Four wires of identical lengths, diameters and materials are stretched on a sonometer box. The ratio of their tension `1:4:9:16`. The ratio of their fundamental frequencies is

Answer

To find the ratio of the fundamental frequencies of the four wires, we can follow these steps:

### Step 1: Understand the relationship between frequency and tension
The fundamental frequency \( f \) of a wire is given by the formula:
\[ f = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \]
where:
- \( L \) = length of the wire
- \( T \) = tension in the wire
- \( \mu \) = mass per unit length of the wire

Since the wires are of identical lengths, diameters, and materials, the mass per unit length \( \mu \) is the same for all wires.

### Step 2: Establish the relationship between frequency and tension
From the formula, we can see that the frequency is directly proportional to the square root of the tension:
\[ f \propto \sqrt{T} \]
This means that if we have different tensions, the frequencies will be in the ratio of the square roots of those tensions.

### Step 3: Write down the given tension ratios
According to the question, the tensions in the wires are given in the ratio:
\[ T_1 : T_2 : T_3 : T_4 = 1 : 4 : 9 : 16 \]

### Step 4: Calculate the square roots of the tensions
Now, we take the square root of each tension ratio:
\[ \sqrt{T_1} : \sqrt{T_2} : \sqrt{T_3} : \sqrt{T_4} = \sqrt{1} : \sqrt{4} : \sqrt{9} : \sqrt{16} \]
Calculating these values gives:
\[ 1 : 2 : 3 : 4 \]

### Step 5: Conclude the ratio of fundamental frequencies
Thus, the ratio of the fundamental frequencies of the wires is:
\[ f_1 : f_2 : f_3 : f_4 = 1 : 2 : 3 : 4 \]

### Final Answer:
The ratio of their fundamental frequencies is \( 1 : 2 : 3 : 4 \).

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