Question

Find the percentage change in height risen by liquid if density of fluid, radius of capillary, and surface tension of liquid are decreased by 1%. Assume contact angle doesn’t change and capillary is of sufficient length.

• A. +1%
• B. -1%
• C. +3%
• D. -3%

Hint:

In capillary rise, the height depends on the surface tension, radius, and density, with inversely proportional dependence on density and radius.

Answer 1

The Correct Option is A

The problem involves calculating the percentage change in the height of liquid risen in a capillary when there is a decrease in density, radius, and surface tension by 1%. Let’s solve this step-by-step.

The height of the liquid column in a capillary tube is given by:

\(h = \frac{2T \cos \theta}{r \rho g}\)

• \(T\) is the surface tension of the liquid.
• \(\theta\) is the contact angle.
• \(r\) is the radius of the capillary tube.
• \(\rho\) is the density of the liquid.
• \(g\) is the acceleration due to gravity.

Given that the contact angle doesn't change, \(\cos \theta\) remains constant.

If surface tension, density, and radius are decreased by 1%, then:

• Change in surface tension, \(\Delta T = -0.01 \times T\)
• Change in density, \(\Delta \rho = -0.01 \times \rho\)
• Change in radius, \(\Delta r = -0.01 \times r\)

The new height \(h'\) is given by:

\(h' = \frac{2(T + \Delta T) \cos \theta}{(r + \Delta r)(\rho + \Delta \rho) g}\)

Approximate \(h'\) using the first-order derivative, ignoring higher-order terms:

\(\frac{\Delta h}{h} \approx \frac{\Delta T}{T} - \frac{\Delta r}{r} - \frac{\Delta \rho}{\rho}\)

Substitute the changes:

• \(\frac{\Delta T}{T} = -0.01\)
• \(\frac{\Delta r}{r} = -0.01\)
• \(\frac{\Delta \rho}{\rho} = -0.01\)

Thus:

\(\frac{\Delta h}{h} \approx -0.01 + 0.01 + 0.01 = 0.01\)

This indicates a positive 1% change in the height of the liquid risen.

Therefore, the percentage change in the height of the liquid risen is +1%. 

Option	Comments
+1%	Correct
-1%	Incorrect because calculation shows a net positive change.
+3%	Incorrect since there's no net increase of this magnitude.
-3%	Incorrect because the decrease in parameters results in net increase of 1%.