A stretched string of length ` 1 m` fixed at both ends , having a mass of `5 xx 10^(-4) kg` is under a tension of `20 N`. It is plucked at a point situated at `25 cm` from one end . The stretched string would vibrate with a frequency of

Options

A. `400 Hz`
B. `100 Hz`
C. `200 Hz`
D. `256 Hz`

Explanation

To find the frequency of the vibrating string, we can follow these steps:

### Step 1: Determine the wavelength (λ)
The string is fixed at both ends and is plucked at a point 25 cm from one end. This point acts as the first antinode, while the ends act as nodes. The distance from the fixed end to the first antinode is given as 25 cm (0.25 m).

Since this distance represents a quarter of the wavelength (λ/4), we can express this relationship as:
\[ \frac{\lambda}{4} = 0.25 \text{ m} \]
To find the wavelength (λ), we multiply both sides by 4:
\[ \lambda = 4 \times 0.25 = 1 \text{ m} \]

### Step 2: Calculate the linear mass density (μ)
The linear mass density (μ) is defined as the mass per unit length of the string. Given that the mass of the string is \(5 \times 10^{-4} \text{ kg}\) and its length is 1 m, we can calculate μ as follows:
\[ \mu = \frac{\text{mass}}{\text{length}} = \frac{5 \times 10^{-4} \text{ kg}}{1 \text{ m}} = 5 \times 10^{-4} \text{ kg/m} \]

### Step 3: Calculate the wave speed (v)
The wave speed (v) on a string under tension (T) can be calculated using the formula:
\[ v = \sqrt{\frac{T}{\mu}} \]
Given that the tension \(T = 20 \text{ N}\) and \(\mu = 5 \times 10^{-4} \text{ kg/m}\), we can substitute these values into the formula:
\[ v = \sqrt{\frac{20}{5 \times 10^{-4}}} \]
Calculating the right-hand side:
\[ v = \sqrt{40000} = 200 \text{ m/s} \]

### Step 4: Calculate the frequency (f)
The frequency (f) of the wave can be calculated using the relationship between wave speed, frequency, and wavelength:
\[ f = \frac{v}{\lambda} \]
Substituting the values we have:
\[ f = \frac{200 \text{ m/s}}{1 \text{ m}} = 200 \text{ Hz} \]

### Conclusion
The frequency of the vibrating string is \(200 \text{ Hz}\).