Consider the ring-shaped body of radius $a$ and mass $M$ shown in the figure. A particle of mass $m$ is placed a distance $x$ from center of the ring, along a line through the center of the ring and perpendicular to it.

(a) Calculate the gravitational potential energy $U$ of this system. Take the potential energy to be zero when the two objects are very far apart. Recall that we can write $dU=-\frac{Gm}{r}dM$ where $r={\left(x^2+a^2\right)}^{\frac{1}{2}}$ as we integrate around the ring.

(b) Show that your answer in part (a) reduces to the point masses result when $x$ is much larger than the radius $a$ of the ring.

(c) Use $F_x=-dU/dx$ to find the magnitude and direction of the force on the particle.

(d) Show that your answer in part (c) reduces to the expected result when $x$ is much larger than $a$. What are the values of $U$ and $F_x$ when $x=0$ and why?