Question

A layer of oil with density 724 $kg{{m}^{-3}}$ floats on water of density 1000$kg{{m}^{-3}}$. A block floats at the oil-water interface with $\dfrac{1}{6}th$ of its volume in oil and $\dfrac{5}{6}th$ of its volume in water, as shown in the figure. What is the density of the block?

Answer 1

We need to understand the concept of relative density in order to understand the dependence of the densities of each of the materials involved in this problem which results in the floating of the block in such a manner with unequal halves in the two media.

Complete Step-by-Step Solution:
We are given the situation in which oil is poured over water in a beaker, which does not mis due to the difference in their densities. Now, a block of mass is dipped in this beaker. The block is said to be partially sunk in water and another portion to be in the oil portion at the boundary as we can see in the figure.

We can use the Archimedes principle to find the relation between the density of the block and the upthrust provided by the water on the block which only allows $\dfrac{5}{6}th$ of the volume of the block to be sinking in the water.

According to the Archimedes principle, the volume of liquid displayed by an object is equal to the volume of the object in the liquid. We can find the density of the block using the relation between the mass and volume from this principle. i.e.,

& {{V}_{b}}={{V}_{o}}+{{V}_{w}} \\\ & \Rightarrow {{V}_{b}}.{{\rho }_{b}}={{V}_{o}}.{{\rho }_{o}}+{{V}_{w}}.{{\rho }_{w}} \\\ & \Rightarrow {{V}_{b}}.{{\rho }_{b}}=\dfrac{{{V}_{b}}}{6}(724)kg+\dfrac{5{{V}_{b}}}{6}(1000)kg \\\ & \Rightarrow {{\rho }_{b}}=\dfrac{724+5000}{6}kg{{m}^{-3}} \\\ & \therefore {{\rho }_{b}}=954kg{{m}^{-3}} \\\ \end{aligned}$$ **So, the density of the block is $$954kg{{m}^{-3}}$$, this is the required solution.** **Note:** The upthrust experienced by a solid in a liquid medium is the force exerted by the fluid molecules against the gravitational force which provides enough force in order to let an object float if the object has a density much lesser than the liquid in which it is dipped.