A fluid container is containing a liquid of density $\rho $ is accelerating upward with acceleration a along the inclined place of inclination $\alpha$  as shown. Then the angle of inclination $ \theta $ of free surface is :

A dumbbell is placed in water of density $\rho$ . It is observed that by attaching a mass $m$ to the rod, the dumbbell floats with the rod horizontal on the surface of water and each sphere exactly half submerged as shown in the figure. The volume of the mass $m$ is negligible. The value of length $l$ is

A block of ice floats on a liquid of density $1.2$ in a beaker then level of liquid when ice completely melt

A vertical triangular plate $ABC$ is placed inside water with side $BC$ parallel to water surface as shown. The force on one surface of plate by water is (density of water is $\rho $ and atmospheric pressure $P_0$ ) 

A solid sphere of radius $R$ and density $\rho$ is attached to one end of a mass-less spring of force constant $k$. The other end of the spring is connected to another solid sphere of radius $R$ and density $3 p$. The complete arrangement is placed in a liquid of density $2 p$ and is allowed to reach equilibrium. The correct statement$(s)$ is (are)

$(A)$ the net elongation of the spring is $\frac{4 \pi R^3 \rho g}{3 k}$

$(B)$ the net elongation of the spring is $\frac{8 \pi R^3 \rho g}{3 k}$

$(C)$ the light sphere is partially submerged.

$(D)$ the light sphere is completely submerged.