A capillary tube of radius 0.6 mm is dipped vertically in a liquid of surface tension `0.04 Nm^(-1) `and relative density 0.8 . Calculate the height of capillary rise if the angle of contact is `15^(@)`.

Explanation

To solve the problem of calculating the height of capillary rise in a liquid, we will use the formula for capillary rise, which is given by:

\[ h = \frac{2T \cos \theta}{\rho g r} \]

Where:
- \( h \) = height of capillary rise
- \( T \) = surface tension of the liquid
- \( \theta \) = angle of contact
- \( \rho \) = density of the liquid
- \( g \) = acceleration due to gravity
- \( r \) = radius of the capillary tube

### Step 1: Convert the radius from mm to meters
Given the radius \( r = 0.6 \, \text{mm} \):
\[ r = 0.6 \, \text{mm} = 0.6 \times 10^{-3} \, \text{m} = 0.0006 \, \text{m} \]

### Step 2: Calculate the density of the liquid
The relative density (RD) is given as \( 0.8 \). The density of water is approximately \( 1000 \, \text{kg/m}^3 \). Therefore, the density of the liquid \( \rho \) can be calculated as:
\[ \rho = \text{RD} \times \text{density of water} = 0.8 \times 1000 \, \text{kg/m}^3 = 800 \, \text{kg/m}^3 \]

### Step 3: Substitute the values into the formula
Now we can substitute the known values into the capillary rise formula. The surface tension \( T = 0.04 \, \text{N/m} \), the angle \( \theta = 15^\circ \), and \( g = 9.8 \, \text{m/s}^2 \).

First, calculate \( \cos \theta \):
\[ \cos 15^\circ \approx 0.9659 \]

Now substitute all values into the formula:
\[ h = \frac{2 \times 0.04 \times 0.9659}{800 \times 9.8 \times 0.0006} \]

### Step 4: Calculate the height \( h \)
Calculating the numerator:
\[ 2 \times 0.04 \times 0.9659 \approx 0.077168 \]

Calculating the denominator:
\[ 800 \times 9.8 \times 0.0006 \approx 4.704 \]

Now, substituting these values into the formula for \( h \):
\[ h = \frac{0.077168}{4.704} \approx 0.0164 \, \text{m} \]

### Final Answer
The height of the capillary rise is approximately:
\[ h \approx 0.0164 \, \text{m} \text{ or } 16.4 \, \text{mm} \]

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