When the tension of a sonometer is increased by 4.5 kg the pitch of the note emitted by a given length of the wire increases in the ratio 4: 5. Calculate the original tension in the wire ?

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the Variables
Let the original tension in the wire be \( T_1 = T \) (in kg). When an additional weight of 4.5 kg is added, the new tension becomes:
\[ T_2 = T + 4.5 \text{ kg} \]

### Step 2: Understand the Relationship Between Frequency and Tension
The problem states that the frequency of the note emitted by the wire changes in the ratio \( \frac{f_1}{f_2} = \frac{4}{5} \). We know that frequency is directly proportional to the square root of the tension in the wire:
\[ \frac{f_1}{f_2} = \sqrt{\frac{T_1}{T_2}} \]

### Step 3: Substitute the Values
Substituting the values of \( T_1 \) and \( T_2 \) into the frequency ratio gives:
\[ \frac{4}{5} = \sqrt{\frac{T}{T + 4.5}} \]

### Step 4: Square Both Sides
To eliminate the square root, we square both sides:
\[ \left(\frac{4}{5}\right)^2 = \frac{T}{T + 4.5} \]
This simplifies to:
\[ \frac{16}{25} = \frac{T}{T + 4.5} \]

### Step 5: Cross-Multiply
Cross-multiplying gives:
\[ 16(T + 4.5) = 25T \]

### Step 6: Expand and Rearrange
Expanding the left side:
\[ 16T + 72 = 25T \]
Rearranging the equation to isolate \( T \):
\[ 25T - 16T = 72 \]
This simplifies to:
\[ 9T = 72 \]

### Step 7: Solve for \( T \)
Dividing both sides by 9:
\[ T = \frac{72}{9} = 8 \text{ kg} \]

### Final Answer
The original tension in the wire is \( 8 \text{ kg} \).

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