Coplanar forces equilibrium- a rod in a sphere

AI Thread Summary
A rod is positioned inside a smooth sphere, inclined at an angle θ to the horizontal, with its center of gravity dividing it into lengths a and b. The relationship to prove is tanθ = (b-a)/(b+a) tanα, where 2α is the angle subtended at the sphere's center. Participants suggest drawing a force triangle to visualize the weight of the rod and the normal forces acting on it. Emphasis is placed on creating a clear diagram to understand the geometry involved. Properly labeling the forces and distances will help clarify the solution process.
enadiz
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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..
 

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hi enadiz! :smile:

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

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

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

what does that give you? :smile:​
 

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