Question

A cylinder having cross sectional area $A$ of density ${\rho }_s$ is floating in two liquids as shown in figure inside a sealed container. (Assume atmospheric pressure as negligible). Liquid-1 has density ${\rho }_1$ and liquid-2 has density ${\rho }_2$. Match the following for given two columns.
Column I Column II

i. Net force on cylinder from liquid-1	1. Zero
ii. Net force on cylinder from liquid-2	2. (${\rho }_1{h}_2$ +${\rho }_2{h}_3$)$gA$
iii. Density of solid (${\rho }_s$ )	3. Net upthrust
iv. Buoyant Force	4. ${\rho }_{s}gA$( $h_1$ +$h_2$+$h_3$)
	5. ${\rho }_{1}g{h}_{2}A$
	6.$\frac{{\rho }_1{h}_2+{\rho }_2{h}_3}{h_1+h_2+h_3}$

• A. i $\to$ 5 ; ii $\to$ 2,3,4 ; iii $\to$ 6 ; iv $\to$ 2,3,4
• B. i $\to$ 1 ; ii $\to$ 2,3 ; iii $\to$ 6 ; iv $\to$ 2,3
• C. i $\to$ 1 ; ii $\to$ 2,3,4 ; iii $\to$ 6 ; iv $\to$ 2,3,4
• D. i $\to$ 5 ; ii $\to$ 2,3,4 ; iii $\to$ 6 ; iv $\to$ 2,3,4

Answer 1

The correct option is C i $\to$ 1 ; ii $\to$ 2,3,4 ; iii $\to$ 6 ; iv $\to$ 2,3,4

Buoyant Force= upthrust = weight of cylinder in equilibrium or floating condition.
From liquid-I, horizontal pair of forces cancel out. So, net force from Liquid-1 on cylinder = 0.
Since, force from liquid 1 cancels out, the only force providing upthrust (buoyant force) and balancing out weight is from liquid 2 which will be (${\rho }_1{h}_2$ +${\rho }_2{h}_3$)$gA$ or ${\rho }_{s}gA$( $h_1$ +$h_2$+$h_3$).
Since weight of solid and buoyant force are equal, we can also represent density of liquid (${\rho }_s$ ) as $\frac{{\rho }_1{h}_2+{\rho }_2{h}_3}{h_1+h_2+h_3}$