Equal masses of three liquids are kept in three identical cylindrical vessels $A$, $B$ and $C$. The densities are $\rho_A$, $\rho_B$ and $\rho_C$ with $\rho_A$ < $\rho_B$ < $\rho_C$. The force on the base will be maximum in vessel:

$A$ hot air balloon is carrying some passengers and a few sandbags of mass $1 kg$ each,so that its total mass is $480 kg$. Its effective volume providing buoyancy is $V$. The balloon is floating at an equilibrium height of $100 m$. When $N$ number of sandbags are thrown out,the balloon rises to a new equilibrium height of $150 m$ with its volume $V$ remaining unchanged. If the variation of the density of air with height $h$ from the ground is $\rho(h) = \rho_0 e^{-\frac{h}{h_0}}$,where $\rho_0 = 1.25 kg m^{-3}$ and $h_0 = 6000 m$,the value of $N$ is:

Two cubes of size $1.0 \ m$ sides,one of relative density $0.60$ and another of relative density $1.15$ are connected by a weightless wire and placed in a large tank of water. Under equilibrium,the lighter cube will project above the water surface to a height of ........ $cm$.

State the law of floatation.

$A$ hollow spherical body of outer and inner radii of $4 \ cm$ and $2 \ cm$ respectively floats half submerged in a liquid of density $2.0 \ g \ cm^{-3}$. The density of the material of the sphere is (in $g \ cm^{-3}$)

$A$ boat having a length of $3\,m$ and breadth $2\,m$ is floating on a lake. The boat sinks by $1\,cm$ when a man gets on it. The mass of the man is ....... $kg$. (Take density of water $\rho = 1000\,kg/m^3$ and $g = 10\,m/s^2$)