What is the equilibrium angle of a rod in a hemispherical bowl?

[roltzje]

Homework Statement

A hemispherical bowl is fixed with its rim in a horizontal plane. Into the bowl is laid a rod of length equal to the diameter of the bowl. The rod is uniform. There is no friction at either point of contact. Find the angle between the rod and the horizontal when the rod assumes its equilibrium position.

(The diagram shows a bowl from the side cutaway view with the rod partway into the bowl, with one end resting inside, and another part resting on the rim)

Homework Equations

Sum of forces

The Attempt at a Solution

I attempted taking normals but my professor did not teach us about hemispherical bowls. I have no idea what angle the normals are at with respect to the rod. I also have no idea how to figure out how deep the rod has gone into the bowl.

 

[AvroArrow]

The only equation I can think of is F=ma, but there is no acceleration or external force. Any help would be greatly appreciated.