The surface tension of two liquids are 30 and 60 dyne `cm^(-1)` respectively. The liquid drop form at the ends of two tube of the same radius. The ratio of the weight of the two drops is

Options

A. `1 : 2`
B. `1 : 3`
C. `2 : 3`
D. `3 : 4`

Answer

To solve the problem, we need to find the ratio of the weights of two liquid drops formed at the ends of two tubes with the same radius, given their surface tensions.

### Step-by-Step Solution:

1. **Understanding Surface Tension and Weight of the Drop:**
- The surface tension (T) of a liquid is defined as the force per unit length. For a drop of liquid, the surface tension acts along the circumference of the drop.
- The weight (W) of the drop can be balanced by the surface tension force.

2. **Calculating the Surface Tension Force:**
- The force due to surface tension can be expressed as:
\[ F = T \times L \]
where \(L\) is the length around the drop, which is the circumference of the drop.
- For a drop with radius \(R\), the circumference \(L\) is given by:
\[ L = 2\pi R \]
- Therefore, the force due to surface tension becomes:
\[ F = T \times 2\pi R \]

3. **Balancing the Forces:**
- The weight of the drop (W) is given by:
\[ W = m \cdot g \]
where \(m\) is the mass of the drop and \(g\) is the acceleration due to gravity.
- For small drops, we can assume that the weight is balanced by the surface tension force:
\[ T \times 2\pi R = W \]

4. **Weight Proportionality:**
- Since both drops have the same radius \(R\) and the same circumference factor \(2\pi\), we can state that the weight of the drops is directly proportional to their surface tension:
\[ W \propto T \]

5. **Finding the Ratio of Weights:**
- Let \(W_1\) be the weight of the first drop and \(W_2\) be the weight of the second drop. Then:
\[ \frac{W_1}{W_2} = \frac{T_1}{T_2} \]
- Given that the surface tension of the first liquid \(T_1 = 30 \, \text{dyne/cm}^2\) and for the second liquid \(T_2 = 60 \, \text{dyne/cm}^2\), we can substitute these values:
\[ \frac{W_1}{W_2} = \frac{30}{60} = \frac{1}{2} \]

6. **Conclusion:**
- Therefore, the ratio of the weights of the two drops is:
\[ W_1 : W_2 = 1 : 2 \]

### Final Answer:
The ratio of the weight of the two drops is \(1:2\).

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