A sonometer wire has a length of 114 cm between its two fixed ends. Where should the two bridges be places so as to divide the wire into three segments, whose fundamental frequencies are in the ratio `1:3:4?`

Options

A. `l_1 = 72 cm, l_2 = 24 cm, l_3 = 18 cm `
B. `l_1 = 60 cm , l_2 = 40 cm, l_3 = 14 cm.`
C. `l_1 = 52 cm, l_2 = 30 cm , l_3 = 32 cm `.
D. `l_1 = 65 cm, l_2 = 30 cm, l_3 = 19 cm .`

Explanation

To solve the problem of where to place the two bridges on a sonometer wire of length 114 cm, dividing it into three segments with fundamental frequencies in the ratio of 1:3:4, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Relationship Between Frequency and Length:**
The fundamental frequency \( f \) of a vibrating wire is given by the formula:
\[ f = \frac{1}{2L} \cdot V \]
where \( L \) is the length of the wire segment and \( V \) is the velocity of the wave in the wire. Since frequency is inversely proportional to the length, we can express the relationship between the frequencies and lengths as:
\[ \frac{1}{L_1} : \frac{1}{L_2} : \frac{1}{L_3} = 1 : 3 : 4 \]

2. **Setting Up the Ratios:**
From the frequency ratio \( 1:3:4 \), we can write:
\[ L_1 : L_2 : L_3 = \frac{1}{1} : \frac{1}{3} : \frac{1}{4} \]
To express this in terms of lengths, we can find a common factor. The least common multiple of the denominators (1, 3, 4) is 12. Thus, we can express the lengths as:
\[ L_1 = 12x, \quad L_2 = 4x, \quad L_3 = 3x \]

3. **Setting Up the Total Length Equation:**
The total length of the wire is given as 114 cm. Therefore, we can set up the equation:
\[ L_1 + L_2 + L_3 = 114 \]
Substituting the expressions for \( L_1, L_2, \) and \( L_3 \):
\[ 12x + 4x + 3x = 114 \]

4. **Solving for \( x \):**
Combining the terms gives:
\[ 19x = 114 \]
Dividing both sides by 19:
\[ x = \frac{114}{19} = 6 \]

5. **Finding the Lengths of Each Segment:**
Now we can find the lengths of each segment:
\[ L_1 = 12x = 12 \times 6 = 72 \text{ cm} \]
\[ L_2 = 4x = 4 \times 6 = 24 \text{ cm} \]
\[ L_3 = 3x = 3 \times 6 = 18 \text{ cm} \]

6. **Determining the Positions of the Bridges:**
To find where to place the bridges, we start from one end of the wire:
- The first bridge is placed after \( L_1 = 72 \) cm.
- The second bridge is placed after \( L_1 + L_2 = 72 + 24 = 96 \) cm.

Thus, the bridges should be placed at 72 cm and 96 cm from one end of the wire.

### Final Answer:
The two bridges should be placed at 72 cm and 96 cm from one end of the sonometer wire.

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