The Physics
Hypertextbook
Opus in profectus

# Rotational Energy

## Problems

### practice

1. Write something.
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) = ⎧⎪⎨⎪⎩ vmax ⎛⎝ r ⎞⎠ 0 ≤ r ≤ reye reye vmax ⎛⎝ reye ⎞½⎠ r ≥ reye r

Where…

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

and…

 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. problem