- Write something.
- 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…
- the final angular speed of the roll
- the final translational speed of roll
- the angular acceleration of the roll
- the translational acceleration of the roll
- the tension in the sheets
- 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…
- the tension T in the string
- something else
- maybe something else
- 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 vmax ⎛
r ≥ reye r
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…
- Graph tangential wind speed as a function of radius.
- Draw a velocity field diagram.
- Derive an expression for the total kinetic energy of a storm.
- 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.)