$A$ thin vertical uniform wooden rod is pivoted at the top and immersed in water as shown. The container is slowly raised. At a certain moment,the equilibrium becomes unstable. If the density of water is $9/5$ times the density of wood,then the ratio of the total length of the rod to the submerged length of the rod at that moment is:

$A$ liquid is taken in a long vertical cylindrical vessel and the cylinder is rotated about its vertical axis as shown in the figure. During rotation, the liquid rises along its sides. If the radius of the vessel is $0.05 \,m$ and the speed of rotation is $10 \,rad \,s^{-1}$, then the height difference between the liquid at the centre of the vessel and its sides is $\left(g=10 \,m \,s^{-2}\right)$.

$A$ spray gun is shown in the figure where a piston pushes air out of a nozzle. $A$ thin tube of uniform cross section is connected to the nozzle. The other end of the tube is in a small liquid container. As the piston pushes air through the nozzle,the liquid from the container rises into the nozzle and is sprayed out. For the spray gun shown,the radii of the piston and the nozzle are $20 \ mm$ and $1 \ mm$ respectively. The upper end of the container is open to the atmosphere.
$1.$ If the piston is pushed at a speed of $5 \ mm \ s^{-1}$,the air comes out of the nozzle with a speed of
$(A)$ $0.1 \ m \ s^{-1}$ $(B)$ $1 \ m \ s^{-1}$ $(C)$ $2 \ m \ s^{-1}$ $(D)$ $8 \ m \ s^{-1}$
$2.$ If the density of air is $\rho_{a}$ and that of the liquid is $\rho_{\ell}$,then for a given piston speed,the rate (volume per unit time) at which the liquid is sprayed will be proportional to
$(A)$ $\sqrt{\frac{\rho_{a}}{\rho_{\ell}}}$ $(B)$ $\sqrt{\rho_{a} \rho_{\ell}}$ $(C)$ $\sqrt{\frac{\rho_{\ell}}{\rho_{a}}}$ $(D)$ $\rho_{\ell}$

$A$ river of salty water is flowing with a velocity $2 \,m/s$. If the density of the water is $1.2 \,g/cc$, then the kinetic energy of each cubic metre of water is

Drops of liquid of density $d$ are floating half immersed in a liquid of density $\rho$. If the surface tension of the liquid is $T$, then the radius of the drop is

$A$ vertical $U-$tube of uniform inner cross-section contains mercury in both sides of its arms. $A$ glycerin (density = $1.3 \text{ g/cm}^3$) column of length $10 \text{ cm}$ is introduced into one of its arms. Oil of density $0.8 \text{ g/cm}^3$ is poured into the other arm until the upper surfaces of the oil and glycerin are in the same horizontal level. Find the length of the oil column in $\text{cm}$. (Density of mercury = $13.6 \text{ g/cm}^3$)