$A$ uniform cylinder of length $L$ and mass $M$ having cross-sectional area $A$ is suspended,with its length vertical,from a fixed point by a massless spring such that it is half submerged in a liquid of density $\sigma$ at equilibrium position. The extension $x_0$ of the spring when it is in equilibrium is

$A$ wooden block,with a coin placed on its top,floats in water as shown in the figure. The distances $l$ and $h$ are shown. After some time,the coin falls into the water. Then:

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 ........ $cm$.

$A$ 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:

$A$ hemispherical bowl just floats without sinking in a liquid of density $1.2 \times 10^3 \ kg/m^3$. If the outer diameter and the density of the bowl are $1 \ m$ and $2 \times 10^4 \ kg/m^3$ respectively,then the inner diameter of the bowl will be........ $m$.

$A$ small solid ball is dropped from a height above the free surface of a liquid. It strikes the surface of the liquid at $t = 0$. The density of the material of the ball is $500 \ kg/m^3$ and that of the liquid is $1000 \ kg/m^3$. If the ball comes momentarily to rest at $t = 2 \ s$,then the initial height of the ball from the surface of the liquid was ..... $m$ (neglect viscosity).