A hollow sphere of volume $V$ is floating on water surface with half immersed in it. What should be the minimum volume of water poured inside the sphere so that the sphere now sinks into the water

Four identical beakers contain same amount of water as shown below. Beaker $A$ contains only water. A lead ball is held submerged in the beaker $B$ by string from above. A same sized plastic ball, say a table tennis $(TT)$ ball, is held submerged in beaker $C$ by a string attached to a stand from outside. Beaker $D$ contains same sized $TT$ ball which is held submerged from a string attached to the bottom of the beaker. These beakers (without stand) are placed on weighing pans and register readings $w_{A}, w_{B}, w_{C}$ and $w_{D}$ for $A, B, C$ and $D$, respectively. Effects of the mass and volume of the stand and string are to be neglected.

A very large block of ice of the size of a volleyball court and of uniform thickness of $8 \,m$ is floating on water. A person standing near its edge wishes to fetch a bucketful of water using a rope. The smallest length of rope required for this is about …………… $m$

A cube of ice floats partly in water and partly in kerosene oil. The radio of volume of ice immersed in water to that in kerosene oil (specific gravity of Kerosene oil $=0.8$, specific gravity of ice $=0.9$ )

The free surface of oil in a tanker, at rest, is horizontal. If the tanker starts accelerating the free surface will be titled by an angle $\theta $. If the acceleration is $\mathrm{a}$ $\mathrm{ms}^{-2}$, what will be the slope of the free surface ?