The speed of transverse waves in a stretched string is `700cm//s`. If the string is `2m` long, the frequency with which it resonates in fundamental mode is

Options

A. `(7//12)Hz`
B. `(7//4)Hz`
C. `14Hz`
D. `(2//7)Hz`

Explanation

To find the frequency with which a stretched string resonates in its fundamental mode, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Values:**
- Speed of transverse waves in the string (v) = 700 cm/s = 7 m/s (conversion from cm/s to m/s)
- Length of the string (L) = 2 m

2. **Understand the Fundamental Mode:**
- In the fundamental mode (first harmonic), the length of the string is equal to half the wavelength (λ) of the wave. Therefore, we can express this relationship as:
\[ L = \frac{\lambda}{2} \]
- From this, we can derive the wavelength:
\[ \lambda = 2L \]

3. **Substitute the Length:**
- Substitute the length of the string into the wavelength equation:
\[ \lambda = 2 \times 2 \text{ m} = 4 \text{ m} \]

4. **Relate Speed, Wavelength, and Frequency:**
- The relationship between speed (v), wavelength (λ), and frequency (f) is given by:
\[ v = f \cdot \lambda \]
- Rearranging this equation to solve for frequency gives:
\[ f = \frac{v}{\lambda} \]

5. **Substitute Known Values:**
- Now substitute the values of speed and wavelength into the frequency equation:
\[ f = \frac{7 \text{ m/s}}{4 \text{ m}} = \frac{7}{4} \text{ Hz} \]

6. **Final Answer:**
- The frequency with which the string resonates in the fundamental mode is:
\[ f = 1.75 \text{ Hz} \]