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Three liquids of densities ${ho _1},{ho _2}$ and ${ho _3}$ (with ${ho _1} > {ho _2} > {ho _3}$), having the same value of surface tension T, rise to the same height in three identical capillaries. The angles of contact ${	heta _1},{	heta _2}\,$ and ${	heta _3}$ obey

VENKATESWARARAO · Accepted Answer

## CORE INFORMATION ✓ **Answer**: C ## CONCEPT FRAMEWORK **Primary Concept**: Capillary Rise and Angle of Contact **Related Topics**: Surface Tension, Adhesion, Cohesion **Prerequisites**: Understanding of capillary action and the relationship between contact angle and liquid properties. ## SOLUTION PATHWAY **Given Information**: * Three liquids with densities $ ho_1 > ho_2 > ho_3$. * Same surface tension (T) for all liquids. * Same capillary height (h) for all liquids. * Identical capillaries (same radius r). **Method & Steps**: 1. **Capillary Rise Formula**: The height of capillary rise is given by $h = \frac{2T\cos heta}{r ho g}$, where h is the height, T is surface tension, $ heta$ is the contact angle, r is the capillary radius, $ ho$ is the liquid density, and g is acceleration due to gravity. 2. **Analyzing the Formula**: Since h, T, r, and g are the same for all three liquids, we have $h \propto \frac{\cos heta}{ ho}$. 3. **Relationship between Density and Contact Angle**: Since $ ho_1 > ho_2 > ho_3$ and h is constant, we must have $\cos heta_1 < \cos heta_2 < \cos heta_3$. 4. **Contact Angle Range**: Since the liquids rise in the capillary, the contact angle must be between 0 and $\frac{\pi}{2}$ (i.e., $0 \le heta < \frac{\pi}{2}$). In this range, as $ heta$ increases, $\cos heta$ decreases. 5. **Final Relationship**: Since $\cos heta_1 < \cos heta_2 < \cos heta_3$, it implies $ heta_1 > heta_2 > heta_3$. Result: $0 \le heta_1 < heta_2 < heta_3 < \frac{\pi}{2}$ ## OPTION ANALYSIS [A] :x: $\pi > { heta _1} > { heta _2} > { heta _3} < \frac{\pi }{2}$: Incorrect. Doesn't satisfy the condition that all angles must be less than $\frac{\pi}{2}$. [B] :x: $\frac{\pi }{2} > { heta _1} > { heta _2} > { heta _3} \ge 0$: Incorrect. The order of angles is reversed. [C] ✓: $0 \le { heta _1} < { heta _2} < { heta _3} < \frac{\pi }{2}$: Correct. Satisfies the derived relationship between densities and contact angles. [D] :x: $\frac{\pi }{2} < { heta _1} > { heta _2} > { heta _3} < \pi $: Incorrect. Implies some angles are greater than $\frac{\pi}{2}$, which contradicts the capillary rise condition. ## LEARNING AIDS **Common Mistakes**: 1. **Confusing the relationship between contact angle and density**: Remember that for a constant height, higher density implies a smaller contact angle. 2. **Ignoring the range of contact angles for capillary rise**: The contact angle must be between 0 and $\frac{\pi}{2}$ for the liquid to rise. **Quick Tips**: 1. **Focus on the proportionality**: $h \propto \frac{\cos heta}{ ho}$. This helps to understand the relationship between h, $ heta$ and $ ho$ directly. 2. **Visualize**: Imagine the liquid meniscus in the capillary. A smaller contact angle means a more 'wetting' liquid, which is consistent with a higher density liquid rising to the same height.

VENKATESWARARAO · Answer

$\pi  > {	heta _1} > {	heta _2} > {	heta _3} < \frac{\pi }{2}$

VENKATESWARARAO · Answer

$\frac{\pi }{2} > {	heta _1} > {	heta _2} > {	heta _3} \ge 0$

VENKATESWARARAO · Answer

The correct Option is "$0 \le {	heta _1} < {	heta _2} < {	heta _3} < \frac{\pi }{2}$"

VENKATESWARARAO · Answer

$\frac{\pi }{2} < { heta _1} > { heta _2} > { heta _3} < \pi $

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