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

None of these



diffcult
Solution
For the stick to be vertical for rotational equilibrium, centre of gravity should be below in a vertical line through the centre of buoyancy. For minimum m, the two will coincide.
Let h be the length of immersed portion. For translatory equilibrium,
Wt. of rod + mass attached = force of buoyancy
The height of centre of mass from bottom
For rotator equilibrium and for minimum m, this should be equal to h/2.
Substituting for h in eq. (i), we get
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