A capillary tube made of glass of radius 0.15 mm is dipped vertically in a beaker filled with methylene iodide (surface tension = `0.05 "Nm"^(-1) ` density = 667 kg `m^(-3)` ) which rises to height h in the tube. It is observed that the two tangents drawn from liquid-glass interfaces (from opp. sides of the capillary) make an angle of `60^(@)` with one another. Then h is close to `( g = 10 "ms"^(-2))`

Options

A. `0.172`m
B. `0.049`m
C. `0.087`m
D. `0.137` m

Explanation

To solve the problem, we will use the formula for capillary rise, which is derived from the balance of forces acting on the liquid column in the tube. The formula is given by:

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

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

### Step 1: Identify the given values
- Radius of the capillary tube, \( R = 0.15 \, \text{mm} = 0.15 \times 10^{-3} \, \text{m} \)
- Surface tension, \( T = 0.05 \, \text{N/m} \)
- Density of methylene iodide, \( \rho = 667 \, \text{kg/m}^3 \)
- Acceleration due to gravity, \( g = 10 \, \text{m/s}^2 \)
- The angle between the tangents from the liquid-glass interface is \( 60^\circ \), which implies that the contact angle \( \theta = 30^\circ \) (since the tangents make an angle of 60°).

### Step 2: Calculate \( \cos \theta \)
We need to calculate \( \cos 30^\circ \):
\[ \cos 30^\circ = \frac{\sqrt{3}}{2} \approx 0.866 \]

### Step 3: Substitute the values into the formula
Now, we can substitute the values into the capillary rise formula:
\[ h = \frac{2 \times 0.05 \times 0.866}{0.15 \times 10^{-3} \times 667 \times 10} \]

### Step 4: Calculate the numerator
Calculating the numerator:
\[ 2 \times 0.05 \times 0.866 = 0.0866 \]

### Step 5: Calculate the denominator
Calculating the denominator:
\[ 0.15 \times 10^{-3} \times 667 \times 10 = 0.15 \times 667 \times 10^{-2} = 1.0005 \times 10^{-2} \approx 0.010005 \]

### Step 6: Calculate \( h \)
Now, substituting the values into the equation for \( h \):
\[ h = \frac{0.0866}{0.010005} \approx 8.66 \, \text{m} \]

### Final Answer
The height \( h \) is approximately \( 0.087 \, \text{m} \) or \( 8.7 \, \text{cm} \).

### Summary of Steps
1. Identify given values.
2. Calculate \( \cos \theta \).
3. Substitute values into the capillary rise formula.
4. Calculate the numerator.
5. Calculate the denominator.
6. Compute \( h \).