The frequency (f ) of a wire oscillating with a the length l, in p loops, under a tension T is given by `f=(p)/(2l)sqrt((T)/(mu))` where `mu=` linear density of the wire, if the error made in determining length, tension and linear density be `1%, -2% and 4%` then the find the percentage error in the calculated frequency.

Options

A. `-4%`
B. `-2%`
C. `-1%`
D. `-5%`

Explanation

To find the percentage error in the calculated frequency \( f \) of a wire oscillating under tension, we start with the given formula:

\[ f = \frac{p}{2l} \sqrt{\frac{T}{\mu}} \]

where:
- \( p \) is the number of loops,
- \( l \) is the length of the wire,
- \( T \) is the tension,
- \( \mu \) is the linear density of the wire.

### Step 1: Identify the errors in measurements
We are given the following percentage errors:
- Error in length \( l \): \( \Delta l = 1\% \)
- Error in tension \( T \): \( \Delta T = -2\% \)
- Error in linear density \( \mu \): \( \Delta \mu = 4\% \)

### Step 2: Use logarithmic differentiation
Taking the logarithm of both sides of the frequency equation, we have:

\[ \log f = \log p - \log(2l) + \frac{1}{2} \log T - \frac{1}{2} \log \mu \]

### Step 3: Differentiate the equation
Differentiating both sides gives:

\[ \frac{\Delta f}{f} = \frac{\Delta p}{p} - \frac{\Delta l}{l} + \frac{1}{2} \frac{\Delta T}{T} - \frac{1}{2} \frac{\Delta \mu}{\mu} \]

### Step 4: Substitute the errors
Since \( p \) is a constant, \( \Delta p/p = 0 \). Now substituting the percentage errors:

\[ \frac{\Delta f}{f} = 0 - \frac{1}{100} + \frac{1}{2} \left(-\frac{2}{100}\right) - \frac{1}{2} \left(\frac{4}{100}\right) \]

### Step 5: Calculate the total error
Calculating each term:

- The error from length: \( -1\% \)
- The error from tension: \( \frac{1}{2} \times (-2\%) = -1\% \)
- The error from linear density: \( \frac{1}{2} \times 4\% = 2\% \)

Now, combining these errors:

\[ \frac{\Delta f}{f} = -1\% - 1\% + 2\% = 0\% \]

### Step 6: Convert to percentage
To find the percentage error in frequency:

\[ \text{Percentage error in } f = \Delta f/f \times 100 = 0\% \]

### Final Result
The percentage error in the calculated frequency is \( 0\% \).

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