Question

A wooden cylinder of diameter 4r, height h and density $\rho /3$ is kept on a hole of diameter 2r of a tank, filled with water of density $\rho$ as shown in the figure. The height of the base of cylinder from the base of tank is H.
If level of liquid starts decreasing slowly when the level of liquid is at a height $h_1$ above the cylinder, the block just starts moving up. Then, value of $h_1$ is

• A. $\frac{2h}{3}$
• B. $\frac{5h}{4}$
• C. $\frac{5h}{3}$
• D. $\frac{5h}{2}$

Answer 1

The correct option is C $\frac{5h}{3}$

$\pi r^2=$ Area of base of cylinder in air

$3\pi r^2=$ Area of base of cylinder inn water

$4\pi r^2=$ Cross-sectional area of the cylinder

From the figure, equating the forces, we get,

$[P_0+\rho g{h}_1]\pi \left(4{\pi }^2\right)+\left(\frac{\rho }{3}\right)\pi 4r^{2}hg$ $=[P_0+\rho g(h_1+h\left)\right]\pi \left(3r^2\right)+P_0\pi r^2$

or,

$\rho g{h}_{1}4\pi r^2+\frac{\rho gh4\pi r^2}{3}=\rho g{h}_{1}3\pi r^2+\rho gh3\pi r^2$

or

$\rho g{h}_1=\rho gh\pi r^2=\rho gh\pi r^2(3-\frac{4}{3})$

or

$h_1=\frac{5h}{3}$