A spherical asteroid of radius \(R\) and uniform density \(ho\) rotates with angular velocity \(\Omega\) about an axis through its center. Visiting astronauts drill a smooth hole from one point on the equator clear through the asteroid's center to a point on the opposite side.

(a) If an astronaut falls into one end of the hole, how long does it take her to reach the opposite side?

(b) During the trip she is pressed against the side of the hole with a force \(N\). Find \(N\) in terms of \(W\), her weight on the asteroid's surface at one of the poles, as a function of her distance \(r\) from the center of the asteroid. Can \(N\) exceed \(W\) ?