A cylinderical block of density d stays fully immersed in a beaker filled with two immiscible liquids of different densities `d_1` and `d_2` The block is in equilibrium with half of it in liquid 1 and the other half in liquid 2 as shown in the Fig. If the block is given a displacement downwards released, then neglecting friction study the following statements.

Options

A. It executes simple harmonic motion.
B. Its motion is periodic but not simple harmonic.
C. The frequency of oscillation is independent of the size of the cylinder.
D. The displacement of the centre of the cylinder is symmetric about its equilibrium position.

Answer

A, D

Explanation

since liquid 2 is below liquid 1, liquid 2 is denser than liquid 1. Let area of cross section of the cylindrical block be A and it be displaced downwards by y. then volume of liquid 2 displaced will get increases by `Ay` and that of liquid 1 will get decreased by the same amount `Ay`. Hence net increase in upthrust on the block will be equal to `(Ayd_2g-Ayd_1g)`. This additional upthurust tries to restore the block in original position.
It means, the block experiences a restoring force `Ay(d_2-d_1)g`. Since this force is restoring and directly proportional to displacement `y`, it will execute SHM along a vertical line.
Hence, option (a) is correct and option (b) is wrong. If mass of the block is equal to `m`, then its acceleration will be equal to ltbr. `(Ayg(d_2-d_1))/(m)`.
Since its acceleration depends on mass `m`, frequency of oscillation will depend on size of the cylinder. Hence option (c ) is wrong. If the cylinder is displaced upward through y from equilibrium position, then it will experience a net downward force, equal to calculated above. This shows that its motion will be symmetric about its equilibrium position.