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 use the formula that relates wave speed (v), frequency (f), and wavelength (λ):

\[ v = f \cdot \lambda \]

However, in the fundamental mode of vibration, the wavelength (λ) is related to the length (L) of the wire. For a wire fixed at both ends, the fundamental mode has a wavelength given by:

\[ \lambda = 2L \]

We will also need to find the length of the wire using its mass and linear density. The linear density (μ) is given by:

\[ \mu = \frac{m}{L} \]

Where:
- \( m \) is the mass of the wire (in kg)
- \( L \) is the length of the wire (in meters)

Given:
- Mass of the wire, \( m = 30 \, \text{g} = 0.03 \, \text{kg} \)
- Linear density, \( \mu = 4 \times 10^{-2} \, \text{kg/m} \)
- Frequency, \( f = 50 \, \text{Hz} \)

### Step 1: Calculate the length of the wire (L)

Using the formula for linear density:

\[ \mu = \frac{m}{L} \implies L = \frac{m}{\mu} \]

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 2: Calculate the wavelength (λ)

Using the relationship for the fundamental mode:

\[ \lambda = 2L = 2 \times 0.75 \, \text{m} = 1.5 \, \text{m} \]

### Step 3: Calculate the speed of the wave (v)

Now we can find the speed using the wave speed formula:

\[ v = f \cdot \lambda \]

Substituting the values:

\[ v = 50 \, \text{Hz} \cdot 1.5 \, \text{m} = 75 \, \text{m/s} \]

### Final Answer

The speed of the transverse wave in the wire is \( 75 \, \text{m/s} \).

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