A wire is stretched between two rigid supports vibrates in its fundamental mode with a frequency of 50 Hz. The mass of the wire is 30 g and its linear density is `4 xx 10^(-2)` kg m `s^(-1)`. The speed of the transverse wave at the string is

Options

A. `25ms^(-1)`
B. `50ms^(-1)`
C. `75ms^(-1)`
D. `100ms^(-1)`

Explanation

To find the speed of the transverse wave on the wire, we can follow these steps:

### Step 1: Identify the given data
- Frequency (f) = 50 Hz
- Mass of the wire (m) = 30 g = 30 x 10^(-3) kg = 0.03 kg
- Linear density (µ) = 4 x 10^(-2) kg/m

### Step 2: Calculate the length of the wire
The linear density (µ) is defined as the mass per unit length:
\[ \mu = \frac{m}{L} \]
Rearranging this gives:
\[ L = \frac{m}{\mu} \]
Substituting the values:
\[ L = \frac{0.03 \text{ kg}}{4 \times 10^{-2} \text{ kg/m}} = \frac{0.03}{0.04} = 0.75 \text{ m} \]

### Step 3: Use the relationship between frequency, length, and wave speed
For a wire vibrating in its fundamental mode, the frequency (f) is related to the speed (v) and length (L) by the formula:
\[ f = \frac{1}{2L} \cdot v \]
Rearranging this gives:
\[ v = 2Lf \]

### Step 4: Substitute the values to find the speed
Now, substituting the values of L and f into the equation:
\[ v = 2 \times 0.75 \text{ m} \times 50 \text{ Hz} \]
Calculating this gives:
\[ v = 2 \times 0.75 \times 50 = 75 \text{ m/s} \]

### Conclusion
The speed of the transverse wave on the wire is **75 m/s**.

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