A solid cylinder of radius R and length L is placed on a smooth surface. Space left side of the cylinder is filled with liquid of density $ρ_{1}$ and on the right side of the cylinder, there two liquid layers each of thickness R and having density $ρ$ and $2 ρ$ are placed as shown in the diagram. What should be the value of $ρ_{1}$ so that cylinder can stay in equilibrium.

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$ρ_{1} = \frac{5}{2} ρ$

$ρ_{1} = \frac{3}{2} ρ$

$ρ_{1} = \frac{5 ρ}{4}$

$ρ_{1} = \frac{3}{4} ρ$

answer is C.

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Detailed Solution

Horizontal force can be given by the area of pressure Vs depth graph.
$\frac{1}{2} 2 R . ρ_{1} g 2 R = \frac{1}{2} R . ρ g R + \frac{1}{2} R .4 ρ g R$
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$2 ρ_{1} g R^{2} = \frac{5}{2} ρ g R^{2}$
$ρ_{1} = \frac{5}{4}$

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