A sonometer wire having a length of 1.50 m between the bridges vibrates in its second harmonic in resonance with a tuning fork of frequency 256 Hz. What is the speed of the transverse wave on the wire ?

Explanation

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

### Step 1: Understand the Harmonics
In the second harmonic, the wire vibrates in such a way that it forms two loops (or antinodes). The length of the wire is equal to one and a half wavelengths (λ) of the wave.

### Step 2: Relate Length to Wavelength
The relationship between the length of the wire (L) and the wavelength (λ) in the second harmonic can be expressed as:
\[ L = \frac{3}{2} \lambda \]
Given that the length of the wire (L) is 1.50 m, we can set up the equation:
\[ 1.50 = \frac{3}{2} \lambda \]

### Step 3: Solve for Wavelength
To find the wavelength (λ), we can rearrange the equation:
\[ \lambda = \frac{2}{3} \times 1.50 \]
Calculating this gives:
\[ \lambda = 1.00 \, \text{m} \]

### Step 4: Use the Wave Speed Formula
The speed (v) of a wave is given by the formula:
\[ v = f \cdot \lambda \]
where \( f \) is the frequency and \( \lambda \) is the wavelength. We know the frequency (f) is 256 Hz and we just calculated the wavelength (λ) to be 1.00 m.

### Step 5: Calculate the Speed
Now, substituting the values into the wave speed formula:
\[ v = 256 \, \text{Hz} \times 1.00 \, \text{m} \]
Calculating this gives:
\[ v = 256 \, \text{m/s} \]

### Final Answer
The speed of the transverse wave on the wire is **256 m/s**.

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