11 11/14

1. Do bathtub vortices have opposite spin in the southern hemisphere as they have in the northern one? Derive some ballpark number for the exit speed of a bathtub vortex at the north pole and one at the south pole, assuming the bath water is initially at rest compared to the earth. What do you conclude about the starting question?

2. A Boeng 747 has a maximum take-off weight of about 400,000 kg and take-off speed of about 75 m/s. The wing span is 65 m. Estimated the circulation in the trailing vortices, and from that, ballpark the typical circulatory velocities around the trailing vortices. Compare to the typical take-off speed of a Cessna 52, 50 mph.

3. Solve the following partial differential equation around a circle:
   
   $$\nabla^2 \psi = 0 \mbox{ for all $r_0<r<\infty$, $\theta$}$$
   
   
   with boundary conditions:
   
   $$\psi(r_0,\theta)= 0 \qquad \psi(\infty,\theta)\sim U r\sin(\theta)$$
   
   
   Hint: this is similar, but not the same as the problem solved in class. The physical meaning of the problem will become clear in next week's homework.