A hollow spherical shell at outer radius $R$ floats just submerged under the water surface. The inner radius of the shell is $r$. If the specific gravity of the shell material is $\frac{27}{8}$ $w.r.t.$ water, the value of $r$ is$......R$

A log of wood of mass $120 Kg$  floats in water. The weight that can be put on the raft to make it just sink, should be ....... $Kg$ (density of wood = $600 Kg/m^3$)

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

The fraction of a floating object of volume ${V_0}$ and density ${d_0}$ above the surface of a liquid of density $d$  will be

A metallic block of density $5\,gm \,cm^{-3}$ and having dimensions $5 cm × 5 cm × 5 cm$ is weighed in water. Its apparent weight will be

In Guericke's experiment to show the effect of atmospheric pressure, two copper hemispheres were tightly fitted to each other to form a hollow sphere and the air from the sphere was pumped out to create vacuum inside. If the radius of each hemisphere is $R$ and the atmospheric pressure is $p$, then the minimum force required (when the two hemispheres are pulled apart by the same force) to separate the hemispheres is