Three liquids of densities $\rho_1, \rho_2$ and $\rho_3$ (with $\rho_1\gt\rho_2\gt\rho_3$ ) having same…

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
  1. $\frac{\pi}{2}\gt\theta_1\gt\theta_2\gt\theta_3\gt0$
  2. $0 \leqslant \theta_1 \lt \theta_2 \lt \theta_3 \lt \frac{\pi}{2}$
  3. $\frac{\pi}{2} \lt \theta_1 \lt \theta_2 \lt \theta_3 \lt \pi$
  4. $\pi\gt\theta_1\gt\theta_2\gt\frac{\pi}{2}$

Solution

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. $0 \leqslant \theta_1 \lt \theta_2 \lt \theta_3 \lt \frac{\pi}{2}$

Asked in: MHT CET 2024 (09 May Shift 2)

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