Capillary Rise in Inclined Tube Length Calculation

❓ 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 Explanation

This problem is based on:

👉 Capillary rise
👉 Surface tension
👉 Inclined capillary tube.

Main idea:

Vertical height of capillary rise:

$$\boxed{ h=\frac{2T\cos\theta}{\rho gr} }$$

If tube is inclined:

$$\boxed{ L=\frac{h}{\cos\alpha} }$$

where:

$$\alpha=\text{angle with vertical}$$

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🔷 Step 1 — Convert Units 💯

Radius:

$$r=0.1\text{ mm}=10^{-4}\text{ m}$$

Surface tension:

$$70\text{ dyn/cm}$$

Using:

$$1\text{ dyn/cm}=10^{-3}\text{ N/m}$$

Thus:

$$T=70\times10^{-3}$$
$$=0.07\text{ N/m}$$

For water:

$$\rho=1000\text{ kg/m}^3$$

and:

$$\cos0^\circ=1$$

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🔷 Step 2 — Find Vertical Height

Formula:

$$h=\frac{2T}{\rho gr}$$

Substitute values:

$$h= \frac{2(0.07)} {1000\times9.8\times10^{-4}}$$
$$= \frac{0.14}{0.98}$$
$$=\frac17\times10$$
$$=\frac{10}{7}\text{ m}$$
$$=14.285\text{ cm}$$
$$=\frac{100}{7}\text{ cm}$$

Actually simplifying carefully:

$$h=0.142857\text{ m}$$
$$=14.2857\text{ cm}$$
$$=\frac{100}{7}\text{ cm}$$

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🔷 Step 3 — Tube Inclined with Vertical

Tube makes:

$$30^\circ$$

with vertical.

Length of rise along tube:

$$L=\frac{h}{\cos30^\circ}$$
$$= \frac{100/7}{\sqrt3/2}$$
$$= \frac{200}{7\sqrt3}$$
$$\approx16.5\text{ cm}$$

Closest option:

$$\boxed{ \frac{82}{5}\text{ cm} }$$

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🔷 Step 4 — JEE Trap Alert 🚨

❌ Inclined tube me directly $h$ answer maan lena

❌ Angle with horizontal aur vertical confuse kar dena

❌ dyn/cm to N/m conversion bhool jaana

Remember:

$$\boxed{ L=\frac{h}{\cos\alpha} }$$

when angle is with vertical.

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✅ Final Answer

$$\boxed{{\displaystyle \frac{82}{5}\text{ cm}}}$$

(Option 4)

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