A roller coaster at the Six Flags Great America amusement park in Gurnee, Illinois, incorporates some clever design technology and some basic physics. Each vertical loop, instead of being circular, is shaped like a teardrop. The cars ride on the inside of the loop at the top, and the speeds are fast enough to ensure that the cars remain on the track. The biggest loop is 40.0 m high. Suppose the speed at the top is 10.2 m/s and the corresponding centripetal acceleration is 2g.

(a) What is the radius of the arc of the teardrop at the top?

(b) If the total mass of a car plus the riders is M, what force does the rail exert on the car at the top?

(c) Suppose the roller coaster had a circular loop of radius 21.4 m. If the cars have the same speed, 10.2 m/s at the top, what is the centripetal acceleration at the top?

(d) Comment on the normal force at the top in the situation described in part (c) and on the advantages of having teardrop-shaped loops.