A one meter long string of mass `4.9 xx 10^(-4)` kg is held under a tension of 19.6 N. IF the string vibrates in one segment, then the frequency of vibration will be

Options

A. 200 Hz
B. 100 Hz
C. 50 Hz
D. 400 Hz

Answer

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

### Step 1: Identify the given values
- Length of the string (L) = 1 meter
- Mass of the string (m) = \(4.9 \times 10^{-4}\) kg
- Tension in the string (T) = 19.6 N

### Step 2: Calculate the mass per unit length (μ)
The mass per unit length (μ) is calculated using the formula:
\[ \mu = \frac{m}{L} \]
Substituting the values:
\[ \mu = \frac{4.9 \times 10^{-4} \text{ kg}}{1 \text{ m}} = 4.9 \times 10^{-4} \text{ kg/m} \]

### Step 3: Calculate the velocity of the wave (v)
The velocity of the wave on the string can be calculated using the formula:
\[ v = \sqrt{\frac{T}{\mu}} \]
Substituting the values:
\[ v = \sqrt{\frac{19.6 \text{ N}}{4.9 \times 10^{-4} \text{ kg/m}}} \]
Calculating the value:
\[ v = \sqrt{40000} = 200 \text{ m/s} \]

### Step 4: Determine the wavelength (λ)
Since the string vibrates in one segment, the wavelength (λ) can be expressed in terms of the length of the string:
\[ \lambda = 2L \]
Substituting the length of the string:
\[ \lambda = 2 \times 1 \text{ m} = 2 \text{ m} \]

### Step 5: Calculate the frequency (f)
The frequency of vibration can be calculated using the formula:
\[ f = \frac{v}{\lambda} \]
Substituting the values:
\[ f = \frac{200 \text{ m/s}}{2 \text{ m}} = 100 \text{ Hz} \]

### Final Answer
The frequency of vibration of the string is **100 Hz**.

---