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

### Step 1: Determine the Length of the Wire
The wire is bent into a circular loop, so the length of the wire \( L \) is equal to the circumference of the circle. The circumference \( C \) of a circle is given by the formula:
\[ C = \pi D \]
Thus, the length of the wire \( L \) is:
\[ L = \pi D \]

### Step 2: Identify the Boundary Conditions
The wire is clamped at two points opposite each other. These points act as nodes for the standing wave. In the fundamental mode of resonance, there is one complete wave (one loop) between the two nodes.

### Step 3: Relate Wavelength to Length
In the fundamental mode, the length of the wire corresponds to half the wavelength \( \lambda \):
\[ L = \frac{\lambda}{2} \]
Substituting the expression for \( L \):
\[ \pi D = \frac{\lambda}{2} \]
From this, we can solve for the wavelength \( \lambda \):
\[ \lambda = 2\pi D \]

### Step 4: Use the Wave Speed Relation
The relationship between wave speed \( v \), frequency \( f \), and wavelength \( \lambda \) is given by:
\[ v = f \lambda \]
We can rearrange this to find the frequency:
\[ f = \frac{v}{\lambda} \]

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

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