A wire stretched between two rigid supports vibrates in its fundamental mode with a frequency of 45 Hz. The mass of the wire is `3.5xx10^(-2)kg` and its linear mass density is `4.0xx10^(-2)kgm^(-1)`. What is the speed of a transverse wave on the wire?

Options

A. `69 m s^(-1)`
B. `79 m s^(-1)`
C. `89 m s^(-1)`
D. `99 m s^(-1)`

Explanation

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

### Step-by-Step Solution:

1. **Identify Given Data:**
- Frequency of the wire (fundamental mode), \( f_0 = 45 \, \text{Hz} \)
- Mass of the wire, \( m = 3.5 \times 10^{-2} \, \text{kg} \)
- Linear mass density of the wire, \( \nu = 4.0 \times 10^{-2} \, \text{kg/m} \)

2. **Calculate the Length of the Wire:**
The linear mass density \( \nu \) is defined as:
\[ \nu = \frac{m}{L} \]
Rearranging this gives:
\[ L = \frac{m}{\nu} \]
Substituting the values:
\[ L = \frac{3.5 \times 10^{-2}}{4.0 \times 10^{-2}} = \frac{3.5}{4.0} = 0.875 \, \text{m} \]

3. **Calculate the Speed of the Wave:**
The speed \( v \) of a wave on a stretched string can be calculated using the formula:
\[ v = 2L f_0 \]
Substituting the values of \( L \) and \( f_0 \):
\[ v = 2 \times 0.875 \times 45 \]
Calculating this:
\[ v = 1.75 \times 45 = 78.75 \, \text{m/s} \]

4. **Round the Result:**
Rounding \( 78.75 \, \text{m/s} \) gives approximately \( 79 \, \text{m/s} \).

### Final Answer:
The speed of the transverse wave on the wire is approximately \( 79 \, \text{m/s} \).

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