A spherical shell has a radius R and a mass M.

(a) Write expressions for the force exerted by the shell on a point mass m0 when m0 is outside the shell and when it is inside the shell.

(b) What is the potential-energy function U(r) for this system when the mass m0 is at a distance r (r ≥ R) if U = 0 at r = ∞? Evaluate this function at r = R.

(c) Using the general relation for dU = -F ∙ dr = -F_rdr, show that U is constant everywhere inside the shell.

(d) Using the fact that U is continuous everywhere, including at r = R, find the value of the constant U inside the shell.

(e) Sketch U(r) versus r for all possible values of r.