The length of a sonometer wire AB is 100 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 1:2:6

Options

A. 30 cm, 90 cm
B. 60cm, 90 cm
C. 40 cm, 80 cm
D. 20 cm, 30 cm

Answer

To solve the problem of placing the two bridges on a sonometer wire of length 100 cm such that the fundamental frequencies of the segments are in the ratio of 1:2:6, we can follow these steps:

### Step 1: Understand the relationship between frequency and length
The fundamental frequency \( f \) of a sonometer wire is inversely proportional to its length \( L \). This means:
\[ f \propto \frac{1}{L} \]
If the frequencies are in the ratio \( f_1 : f_2 : f_3 = 1 : 2 : 6 \), we can express the lengths corresponding to these frequencies as:
\[ \frac{1}{L_1} : \frac{1}{L_2} : \frac{1}{L_3} = 1 : 2 : 6 \]
This implies:
\[ L_1 : L_2 : L_3 = 6 : 3 : 1 \]

### Step 2: Calculate the total length ratio
The total of the parts in the ratio \( 6 : 3 : 1 \) is:
\[ 6 + 3 + 1 = 10 \]
Thus, we can express each length in terms of the total length of the wire (100 cm).

### Step 3: Calculate individual lengths
Now we can calculate the lengths \( L_1 \), \( L_2 \), and \( L_3 \):
- For \( L_1 \):
\[ L_1 = \frac{6}{10} \times 100 = 60 \text{ cm} \]
- For \( L_2 \):
\[ L_2 = \frac{3}{10} \times 100 = 30 \text{ cm} \]
- For \( L_3 \):
\[ L_3 = \frac{1}{10} \times 100 = 10 \text{ cm} \]

### Step 4: Determine the positions of the bridges
The first bridge should be placed at the end of \( L_1 \) from point A:
- Position of the first bridge from A:
\[ \text{Position of Bridge 1} = L_1 = 60 \text{ cm} \]
The second bridge should be placed at the end of \( L_1 + L_2 \):
- Position of the second bridge from A:
\[ \text{Position of Bridge 2} = L_1 + L_2 = 60 + 30 = 90 \text{ cm} \]

### Conclusion
The two bridges should be placed at 60 cm and 90 cm from point A.

### Final Answer
The positions of the two bridges are:
- First bridge: 60 cm from A
- Second bridge: 90 cm from A

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