A tunnel is drilled straight through a uniform-density nonrotating sphericallysymmetric airless asteroid of radius \(R\). The tunnel is oriented along the \(x\) axis, with \(x=0\) at the center of the asteroid and of the tunnel. Using the results of the preceding problem,

(a) show that if an astronaut of mass \(m\) steps into one side of the tunnel she will experience a spring-like force \(F=-k x\) as she falls through the tunnel.

(b) Find \(k\) in terms of any or all of \(G\) and the mass \(M\) and radius \(R\) of the asteroid.

(c) Find the time it would take for her to oscillate from one end of the tunnel to the other and back again, in terms of the same parameters.