The length of a sonometer wire AB is 110 cm. Where should the two bridges be placed from A to divide the wire in 3 segments whose fundamental frequencies are in the ratio of ` 1: 2 : 3` ?

Options

A. 30 cm and 90 cm
B. 40 cm and 80 cm
C. 60 cm and 90 cm
D. 30 cm and 60 cm

Explanation

To solve the problem, we need to determine where to place the two bridges on a sonometer wire of length 110 cm so that the wire is divided into three segments with fundamental frequencies in the ratio of 1:2:3.

### 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 the frequencies are in the ratio 1:2:3, the lengths must be in the ratio:
\[ \frac{1}{L_1} : \frac{1}{L_2} : \frac{1}{L_3} = 1 : 2 : 3 \]
This implies:
\[ L_1 : L_2 : L_3 = 1 : \frac{1}{2} : \frac{1}{3} \]

2. **Finding a Common Denominator:**
To express the lengths in a simpler ratio, we can multiply each part of the ratio by a common multiple. The least common multiple of 1, 2, and 3 is 6. Thus, we can write:
\[ L_1 : L_2 : L_3 = 6 : 3 : 2 \]

3. **Calculating the Total Length:**
The total length of the wire is given as 110 cm. The sum of the parts of the ratio is:
\[ 6 + 3 + 2 = 11 \]

4. **Calculating Each Segment Length:**
Now, we can find the actual lengths of each segment:
- For \(L_1\):
\[ L_1 = \frac{6}{11} \times 110 = 60 \text{ cm} \]
- For \(L_2\):
\[ L_2 = \frac{3}{11} \times 110 = 30 \text{ cm} \]
- For \(L_3\):
\[ L_3 = \frac{2}{11} \times 110 = 20 \text{ cm} \]

5. **Determining the Positions of the Bridges:**
- The first bridge (to separate \(L_1\)) should be placed at 60 cm from point A.
- The second bridge (to separate \(L_2\)) should be placed at \(60 + 30 = 90\) cm from point A.

### Final Answer:
The two bridges should be placed at:
- First bridge at 60 cm
- Second bridge at 90 cm