 270 W to levitate the train and 783 W total to move it at 100 mph (fast MagLevs can make closer to 300 mph). The number of passengers each train can hold is comparable. How much less energy does it take to travel one mile in a MagLev train as opposed to one mile in a train pulled by a diesel locomotive?

Maglev Train
Harel Barzilai

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5.6) Kickball

It is a beautiful day, so Jamal and his friends are playing kickball at the park. Just as he winds up for his kick, the wind, which is blowing directly toward him, suddenly picks up speed. Despite his strong kick, the ball lands barely 10 feet in front of him.

What happened? Can you derive an equation to describe the kinetic power of the wind, and explain to Jamal why his kick was not successful? [Hint: Derive an equation for the kinetic energy the wind is delivering to the ball in one second. Consider the kinetic energy in the volume of air that will impact the ball in one second.

5.7) Catching the Wind
Wind Turbines
www.me3.org

Wind turbines are one of the most promising sources of renewable energy today. The problem above shows that the kinetic power of the wind is:

where ris the density of the air, A is the area swept out by the turbine blades, and v is the speed of the wind. Doing some research on wind turbines, you find out the a practical turbine can convert about 40% of that power into electricity. That doesn’t sound like a lot, but is it? If the average household requires 3 kW of power, and the average wind speed is 14 mph, how big a rotor would a household require? (Assume the density of the air is 1.3 kg/m3.)

5.8) Wind Farm

On the bus ride home from school, Ignacio and Mariella pass a wind farm. As they are both taking physics, this is naturally of interest to them. Ignacio says that wind turbines can only convert about 30% of the wind’s power, but Mariella is positive that an efficient turbine can do better than that. Derive an equation for the maximum power a wind turbine can extract, and settle this argument.

5.9) Earthship
Wind Turbine
www.me3.org

Some friends of yours are planning to build an earthship near Lamar, Colorado, and live off the grid. They would like to install a wind turbine to satisfy their electricity needs, and, knowing you are a physics student, have asked you to help them design their system. You look on a wind map and determine that the average wind velocity at 10 m above the ground in Lamar is 5.2 m/s, and that the air density is 1.0 kg/m3. Small, high efficiency windmills can convert 45% of the wind’s power into electricity. Your friends think that they will need 1 kW of power to meet their needs.

  1. How long must the blades on the turbine be to provide them with sufficient power?
  2. How much power does the windmill produce when the windspeed is 1 m/s (about 2.2. mph)? 2 m/s? 10 m/s?

 

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5.10) Biking Power

A trained bicyclist in excellent shape might be able to convert food energy to mechanical energy at a rate of 0.25 hp for a reasonable length of time. Imagine such a person pedaling a stationary bike connected to a perfectly efficient electrical generator.

  1. Could such a person generate enough electrical power to run a toaster?
  2. What about a single ordinary light bulb?

5.11) Hydroelectric Generators

Suppose a hydroelectric generator running on pumped storage is required to provide its full power of 200 MW for a period of 8 hr.

  1. What is the total energy, in joules, of the output?
  2. Assuming that the generating part of the cycle is 85% efficient, how much energy must be stored in the water in the upper reservoir?
  3. If the upper reservoir is 100 m about the power plant, what is the mass of water needed?
  4. What is the volume of the stored water?
  5. If the water level in the reservoir is allowed to drop 2 m in the 8 hour, what is the area of the reservoir and, if the reservoir is square, how long are the sides of the square?

5.12) Power of the Wind

www.nrel.gov

Suppose one has a propeller-type windmill with blades 1 m long. What is the area swept out by the blades? If the overall efficiency of the windmill is 50%, what is the power produced when the wind speed is 1 m/s (about 2.2 mph)? At 2 m/s? At 10 m/s?

5.13) Eliminating CO2

It might be proposed that we eliminate excess CO2 from the use of fossil fuels by shooting it into space. Compare the energy required to do this (that is, the energy required for one kg of CO2 to reach escape velocity of 11 km/s with that obtained from burning the coal that created the CO2 in the first place. Explain why this is not a good idea.

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Energy Problems

 

Posted on 8/1/05

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