18.02A PROBLEM SET 9 Due in recitation Tuesday, February 10

1. (10 points) The Lasker Rink in the northeast corner of Central Park is a perfect circle having a radius of 50 meters. A skater moves counterclockwise around the edge of the rink at a speed of 5 meters per second. We’ll put a coordinate system on the rink with origin at the center of the rink, the x-axis running east (toward Fifth Avenue) and the y-axis running north (toward Central Park North).

a) Suppose (x, y) is a point at the edge of the rink. Write down a non-zero tangent vector

T(x, y) = A(x, y)i + B(x, y)j

to the rink at the point (x, y). (That is, write formulas for A(x, y) and B(x, y).)

b) Calculate the length of your tangent vector T(x, y).

c) The skater’s velocity vector at v(x, y) must be some multiple of your tangent vector:

v(x, y) = c(x, y)T(x, y).

Calculate c(x, y).

d) Find a vector field F with the property that the line integral

∫_{C} F·dR

along the skater’s path measures the distance that the skater has travelled to the west. (For example, over a path beginning at the northern extremity of the rink and continuing to the western extremity, the line integral ought to be 50 meters.)

e) Suppose the skater’s path C begins at the point (50\sqrt{3}/2, 25), and continues exactly halfway around the rink to the point (−50\sqrt{3}/2, −25). Evaluate the line integral

∫_{C} F·dR

using your formulas for F and v = dR/dt. (It’s possible to get an answer without writing down any integrals precisely, but part of the problem is to write them down.)

2. (8 points) This problem is about whirlpools. A vector field F in the plane (possibly not defined at the origin) is called a whirlpool if its direction at every point is perpendicular to the radius. The most general whirlpool vector field looks like

F(x, y) = −f(r)yi + f(r)xj,

with f a function of the radius r = \sqrt{x^{2} + y^{2}}.

a) How much work is done by the whirlpool force field in moving counterclockwise around the circle of radius r centered at the origin? (The answer depends on f(r).)

b) The coordinates of F are

M(x, y) = −f(r)y, N(x, y) = f(r)x.

Suppose f(r) = 1/r^{2}, so that the whirlpool field has magnitude 1/r. Calculate ∂M/∂y and ∂N/∂x.

c) Explain why part (a) shows that the only conservative whirlpool field is the zero field. Explain why part (c) shows that the whirlpool field with f(r) = 1/r is conservative. Explain this apparent contradiction.

3. (7 points) Problem 21.1(12) on page 757.