A light cylindrical tube T of length l and radius r containing air is inverted in water (density `d_(1)`).One end of the tube is open and the other is closed. A block B of density `d_(2)(gt d_(1))` is kept on the tube as shown in the figure. The tube stays in equlibrium in the position shown. Find the volume of B. Density of the air is negligible as compared with the density of water :

Options

A. the volume of block B is `(pi r^(2) l)/(3)`
B. the volume of block B is `(2 pi r^(2) l)/(3)`
C. the pressure of air trapped in the tube is `P_(1) + d g(h + (l)/(3))`
D. the pressure of air trapped in the tube is `P_(0) + d g (h + (2l)/(3))`

Answer

A, C

Explanation

The free body diagram of block + tube inclusive of water is as shown below
Since the block + tube system shown in figure is in equilibrium
`:.` Net weight of system = buoyant force
`2d Vg + pi r^(2) (2l)/(3) dg =(V + pi r^(2) l)dg`
where V is the volume of block B
`rArr (2d -d)V = (pi r^(2) l d)/(3)`
or `V = (pi r^(2) l)/(3)`
The pressure of the air trapped inside the tube is same as pressure at point A in the water as shown in figure.

`:. P = P_(0) +(h + (l)/(3))dg`
`P_(a) =P_(0) + dg(h + (l)/(3))`.