In this problem, you are to find the gravitational potential energy of the stick in Example 11-8 and a point mass m₀ that is on the x axis at x₀.

(a) Show that the potential energy of an element of the stick dm and m₀ is given by

where U = 0 at x₀ = ∞.

(b) Integrate your result for part (a) over the length of the rod to find the total potential energy for the system. Write your result as a general function U(x) by setting x₀ equal to a general point x.

(c) Compute the force on m₀ at a general point x from F_x = -dU/dx and compare your result with m₀g, where g is the field at x₀ calculated in Example 11-8.

GMm L(x ) dU = - Gm,dm dx