Two solids $A$ and $B$ float in water. It is observed that $A$ floats with $\frac{1}{2}$ of its body immersed in water and $B$ floats with $\frac{1}{4}$ of its volume above the water level. The ratio of the density of $A$ to that of $B$ is

$A$ sphere of solid material of relative density $9$ has a concentric spherical cavity and floats just submerged in water. If the radius of the sphere is $R$,then the radius of the cavity $(r)$ is related to $R$ as:

$A$ wooden block of density $0.5 \ g/cc$ is tied to a string. The other end of the string is fixed to the bottom of a tank. The tank is filled with a liquid of density $1 \ g/cc$. If the tension of the string is $20 \ N$,then the mass of the block is (Take $g = 10 \ m/s^2$) (in $kg$)

$A$ cubical block is floating in a liquid with half of its volume immersed in the liquid. When the whole system accelerates upwards with a net acceleration of $g/3$,the fraction of volume immersed in the liquid will be:

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

An object with uniform density $\rho$ is attached to a spring that is known to stretch linearly with applied force. When the spring-object system is immersed in a liquid of density $\rho_1$,the spring stretches by an amount $x_1$ (where $\rho > \rho_1$). When the experiment is repeated in a liquid of density $\rho_2$ (where $\rho_2 < \rho_1$),the spring stretches by an amount $x_2$. Neglecting any buoyant force on the spring,the density of the object is: