Coplanar forces equilibrium- a rod in a sphere

[enadiz]

Homework Statement

A rod whose CG divides it into two portions of length a and b rests inside a smooth sphere in a position inclined to the angle θ to the horizontal. If 2α is the angle it subtends at the centre of the sphere, prove that

tanθ= (b-a)/(b+a) tanα

Picture is attached.

Homework Equations

The Attempt at a Solution

Well I drew the force triangle with the weight of the rod, and the two normal forces. I know the normal force on the left side will be greater than the one on the right... now the thing is I am not quite sure what to do to get tan θ, if you do that with the weight of the rod, what is the adjacent component of that triangle? hmm sorry I am really confused haha..

 

[tiny-tim]

hi enadiz!

(where have you been for the last five years? )

this is just a geometry question, all you need is a proper diagram

draw the three forces onto the diagram, and the distances a and b …

what does that give you? ​