To determine the composition of a bimetallic alloy,a sample is first weighed in air and then in water. These weights are found to be $w_1$ and $w_2$ respectively. If the densities of the two constituent metals are $\rho_1$ and $\rho_2$ respectively,then the weight of the first metal in the sample is (where $\rho_w$ is the density of water):

$A$ cube having a side of $10 \ cm$ with unknown mass $m$ and a $200 \ g$ mass were hung at two ends of a uniform rigid rod of $27 \ cm$ length. The rod along with the masses was placed on a wedge,keeping the distance between the wedge point and the $200 \ g$ weight as $25 \ cm$. Initially,the masses were not in balance. $A$ beaker is placed beneath the unknown mass and water is added slowly to it. At a given point,the masses were in balance and half the volume of the unknown mass was submerged in the water. (Take the density of the unknown mass to be greater than that of water,the mass did not absorb water,and the water density is $1 \ g/cm^3$). The unknown mass $m$ is . . . . . . $kg$.

$A$ solid cube is floating in a liquid. The coefficient of linear expansion of the cube is $\alpha$ and the coefficient of volume expansion of the liquid is $\gamma$. On increasing the temperature of the (liquid + cube) system,the cube will sink if

$A$ metal ball of density $7800 \ kg/m^3$ is suspected to have a large number of cavities. It weighs $9.8 \ kg$ when weighed in air and $1.5 \ kg$ less when immersed in water. The fraction by volume of the cavities in the metal ball is approximately ....... $\%$

$A$ body floats in a liquid contained in a beaker. The whole system,as shown,falls freely under gravity. The upthrust on the body due to the liquid is

Two solids $P$ and $Q$ float in water. It is observed that $P$ floats with half of its volume immersed and $Q$ floats with $\frac{2}{3}$ of its volume immersed. The ratio of densities of $P$ and $Q$ is