A sonometer wire has a total length of 1m between the fixed ends. Where should the two bridges be placed below the wire so that the three segments of the wire have their fundamental frequencies in the ratio 1 : 2 : 3 ?

Explanation

To solve the problem of placing the two bridges on a sonometer wire of total length 1 meter so that the fundamental frequencies of the three segments are in the ratio 1:2:3, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Relationship Between Frequency and Length**:
The fundamental frequency (f) of a vibrating string is inversely proportional to its length (L). This means:
\[ f \propto \frac{1}{L} \]
Therefore, if we have three segments of lengths \(L_1\), \(L_2\), and \(L_3\), their frequencies will be in the ratio:
\[ \frac{f_1}{f_2} = \frac{L_2}{L_1}, \quad \frac{f_2}{f_3} = \frac{L_3}{L_2} \]

2. **Setting Up the Ratios**:
Given that the frequencies are in the ratio 1:2:3, we can express this as:
\[ f_1 : f_2 : f_3 = 1 : 2 : 3 \]
This implies:
\[ \frac{1}{L_1} : \frac{1}{L_2} : \frac{1}{L_3} = 1 : 2 : 3 \]

3. **Expressing Lengths in Terms of a Common Variable**:
From the frequency ratios, we can write:
\[ L_1 = k, \quad L_2 = \frac{k}{2}, \quad L_3 = \frac{k}{3} \]
where \(k\) is a common length factor.

4. **Finding the Total Length**:
The total length of the wire is given as 1 meter, so we can write:
\[ L_1 + L_2 + L_3 = 1 \]
Substituting the expressions for \(L_1\), \(L_2\), and \(L_3\):
\[ k + \frac{k}{2} + \frac{k}{3} = 1 \]

5. **Finding a Common Denominator**:
The common denominator for the fractions is 6. Rewriting the equation:
\[ \frac{6k}{6} + \frac{3k}{6} + \frac{2k}{6} = 1 \]
This simplifies to:
\[ \frac{11k}{6} = 1 \]

6. **Solving for k**:
Solving for \(k\):
\[ k = \frac{6}{11} \]

7. **Calculating the Lengths**:
Now we can find the lengths of the segments:
\[ L_1 = k = \frac{6}{11} \text{ m} \]
\[ L_2 = \frac{k}{2} = \frac{6}{11 \times 2} = \frac{3}{11} \text{ m} \]
\[ L_3 = \frac{k}{3} = \frac{6}{11 \times 3} = \frac{2}{11} \text{ m} \]

8. **Positioning the Bridges**:
To place the bridges, we need to measure the lengths from one end of the wire:
- The first bridge should be placed at \(L_1 = \frac{6}{11} \text{ m}\).
- The second bridge should be placed at \(L_1 + L_2 = \frac{6}{11} + \frac{3}{11} = \frac{9}{11} \text{ m}\).

### Final Answer:
- The first bridge should be placed at \(\frac{6}{11} \text{ m}\) from one end, and the second bridge should be placed at \(\frac{9}{11} \text{ m}\) from the same end.