$A$ boy has a mass of $60\, kg$. He wants to swim in a river with the help of a wooden log. If the relative density of the wood is $0.6$,what is the minimum volume of the wooden log required? (Density of river water is $1000\, kg/m^3$)

$A$ wooden cube first floats inside water when a $200\,g$ mass is placed on it. When the mass is removed,the cube rises $2\,cm$ above the water level. The side of the cube is ........ $cm$.

An empty balloon weighs $1\, g$. The balloon is filled with water to the neck and tied with a massless thread. The weight of the balloon along with water is $101\, g$. The balloon filled with water is weighed when fully immersed in water. Its weight in water is ...... $g$.

$A$ $20 \,g$ copper block is suspended by a vertical spring causing $1 \,cm$ elongation over the natural length of the spring. If a beaker of water is placed below the block so that the copper block is completely immersed in the liquid, the elongation of the spring is (Density of copper = $9000 \,kg/m^3$, Density of water = $1000 \,kg/m^3$, $g = 10 \,m/s^2$) (in $\,cm$)

$A$ tub is filled with water and a wooden cube $10 \ \text{cm} \times 10 \ \text{cm} \times 10 \ \text{cm}$ is placed in the water. The wooden cube is found to float on the water with a part of it submerged in water. When a metal coin is placed on the wooden cube,the submerged part is increased by $3.87 \ \text{cm}$. The mass of the metal coin is . . . . . . gram. (Take water density as $1 \ \text{g/cm}^3$ and density of wood as $0.4 \ \text{g/cm}^3$)

$A$ 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: