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Step-by-Step Solution
Step 1: State the physical principle
Archimedes’ principle tells us that the total buoyant force acting on a submerged or partially submerged body is equal to the weight of the fluid displaced by it. For a floating body, equilibrium requires:
$ d \cdot g \cdot L \cdot A = \text{(Upthrust from lighter fluid)} + \text{(Upthrust from denser fluid)}.$
Step 2: Express the buoyant forces
Let the fraction of the cylinder immersed in the denser liquid (of density $n \rho$) be $p$, so the remaining fraction $(1 - p)$ is in the lighter liquid (of density $\rho$). Then, the volume in the lighter liquid is $(1 - p)L A$ and the volume in the denser liquid is $p L A.$
• Buoyant force due to lighter liquid: $ (1 - p) L A \rho g $
• Buoyant force due to denser liquid: $ p \, L A (n \rho) g $
Step 3: Set up the equilibrium equation
The weight of the cylinder is
$ d \, L A \, g.$
Hence, equating weight to total buoyant force:
$ d \, L A \, g = (1 - p) \, L A \, \rho \, g + p \, L A \, (n \rho) \, g.$
Step 4: Solve for the density $d$ of the cylinder
Divide both sides by $L A g$ to simplify:
$ d = (1 - p) \rho + p \, n \rho.$
Rearrange the expression:
$ d = \rho - p \rho + p n \rho = \rho + p(n - 1)\rho.$
Factor out $\rho$:
$ d = \rho \bigl[1 + p(n - 1)\bigr].$
Step 5: Match with the correct answer
This final expression for $d$ matches the form
$ \left\{1 + (n - 1)p\right\}\rho,$
which is the correct answer.
