Physics 311 - Fall 97

Problem set 7 - Due Thursday, Nov. 6

1. Quick questions, mostly from How Things Work:

   a.

   Is it better for an airplane to takeoff into the wind or with the wind? Why?

   b.

   How would the take-off speed of a jet at Denver's Stapleton airport compare to speed for the same jet at Washington's Dulles Airport?

   c.

   HTW page 205, number 5.

   d.

   HTW page 205, number 10. How does the lift depend on the span of the blades (I am asking for an approximate power law)?

   e.

   HTW page 205, number 13 (see also number 12).

   f.

   HTW page 205, number 26. Compute the lift on such a plane when it flies at 10 m/sec if the wingspan is 20 cm and the chord is 7.5 cm. What angle of attack is needed to generate a lift of 1 kg?

   g.

   HTW page 205, number 27.

2. Gliding flight. (For inspiration, read Case 1, page 206 of HTW.) The glide angle is the angle that the trajectory of a plane (or bird) makes with the horizontal during gliding flight (no thrust). Draw a clearly labeled free body diagram for gliding flight, and use this (and Newton's Laws) to calculate the glide angle in terms of the coefficients of lift and drag. If you want the plane to glide farther, does it help to throw the passengers off the plane? Also find the terminal speed in gliding flight. How does it depend upon mass?

3. Your arm, the airfoil. When I was a kid I used to annoy my parents by sticking my arm out of the car window; I discovered that by flattening my hand and tipping it at a certain angle the wind would actually support the entire weight of my arm. Can this actually work? Make some estimates (for your arm, not mine). Read HTW, page 205, numbers 20 and 21, and answer number 22.

4. Sailing into the wind. The first time I went sailing I was amazed that it was actually possible to sail into the wind, by setting a zigzag course (called tacking in sailor jargon). How can this work? Assuming a simple sailboat (with a single mainsail), discuss the aerodynamics of the sail. By drawing a careful force diagram for the sailboat, explain how you can sail into the wind. How much thrust (to order of magnitude) can be generated by a small sailboat in a moderate wind? What happens if you remove the keel (or centerboard)? Why are sails shaped like triangles rather than squares?

5. Sound. A few problems on sound propagation:

   a.

   The speed of sound in an ideal gas is

   

   where is the specific heat ratio for the gas. What is for air? Why? [Hint: remember the equipartition theorem.]

   b.

   Assuming air to be an ideal gas, calculate c. How does it compare against measured values?

   c.

   What are the upper and lower limits of frequencies which can be heard by the average person? What are the corresponding wavelengths for these frequencies? Why can we hear, but not see, around corners?

   d.

   Sound in a gas (air, say) propagates adiabatically, so that there is negligible heat flow as the sound propagates. This accounts for the factor of in the expression for the sound speed--if the sound propagated isothermally (at constant temperature), there would be no . Why does sound propagate adiabatically? To understand this, imagine dividing up the sound wave into regions of size of order the wavelength . If a fluid has a thermal diffusivity D, how much time does it take for heat to diffuse a distance ? How much time does it take for a pressure disturbance to propagate through the same distance? How do these two times compare (i.e., put in some numbers)? Does the heat have time to diffuse?