A vessel containing water is given a constant acceleration a towards the right, along a straight horizontal path. Which of the following diagram represents the surface of the liquid

Asphere of radius $R$ and made of material of relative density $\sigma$ has a concentric cavity of radius $r$. It just floats when placed in a tank full of water. The value of the ratio $R/r$ will be 

Two solid spheres $A$ and $B$ of equal volumes but of different densities $d_A$ and $d_B$ are connected by a string. They are fully immersed in a fluid of density $d_F$. They get arranged into an equilibrium state as shown in the figure with a tension in the string. The arrangement is possible only if

$(A)$ $d_Ad_F$  $(B)$ $d_B > d_F$ $(C)$ $d_A>d_F$ $(D)$ $d_A+d_B=2 d_F$

A wooden block floating in a bucket of water has $\frac{4}{5}$ of its volume submerged. When certain amount of an oil is poured into the bucket, it is found that the block is just under the oil surface with half of its volume under water and half in oil. The density of oil relative to that of water is

A ball is made of a material of density $\rho$ where $\rho_{oil} < \rho < \rho_{water}$ with $\rho_{oil}$ and $\rho_{water}$ representing the densities of oil and water, respectively. The oil and water are immiscible. If the above ball is in equilibrium in a mixture of this oil and water, which of the following pictures represents its equilibrium position?

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