Two bodies having volumes $V$ and $2V$ are suspended from the two arms of a common balance and they are found to balance each other. If the larger body is immersed in oil (density $d_1 = 0.9 \ gm/cm^3$) and the smaller body is immersed in an unknown liquid,the balance remains in equilibrium. The density of the unknown liquid is given by ......... $gm/cm^3$.

$A$ spherical ball of radius $r$ and relative density $0.5$ is floating in equilibrium in water with half of it immersed in water. The work done in pushing the ball down so that the whole of it is just immersed in water is: (where $\rho$ is the density of water)

$A$ 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.]

$A$ container of liquid is in free fall. Without any liquid spilling out, does this container obey the principle of Archimedes?

An aluminium sphere is dipped into water. Which of the following is true?

As shown in the figure,a container is filled with a liquid of density $2d$ up to a height of $H/2$ and a liquid of density $d$ above it up to a height of $H/2$. $A$ solid cylinder of cross-sectional area $A/5$ and length $L$ $(L < H/2)$ is placed vertically in this container. The cylinder floats vertically such that its lower end is at a distance of $L/4$ from the interface of the two liquids. Find the density $D$ of the cylinder. (The atmospheric pressure at the surface of the upper liquid is $P_0$.)