Physics 311 - Fall 97

Problem set 5 - PLEDGED

Due Thursday, Oct. 16

1. Consider a fluid rotating rigidly with angular velocity w₀ around the z axis, so that the velocity at the point of coordinates (x,y,z) is w₀ × r, with w₀ = (0,0,w₀) and r = (x,y,0).

   a. Write down the cartesian components of v.

   b. Write down the components of v in cylindrical coordinates

   c. Show that , using cartesian coordinates.

   d. Show that , using cylindrical coordinates. Hint: use the equation on page 164 of PQRG, but note that the notation is different: we use for the cylindrical coordinates, while PQRG uses .

   e. Compute , using cartesian coordinates.

   f. Compute , using cylindrical coordinates. Hint: see part (d).

   g. Make a field plot of v. You can do this by hand or by MAPLE, using fieldplot

2. Consider a fluid rotating (non-rigidly) with a single vortex line along the z axis and circulation Using the same notation as in the previous problem,

   a. Write down the cartesian components of v.

   b. Write down the components of v in cylindrical coordinates.

   c. Show that , using cartesian coordinates.

   d. Show that , using cylindrical coordinates.

   e. Compute , using cartesian coordinates. Note that something singular happens at x = y =0, or r = 0.

   f. Compute , using cylindrical coordinates.

   g. Make a field plot of . You can do this by hand or by MAPLE, using fieldplot.

   
   
3. We learned that the time to cook a turkey of mass m in a conventional oven is proportional to m^2/3, but this cannot be right for a very small bird: a more accurate formula is t = t₀[1+(m/m₀)^2/3], where t₀ is a minimum cooking time. The same formula, with a longer t₀, applies to potatoes. Also, it was assumed that all turkeys have the same shape. Here are some related (but different) questions:

   a. In a microwave oven, it takes 5 minutes to cook one standard potato. If you put two standard potatoes in the oven, how long does it take to cook them? Explain.

   b. In a conventional oven, it takes one hour to cook one standard potato. If you put two standard potatoes in the oven, how long does it take to cook them? Explain.

   c. In a conventional oven, how does the cooking time for a sausage depend on the length, l, and the diameter, d, of the sausage when l » d?

   d. In a conventional oven, it takes 20 minutes to cook a long thin sausage. How long does it take to cook a sausage of the same diameter, but twice as long? How long does it take to cook a sausage of the same length, but twice the diameter? [For this part, you can neglect t₀].

   e. Explain why kindling burns much faster than a log. If a log is split lengthwise into 8 equal splinters, how much faster will the wood burn?

4. Miscellaneous estimates and values:
   a. Estimate the R value of a down comforter that is 2.5 cm thick by assuming that the down traps air so that convection is completely suppressed. You can take the D for air from the PQRG.

   b. The water jet from your garden hose can be controlled by a nozzle. The jet can rise to a maximum height of 5 meters, but does not rise very high if there is no nozzle and barely trickles out if the nozzle is nearly closed. Explain this and put a lower limit to the pressure in your pipes from the problem's data.

   c. We learned how to estimate the pressure in the center of the earth. By the same argument, estimate the pressure in the center of the sun.

   d. What is the formula for the Bohr radius, a_B, in terms of basic constants? What is the formula for the fine structure constant a? If you multiply a_B by a, you obtain another length: what is it called? If you multiply a_B by a², you obtain another length yet: what is it called? [Consult the PQRG].

   e. Out of G, c and the mass M of a star (or anything), you can make a length, known as the Schwarzschild radius (within a factor of 2) by dimensional analysis alone. Do so and compute the Schwarzschild radius of the sun.