$A$ 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:

$A$ certain block weighs $15 \ N$ in air. It weighs $12 \ N$ when immersed in water. When immersed in another liquid,it weighs $13 \ N$. The relative density of the block is:

Four identical beakers contain the same amount of water as shown below. Beaker $A$ contains only water. $A$ lead ball is held submerged in beaker $B$ by a 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 a same-sized $TT$ ball which is held submerged by a string attached to the bottom of the beaker. These beakers (without the 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.

The spring balance $A$ reads $2 \, kg$ with a block $m$ suspended from it. $A$ balance $B$ reads $5 \, kg$ when a beaker filled with liquid is put on the pan of the balance. The two balances are now arranged such that the hanging mass is inside the liquid as shown in the figure. In this situation:

$A$ hollow spherical shell of 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 $\frac{27}{8}$ with respect to water,the value of $r$ is $......R$.

$A$ spring balance $A$ reads $2 \,kg$ with a block of mass $m$ suspended from it. Another balance $B$ reads $3 \,kg$ when a beaker with a liquid is put on its pan. The two balances are now so arranged that the hanging mass $m$ is fully immersed inside the liquid in the beaker as shown in the figure. In this situation,