Question

A wooden stick of length l and radius R and density ρ has a small metal piece of mass m (of negligible volume) attached to its one end. Find the minimum value for the mass m (in terms of given parameters) that would make the stick float vertically in equilibrium in a liquid of density σ(>ρ).

Answer 1

For the rod to be in equilibrium centre of mass of (rod + mass m) system should be below centre of gravity of the volume displaced by the rod.

For minimum m should coincide so.

Suppose h length of rod is below water then, by force balance

Reaction of centre of mass should be at h/2 distance from bottom