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station with artificial gravity! The space station will be in the shape of a

spinning ring, with a generator at the center. Experimental simulations have

found that the local gravity experienced by a person in this setup will be equal

to:

z=2-4 cosz+e+e="

where is a multiplier relative to the gravity of the setup (which is con-

figurable) and r and y are the position relative to the generator in kilometers.

(Again don't worry about unit conversion on this problem.)

This is highly non-uniform, so the goal will be to create a track, roughly

elliptical, along which the gravity will be constant. Your goal is to determine

some basic information about the shape of that track.

(a) (1 point) What is the value of the gravity at position (0,0)? That is,

what is 2?

(b) (2 points) Find the first derivatives 2 and 2, (for all positions).

9/n(c) (3 points) What are the second derivatives Zz, zy, and yy at the

point (0,0)?

(d) (2 points) Just for sanity, find the tangent plane for zat (0,0). Given

that the generator itself is a local minimum, this should make sense (but

compute it the long way).

(e) (2 points) Find the equation of the second-order approximation to

z around (0,0) using Taylor polynomials./n10

(f) (1 point) We said earlier that the path of uniform gravity would be roughly

elliptical. According to the second-order approximation, what will the

aspect ratio of this ellipse be? HINT: When an ellipse is written in the

form + = 1, the aspect ratio is a : b. Use a level curve!

Fig: 1

Fig: 2

Fig: 3