The interanal radius of one limb of a capillary `U-`tube is `r_(1) = 1 mm` and the internal radius of the second limb is `r_(2) = 2 mm`, the tube is filled with same mercury, and one of the limbs is connected to a vecuum pump. The surface tension & density of mercury are `480 dyn//cm` & `13.6 gm//cm^(3)` respectively (assume contact angle to be `theta = 180^(@)`)`(g = 9.8 m//s^(2)`)
Which limb of the should be connected to the pump ?

Options

A. Limb having radius `2 mm`
B. Limb having radius `1 mm`
C. Any of the limb
D. none of these

Explanation

To determine which limb of the capillary U-tube should be connected to the vacuum pump, we can analyze the pressure difference caused by the surface tension of mercury in the two limbs.

### Step-by-Step Solution:

1. **Identify the Parameters**:
- Internal radius of limb 1: \( r_1 = 1 \, \text{mm} = 0.1 \, \text{cm} \)
- Internal radius of limb 2: \( r_2 = 2 \, \text{mm} = 0.2 \, \text{cm} \)
- Surface tension of mercury: \( T = 480 \, \text{dyn/cm} \)
- Density of mercury: \( \rho = 13.6 \, \text{g/cm}^3 \)
- Acceleration due to gravity: \( g = 9.8 \, \text{m/s}^2 \)
- Contact angle: \( \theta = 180^\circ \)

2. **Calculate the Pressure in Each Limb**:
- The pressure at the surface of the mercury in limb 1 (connected to the vacuum pump) can be expressed as:
\[ P_a = P + \frac{T}{2r_1} \]
- The pressure in limb 2 can be expressed as:
\[ P_b = P' + \frac{T}{2r_2} \]

3. **Set Up the Pressure Difference Equation**:
- The difference in pressure between the two limbs can be given by:
\[ P_a - P_b = \left( P + \frac{T}{2r_1} \right) - \left( P' + \frac{T}{2r_2} \right) \]
- Rearranging gives:
\[ P - P' = \frac{T}{2r_2} - \frac{T}{2r_1} \]

4. **Substituting Values**:
- Since \( r_1 < r_2 \), we know that \( \frac{1}{r_1} > \frac{1}{r_2} \). Thus:
\[ P - P' < 0 \quad \text{(because } \frac{T}{2r_1} > \frac{T}{2r_2}\text{)} \]
- This indicates that \( P < P' \), meaning the pressure in limb 1 (the thinner limb) is lower than in limb 2 (the thicker limb).

5. **Conclusion**:
- To create a vacuum, we should connect the limb with the higher pressure to the vacuum pump. Since limb 2 has a higher pressure than limb 1, the vacuum pump should be connected to limb 1 (the one with radius \( r_1 = 1 \, \text{mm} \)).

### Final Answer:
Connect the vacuum pump to the limb with radius \( r_1 = 1 \, \text{mm} \).