The Physics
Opus in profectus

Rotational Energy

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  1. A flywheel is a rotating mechanical device used to store mechanical energy. When attached to a combined electric motor-generator, flywheels are a practical way to store excess electrical energy. Solar farms only generate electricity when it's sunny and wind turbines only generate electricity when it's windy. Combining energy sources like solar and wind with flywheel energy storage devices like a flywheel is one way to create a renewable energy system that is load balanced.

    Given the energy storage flywheel described in the table to the right, answer the following questions.

    1. What is the mass of the flywheel?
    2. What is the top angular speed of the flywheel?
    3. For how long could a fully charged flywheel deliver maximum power before it needed recharging?
    4. What is the average angular acceleration of the flywheel when it is being discharged?
  2. A roll of toilet paper is held by the first piece and allowed to unfurl as shown in the diagram to the right. The roll has an outer radius R = 6.0 cm, an inner radius r = 1.8 cm, a mass m = 200 g, and falls a distance s = 3.0 m. Assuming the outer diameter of the roll does not change significantly during the fall, determine…
    1. the final angular speed of the roll
    2. the final translational speed of roll
    3. the angular acceleration of the roll
    4. the translational acceleration of the roll
    5. the tension in the sheets
  3. The top shown below consists of a cylindrical spindle of negligible mass attached to a conical base of mass m = 0.50 kg. The radius of the spindle is r = 1.2 cm and the radius of the cone is R = 10 cm. A string is wound around the spindle. The top is thrown forward with an initial speed of v0 = 10 m/s while at the same time the string is yanked backward. The top moves forward a distance s = 2.5 m, then stops and spins in place.

    Using energy considerations determine…

    1. the tension T in the string
    2. something
    3. something else
    4. maybe something else
  4. The simplest mathematical models of hurricanes and typhoons (collectively known as tropical cyclones) describe a cylindrical mass of rotating air with no updrafts, downdrafts, or turbulence. In these vortex models, the air in a central region called the eye is often assumed to rotate as if it was one solid piece of material — slowest at the center and fastest at the outer edge or eye wall. (The eye wall, not the center, is the region of maximal wind speed in a hurricane.) Beyond the eye wall, wind speeds decay away according to a simple power law.

    Here's an example of a vortex model of a hurricane with an outer region described by an inverse square root power law.

    v(r) = 

    0 ≤  r  ≤ reye
    reye ½
      r  ≥ reye


    v(r) =  tangential wind speed
    vmax =  wind speed at eye wall
    r =  distance from center of hurricane
    reye =  radius of eye wall


    h =  height of hurricane
    R =  radius of hurricane (R > reye)
    ρ =  average density of air

    Given this model…

    1. Graph tangential wind speed as a function of radius.
    2. Draw a velocity field diagram.
    3. Derive an expression for the total kinetic energy of a storm.
    4. Determine the total kinetic energy of a tropical cyclone 500 km in diameter, 10 km tall, with an eye 10 km in diameter and peak winds speeds of 140 km/h. (Assume the average density of the air is 0.9 kg/m3.)


  1. Scratchy is trapped at the bottom of a vertical shaft. Itchy is rolling a heavy, thin-walled cylindrical shell (I = MR2) of mass 50 kg and radius 0.50 m toward a 5.0 m long, 30° ramp that leads to the shaft. The angular velocity of the cylindrical shell is 10 rad/s when Itchy releases it at the base of the ramp. All inanimate objects in this "experiment" obey the laws of physics. Itchy 'n' Scratchy are cartoon characters and are subject to the laws of cartoon physics.

    Itchy rolling a cylindrical shell n' Scratchy stuck in a well.

    1. Who gets squashed in the end? That is, will the cylindrical shell make it to the top of the shaft and fall on Scratchy or will it turn around and roll back on Itchy? Show all work used to arrive at your answer.
    2. Would your answer to part a. change if the "experiment" took place on the moon where g = 1.6 m/s2? Explain your reasoning.
    3. Would your answer to part a. change if Itchy rolled a different hoop with the same radius and initial angular velocity but a mass of 100 kg? Explain your reasoning.
    4. Would your answer to part a. change if Itchy rolled a solid cylinder (I = ½MR2) with the same mass, radius, and initial angular velocity as the hoop? Explain your reasoning.