A string is fixed at both ends. The tension in the string and density of the string are accurately known but the length and the radius of cross section of the string are known with some errorl If maximum errors made in the measurements of length and radius are `1%` and `0.5%` respectively them what is the maximum possible percentage error in the calculation of fundamental frequencyof the that string ?

Explanation

To solve the problem, we need to find the maximum possible percentage error in the calculation of the fundamental frequency of a string that is fixed at both ends. The fundamental frequency \( f_0 \) of the string is given by the formula:

\[ f_0 = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \]

where:
- \( L \) is the length of the string,
- \( T \) is the tension in the string,
- \( \mu \) is the mass per unit length of the string.

### Step 1: Determine the expression for mass per unit length \( \mu \)

The mass per unit length \( \mu \) can be expressed as:

\[ \mu = \frac{m}{L} = \rho \cdot V \]

where \( V \) is the volume of the string. The volume \( V \) of a cylindrical string is given by:

\[ V = \pi r^2 L \]

Thus, we can write:

\[ \mu = \rho \cdot \pi r^2 \]

### Step 2: Substitute \( \mu \) in the frequency formula

Substituting \( \mu \) back into the frequency formula, we get:

\[ f_0 = \frac{1}{2L} \sqrt{\frac{T}{\rho \cdot \pi r^2}} \]

### Step 3: Analyze the dependencies of \( f_0 \)

The frequency \( f_0 \) depends on \( L \) and \( r \). To find the percentage error in \( f_0 \), we need to consider how errors in \( L \) and \( r \) affect \( f_0 \).

### Step 4: Calculate the percentage error in \( f_0 \)

The percentage error in \( f_0 \) can be expressed in terms of the percentage errors in \( L \) and \( r \):

\[ \frac{\Delta f_0}{f_0} = \frac{\Delta L}{L} + 2 \frac{\Delta r}{r} \]

Where:
- \( \Delta L \) is the error in length,
- \( \Delta r \) is the error in radius.

### Step 5: Substitute the known errors

From the problem, we know:
- The maximum error in length \( \Delta L/L \) is \( 1\% \) or \( 0.01 \).
- The maximum error in radius \( \Delta r/r \) is \( 0.5\% \) or \( 0.005 \).

Substituting these values into the error formula:

\[ \frac{\Delta f_0}{f_0} = 0.01 + 2 \times 0.005 \]

Calculating this gives:

\[ \frac{\Delta f_0}{f_0} = 0.01 + 0.01 = 0.02 \]

### Step 6: Convert to percentage

To convert this to a percentage, we multiply by 100:

\[ \text{Maximum possible percentage error in } f_0 = 0.02 \times 100 = 2\% \]

### Final Answer

The maximum possible percentage error in the calculation of the fundamental frequency of the string is **2%**.

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