Question

Three different liquids are filled in a U-tube as shown. Their densities are \( \rho_1, \rho_2, \rho_3 \) respectively. From the figure we may conclude that 

• A. \( \rho_3 = 4(\rho_2 - \rho_1) \)
• B. \( \rho_3 = 4(\rho_1 - \rho_2) \)
• C. \( \rho_3 = 2(\rho_2 - \rho_1) \)
• D. \( \rho_3 = \frac{\rho_1+\rho_2}{2} \)

Hint:

U-tube problems:
Compare pressures at same level.
Use \( \rho g h \).

Answer 1

The Correct Option is C

Concept: Hydrostatic equilibrium: \[ P = \rho g h \] Pressures at same horizontal level are equal. Step 1: {\color{red}Equate pressures at dotted line.} Left side: \[ \rho_1 g h + \rho_3 g \frac{h}{2} \] Right side: \[ \rho_2 g h \] Step 2: {\color{red}Cancel \( g \).} \[ \rho_1 h + \rho_3 \frac{h}{2} = \rho_2 h \] Step 3: {\color{red}Simplify.} \[ \rho_1 + \frac{\rho_3}{2} = \rho_2 \Rightarrow \rho_3 = 2(\rho_2 - \rho_1) \]