The tension in a wire is decreased by `19%` The
percentage decrease in frequency will be

Options

A. `0.19%`
B. `10%`
C. `19%`
D. `0.9%`

Answer

To solve the problem of finding the percentage decrease in frequency when the tension in a wire is decreased by 19%, we can follow these steps:

### Step 1: Understand the relationship between tension and frequency
The frequency (F) of a wave in a wire is directly proportional to the square root of the tension (T) in the wire. This can be expressed mathematically as:
\[ F \propto \sqrt{T} \]
This means that if the tension changes, the frequency will also change according to this relationship.

### Step 2: Define initial and final tensions
Let the initial tension be \( T_1 \) and the final tension be \( T_2 \). If the initial tension is 100% (or simply T), then a decrease of 19% means:
\[ T_2 = T_1 - 0.19 T_1 = 0.81 T_1 \]

### Step 3: Relate the frequencies to the tensions
Using the relationship defined in Step 1, we can express the frequencies as:
\[ \frac{F_1}{F_2} = \sqrt{\frac{T_1}{T_2}} \]
Substituting \( T_2 \) from Step 2:
\[ \frac{F_1}{F_2} = \sqrt{\frac{T_1}{0.81 T_1}} = \sqrt{\frac{1}{0.81}} \]

### Step 4: Calculate the ratio of frequencies
Now, we can simplify the expression:
\[ \frac{F_1}{F_2} = \sqrt{\frac{1}{0.81}} = \frac{1}{\sqrt{0.81}} = \frac{1}{0.9} \]
This means:
\[ F_2 = 0.9 F_1 \]

### Step 5: Calculate the percentage decrease in frequency
The percentage decrease in frequency can be calculated using the formula:
\[ \text{Percentage decrease} = \left(1 - \frac{F_2}{F_1}\right) \times 100 \]
Substituting \( F_2 = 0.9 F_1 \):
\[ \text{Percentage decrease} = \left(1 - 0.9\right) \times 100 = 0.1 \times 100 = 10\% \]

### Conclusion
The percentage decrease in frequency when the tension in the wire is decreased by 19% is **10%**.

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