Two non-mixing liquids of densities $\rho$ and $n\rho$ ($n1$) are put in a container. The height of each liqu

Question

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:

Accepted Answer

1. **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 .
2. **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 $W = mg = (ALd)g$.
3. **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.
   $$ F_B = (V_1 \cdot n\rho \cdot g) + (V_2 \cdot \rho \cdot g) $$
   $$ F_B = A(pL)(n\rho)g + A L (1-p)(\rho)g $$
4. **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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