MHT CET · Physics · Mechanical Properties of Fluids
Three liquids of densities \(\rho_1, \rho_2\) and \(\rho_3\) (with \(\rho_1\gt\rho_2\gt\rho_3\) ) having same value of surface tension T , rise to the same height in three identical capillaries. Angle of contact \(\theta_1, \theta_2\) and \(\theta_3\) respectively obey
- A \(\frac{\pi}{2}\gt\theta_1\gt\theta_2\gt\theta_3\gt0\)
- B \(0 \leqslant \theta_1 \lt \theta_2 \lt \theta_3 \lt \frac{\pi}{2}\)
- C \(\frac{\pi}{2} \lt \theta_1 \lt \theta_2 \lt \theta_3 \lt \pi\)
- D \(\pi\gt\theta_1\gt\theta_2\gt\frac{\pi}{2}\)
Answer & Solution
Correct Answer
(B) \(0 \leqslant \theta_1 \lt \theta_2 \lt \theta_3 \lt \frac{\pi}{2}\)
Step-by-step Solution
Detailed explanation
Rise of a liquid in a capillary tube:
\(\begin{aligned}
& \mathrm{h}=\frac{2 \mathrm{~T} \cos \theta}{\mathrm{rg} \rho} \\
& \cos \theta=\frac{\mathrm{hr} \rho \mathrm{~g}}{2 \mathrm{~T}}
\end{aligned}\)
\(\ldots .(\because \cos 0=1\) is the max. value for cosine \()\)
\(\therefore \quad\) Here, \(\rho_1\gt\rho_2\gt\rho_3\), so
\(\theta_1 \lt \theta_2 \lt \theta_3\)
The all three angles of contact should be acute.
\(0 \leqslant \theta_1 \lt \theta_2 \lt \theta_3 \lt \frac{\pi}{2}\)
\(\begin{aligned}
& \mathrm{h}=\frac{2 \mathrm{~T} \cos \theta}{\mathrm{rg} \rho} \\
& \cos \theta=\frac{\mathrm{hr} \rho \mathrm{~g}}{2 \mathrm{~T}}
\end{aligned}\)
\(\ldots .(\because \cos 0=1\) is the max. value for cosine \()\)
\(\therefore \quad\) Here, \(\rho_1\gt\rho_2\gt\rho_3\), so
\(\theta_1 \lt \theta_2 \lt \theta_3\)
The all three angles of contact should be acute.
\(0 \leqslant \theta_1 \lt \theta_2 \lt \theta_3 \lt \frac{\pi}{2}\)
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