If two bodies of density `rho_1` and `rho_2` are mixed at same mass, then the density of the mixture will be p = `(2rho_1rho_2)/(rho_1+rho_2)`. [show that].

(i) Prove that the density of the mixture of two substances with densities rho_(1)andrho_(2) of equal mass will be (2rho_(1)rho_(2))/(rho_(1)+rho_(2)) . (ii) Prove that if the two substance with densities rho_(1)andrho_(2) are mixed in equal volumes, then the density of the mixture thus formed will be 1/2(rho_(1)+rho_(2))

When a body of density rho and volume V is floating in a liquid of density sigma

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 solid spherical ball and a hollow spherical ball of two different materials of densities rho_(1) and rho_(2) respectively have same outer radii and same mass. What will be the ratio of the moment of inertia ( about an axis passing through the centre ) of the hollow sphere to that of the solid sphere?

Determine the change in potential energy of a body when it is raised through a height h, inside water. Volume of the body is V, its density is rho and the density of water is rho_(0) . Given your answer for the situations when (i) rhogtrho_(0) and (ii) rholtrho_(0) .

If the wt. of a body in vacuum is W and W_1 and W_2 are weights when it is immersed in a liquid of specific gravity rho_1 and rho_2 respectively, then prove that W=(W_1rho_2-W_2rho_1)/(rho_2-rho_1)

The density of a body is d and that of air is rho . Prove that the real weight of the body, weighed W in air, is W_(0)=W/(1-rho//d) .