$A$ vessel contains oil (density = $0.8 \text{ g/cm}^3$) over mercury (density = $13.6 \text{ g/cm}^3$). $A$ 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 $\text{g/cm}^3$ is

$A$ container of large surface area is filled with a liquid of density $\rho$. $A$ cubical block of side edge $a$ and mass $M$ is floating in it with four-fifths of its volume submerged. If a coin of mass $m$ is placed gently on the top surface of the block,it is just submerged. $M$ is:

The reading of a spring balance when a block is suspended from it in air is $60\,N$. This reading changes to $40\,N$ when the block is immersed in water. The specific density (relative density) of the block is:

$A$ pan balance has a container of water with an overflow spout on the right-hand pan as shown. It is full of water right up to the overflow spout. $A$ container on the left-hand pan is positioned to catch any water that overflows. The entire apparatus is adjusted so that it is balanced. $A$ brass weight on the end of a string is then lowered into the water,but not allowed to rest on the bottom of the container. What happens next?

An ornament weighs $10\, g$ in air and $6\, g$ in water. The density of the material of the ornament is $20\, g/cc$. The volume of the cavity in the ornament is ........ $cc$.

$A$ dumbbell is placed in water of density $\rho$. It is observed that by attaching a mass $m$ to the rod,the dumbbell floats with the rod horizontal on the surface of water and each sphere exactly half submerged as shown in the figure. The volume of the mass $m$ is negligible. The value of length $l$ is