Consider a solid sphere of radius $R$ and mass density $\rho(r) = \rho_{0} \left(1 - \frac{r^{2}}{R^{2}}\right)$ for $0 < r \leq R$. The minimum density of a liquid in which it will float is

$A$ homogeneous solid cylinder of length $L$ $(L < H/2)$ and cross-sectional area $A/5$ is immersed such that it floats with its axis vertical at the liquid-liquid interface with length $L/4$ in the denser liquid as shown in the figure. The lower density liquid is open to the atmosphere having pressure $P_0$. Then,the density $D$ of the solid is given by:

Determine the equation for the volume of the partially immersed part of a body floating in a fluid.

$A$ piece of copper having an internal cavity weighs $264\, g$ in air and $221\, g$ in water. Find the volume (in $cc$) of the cavity. Density of $Cu = 8.8\, g/cc$.

$A$ container contains mercury $(\rho = 13.6 \text{ g/cm}^3)$ and oil $(\rho = 0.8 \text{ g/cm}^3)$. $A$ sphere floats such that half of its volume is in mercury and half of its volume is in oil. What is the density of the material of the sphere in $\text{g/cm}^3$?

$A$ 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?