Solved: PHYSICS Question for NEET

NEET PHYSICSMedium

Two non-mixing liquids of densities $\rho$ and $n\rho$ ( $n>1$ ) are put in a container. The height of each liquid is $h$ . A solid cylinder floats with its axis vertical and length $L$ . The length of the cylinder inside the denser liquid is $pL$ ( $p<1$ ). The density of the cylinder is $d$ . The density $d$ is equal to:

$[2+(n+1)p]\rho$

$[2+(n-1)p]\rho$

$[1+(n-1)p]\rho$

$[1+(n+1)p]\rho$

Step-by-Step Solution

Principle of Floatation: For a body floating in a fluid (or multiple fluids), the total weight of the body is equal to the total buoyant force (upthrust) exerted by the displaced fluids .
Weight of Cylinder: Let $A$ be the cross-sectional area of the cylinder. The volume of the cylinder is $V = AL$ . The weight of the cylinder is $W = mg = (ALd)g$ .
Buoyant Force: The cylinder is partially immersed in the denser liquid (density $n\rho$ ) and partially in the lighter liquid (density $\rho$ ).

Volume in denser liquid: $V_1 = A(pL)$ .
Volume in lighter liquid: Since the total length is $L$ , the remaining length is $(L - pL) = L(1-p)$ . Thus, $V_2 = A L (1-p)$ .
Total Buoyant Force ( $F_B$ ) = Weight of displaced denser liquid + Weight of displaced lighter liquid. $F_B = (V_1 \cdot n\rho \cdot g) + (V_2 \cdot \rho \cdot g)$ $F_B = A(pL)(n\rho)g + A L (1-p)(\rho)g$

Equilibrium Equation: $W = F_B$ $ALdg = ALpn\rho g + AL(1-p)\rho g$ Divide by $ALg$ : $d = pn\rho + (1-p)\rho$ $d = \rho [pn + 1 - p]$ $d = \rho [1 + p(n - 1)]$ $d = [1 + (n-1)p]\rho$

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