A stiff wire is bent into a cylinder loop of diameter `D`. It is clamped by knife edges at two points opposite to each other . A transverse wave is sent around the loop by means resonance frequency (fundamental mode) of the loop in terms of wave speed `v` and diameter `D` is

Options

A. `(v)/(D)`
B. `( 2v)/( pi D)`
C. `(v)/( pi D)`
D. `(v)/( 2 pi D)`

Explanation

To find the resonance frequency (fundamental mode) of a stiff wire bent into a cylindrical loop of diameter \( D \), we can follow these steps:

### Step 1: Understand the Setup
The wire is bent into a circular loop and clamped at two opposite points. This creates a standing wave pattern in the loop.

### Step 2: Identify the Nodes and Antinodes
In the fundamental mode of vibration, there are two nodes at the clamped points and one antinode in between. The distance between the two nodes is half the wavelength (\( \lambda/2 \)).

### Step 3: Relate the Wavelength to the Geometry
The circumference of the loop can be expressed in terms of the diameter \( D \):
\[ C = \pi D \]
In the fundamental mode, the length of the loop corresponds to half a wavelength:
\[ C = \frac{\lambda}{2} \]
Thus, we can write:
\[ \pi D = \frac{\lambda}{2} \]
From this, we can solve for the wavelength:
\[ \lambda = 2\pi D \]

### Step 4: Use the Wave Speed Formula
The wave speed \( v \) is related to the frequency \( f \) and wavelength \( \lambda \) by the formula:
\[ v = f \lambda \]
Rearranging this gives us:
\[ f = \frac{v}{\lambda} \]

### Step 5: Substitute for Wavelength
Now substituting the expression for \( \lambda \) into the frequency formula:
\[ f = \frac{v}{2\pi D} \]

### Conclusion
Thus, the fundamental frequency of the loop in terms of wave speed \( v \) and diameter \( D \) is:
\[ f = \frac{v}{2\pi D} \]

### Final Answer
The resonance frequency (fundamental mode) of the loop is:
\[ f = \frac{v}{\pi D} \]

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