A capillary tube of radius 0.1 mm is partly dipped in water (surface tension 70 dyn/cm and glass-water contact angle ≈ 0°) with 30° inclined with the vertical. The length of water risen in the capillary is:

❓ Question

A capillary tube of radius 0.1 mm is partly dipped in water (surface tension = 70 dyn/cm, contact angle ≈ 0°) with 30° inclination with the vertical. The length of water risen in the capillary is to be found.

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🖼️ Question Image

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✍️ Short Solution

Let’s recall the concept of capillary rise — the upward movement of liquid in a narrow tube due to surface tension.

For a vertical tube,

$$h=\frac{2T\cos \theta }{r\rho \mathrm{g}}$$

But if the tube is inclined at an angle α with the vertical, then the actual length of the water column is longer than the vertical height:

$$l=\frac{h}{\cos \mathrm{\alpha }}$$

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🔹 Given Data

Quantity	Symbol	Value
Surface tension	$T$	70 dyn/cm = 70 × 10⁻³ N/m
Radius of tube	$r$	0.1 mm = 1 × 10⁻⁴ m
Contact angle	$\theta$	0° ⇒ cosθ = 1
Density of water	$\rho$	1000 kg/m³
Acceleration due to gravity	$g$	9.8 m/s²
Inclination angle	$\alpha$	30°

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🧮 Step-by-Step Calculation

Step 1: Calculate vertical height $h$

$$h = \frac{2T \cos \theta}{r \rho g}$$ $$h = \frac{2 \times 70 \times 10^{-3} \times 1}{(1 \times 10^{-4}) \times 1000 \times 9.8}$$ $$h=\frac{0.14}{0.98}=0.1429\text{ m}=14.29\text{ cm}$$

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Step 2: Adjust for inclination

Since the tube is inclined at 30°,

$$l=\frac{h}{\cos 30\mathrm{°}}=\frac{14.29}{0.866}=16.5\text{ cm}$$

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🧮 Image Solution

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✅ Conclusion & Video Solution

✅ Final Answer:

$$\boxed{{\displaystyle l=16.5\text{cm}}}$$

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📘 Concept Recap:

• Capillary rise is caused by the balance of surface tension and weight of the liquid column.

• Vertical rise (h) depends on surface tension, radius, and density.

• When the tube is tilted, the water travels a longer path — hence $l = h / \cos α$.

• Smaller the radius, greater the rise!

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