Find the apparent weight of a metallic block of density $5 \text{ g cm}^{-3}$ and dimensions $5 \text{ cm} \times 5 \text{ cm} \times 5 \text{ cm}$,in water.

$A$ boat carrying a number of large stones is floating in a water tank. What would happen to the water level if a few stones are unloaded into the water?

$A$ solid cylinder of length $\ell$ and cross-sectional area $A$ is immersed such that it floats with its axis vertical at the liquid-liquid interface with length $\ell / 4$ in the denser liquid as shown in the figure. The lower density liquid has density $\varrho$ and the denser liquid has density $3\varrho$. The density $d$ of the solid cylinder is:

State and prove Archimedes' principle.

$A$ boat carrying steel balls is floating on the surface of water in a tank. If the balls are thrown into the tank one by one,how will it affect the level of water?

$A$ 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 ............... $m$.