A string 1m long is drawn by a 300 Hz vibrator attached to its end. The string vibrates in three segments. The speed of transverse waves in the string is equal to

Options

A. `100 m//s`
B. `200 m//s`
C. `300 m//s`
D. `400 m//s`

Explanation

To find the speed of transverse waves in the string, we can follow these steps:

### Step 1: Understand the relationship between frequency, wavelength, and wave speed
The speed of a wave (V) on a string can be calculated using the formula:
\[ V = f \times \lambda \]
where \( f \) is the frequency and \( \lambda \) is the wavelength.

### Step 2: Identify the fundamental frequency
The fundamental frequency (first harmonic) of a string fixed at both ends is given by:
\[ f_0 = \frac{V}{2L} \]
where \( L \) is the length of the string.

### Step 3: Determine the frequency for the third harmonic
Since the string vibrates in three segments, we are dealing with the third harmonic. The frequency of the third harmonic (f3) is:
\[ f_3 = 3f_0 \]

### Step 4: Substitute the expression for \( f_0 \) into \( f_3 \)
Substituting \( f_0 \) into the equation for \( f_3 \):
\[ f_3 = 3 \left( \frac{V}{2L} \right) \]

### Step 5: Rearrange to find the speed of the wave
From the equation for \( f_3 \), we can rearrange it to solve for \( V \):
\[ V = \frac{2L f_3}{3} \]

### Step 6: Substitute the known values
Given that:
- Length of the string \( L = 1 \, \text{m} \)
- Frequency \( f_3 = 300 \, \text{Hz} \)

Substituting these values into the equation:
\[ V = \frac{2 \times 1 \, \text{m} \times 300 \, \text{Hz}}{3} \]

### Step 7: Calculate the speed
Calculating the above expression:
\[ V = \frac{600}{3} = 200 \, \text{m/s} \]

Thus, the speed of transverse waves in the string is:
\[ \boxed{200 \, \text{m/s}} \]

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