$A$ wooden cube floats in water when a $200\,g$ mass is placed on it. When the mass is removed,the cube rises by $2\,cm$ above the water level. The side of the cube is ......... $cm$.

$A$ wide-bottomed cylindrical massless plastic container of height $9 \, cm$ contains $40$ identical coins and is floating on water with $3 \, cm$ of its height submerged. If we start placing additional identical coins on its lid,it is observed that after $N$ coins are added,the equilibrium changes from stable to unstable. Equilibrium in floating is stable if the geometric centre of the submerged portion is above the centre of mass of the object. The value of $N$ is closest to:

Two rods of the same material and length have their electric resistance in the ratio $1:2$. When both rods are dipped in water,the correct statement will be:

$A$ block of ice floats in an oil in a vessel. When the ice melts,the level of oil will ..............

$A$ gas in equilibrium has uniform density and pressure throughout its volume. This is strictly true only if there are no external influences. $A$ gas column under gravity,for example,does not have uniform density (and pressure). As you might expect,its density decreases with height. The precise dependence is given by the so-called law of atmospheres:
$n_{2}=n_{1} \exp \left[-m g\left(h_{2}-h_{1}\right) / k_{B} T\right]$
where $n_{2}, n_{1}$ refer to number density at heights $h_{2}$ and $h_{1}$ respectively. Use this relation to derive the equation for sedimentation equilibrium of a suspension in a liquid column:
$n_{2}=n_{1} \exp \left[-m g N_{A}\left(\rho-\rho^{\prime}\right)\left(h_{2}-h_{1}\right) /(\rho R T)\right]$
where $\rho$ is the density of the suspended particle,and $\rho^{\prime}$ that of the surrounding medium. [$N_{A}$ is Avogadro's number,and $R$ the universal gas constant.]

$A$ cubical block of wood having a mass of $160 \,g$ has a metal piece fastened underneath as shown in the figure. Find the maximum mass of the metal piece which will allow the block to float in water. The specific gravity of wood is $0.8$, the specific gravity of the metal is $10$, and the density of water is $1 \,g/cm^3$. (in $\,g$)