$A$ 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

$A$ uniform rod is suspended horizontally from its mid-point. $A$ piece of metal whose weight is $w$ is suspended at a distance $l$ from the mid-point. Another weight $w_{1}$ is suspended on the other side at a distance $l_{1}$ from the mid-point to bring the rod to a horizontal position. When $w$ is completely immersed in water, $w_{1}$ needs to be kept at a distance $l_{2}$ from the mid-point to get the rod back into a horizontal position. The specific gravity of the metal piece is

$A$ concrete sphere of radius $R$ has a cavity of radius $r$ which is packed with sawdust. The specific gravities of concrete and sawdust are $2.4$ and $0.3$ respectively. For this sphere to float with its entire volume submerged under water,the ratio of the mass of concrete to the mass of sawdust will be:

$A$ $20 \,g$ copper block is suspended by a vertical spring causing $1 \,cm$ elongation over the natural length of the spring. If a beaker of water is placed below the block so that the copper block is completely immersed in the liquid, the elongation of the spring is (Density of copper = $9000 \,kg/m^3$, Density of water = $1000 \,kg/m^3$, $g = 10 \,m/s^2$) (in $\,cm$)

$A$ vessel contains oil (density = $0.8 \; g/cm^3$) over mercury (density = $13.6 \; g/cm^3$). $A$ homogeneous sphere floats with half of its volume immersed in mercury and the other half in oil. The density of the material of the sphere in $g/cm^3$ is:

$A$ block of volume $V$ and density $\sigma_b$ is placed in a liquid of density $\sigma_l$ (where $\sigma_l > \sigma_b$). If the block is moved upward by a height $h$ while remaining completely submerged in the liquid,what is the increase in the gravitational potential energy of the system?