A rod of nickel of length `l` is clamped at its midpoint . The rod is stuck and vibrations are set up in the rod . Find the general expression for the frequency of the longitudinal vibrations of the rod . Young's modulus and density of the rod is `Y and rho` , respectively.

Explanation

To find the general expression for the frequency of the longitudinal vibrations of a nickel rod clamped at its midpoint, we can follow these steps:

### Step 1: Understand the Setup
The rod has a length \( L \) and is clamped at its midpoint. This means that the rod can vibrate in modes where there are nodes at the midpoint.

### Step 2: Determine the Wavelength for the Fundamental Mode
For the fundamental mode of vibration in a rod clamped at its midpoint, the length of the rod \( L \) is equal to half the wavelength (\( \lambda_1 \)) of the fundamental frequency. Thus, we can write:
\[ L = \frac{\lambda_1}{2} \]
This implies:
\[ \lambda_1 = 2L \]

### Step 3: Relate Frequency, Wavelength, and Wave Speed
The speed of longitudinal waves in a rod is given by:
\[ v = \sqrt{\frac{Y}{\rho}} \]
where \( Y \) is the Young's modulus and \( \rho \) is the density of the rod.

The relationship between speed, frequency, and wavelength is given by:
\[ v = f \lambda \]
For the fundamental frequency \( f_1 \):
\[ v = f_1 \lambda_1 \]

### Step 4: Substitute for Wavelength
Substituting \( \lambda_1 = 2L \) into the wave speed equation gives:
\[ v = f_1 (2L) \]
Thus, we can express the frequency as:
\[ f_1 = \frac{v}{2L} \]

### Step 5: Substitute the Speed of Sound in the Rod
Now, substituting the expression for \( v \):
\[ f_1 = \frac{1}{2L} \sqrt{\frac{Y}{\rho}} \]

### Step 6: Generalize for Higher Modes
For the \( n \)-th mode of vibration, the relationship changes. The general expression for the frequency of the \( n \)-th mode can be given by:
\[ f_n = \frac{(2n-1)}{2L} \sqrt{\frac{Y}{\rho}} \]
where \( n \) is a positive integer (1, 2, 3, ...).

### Final Expression
Thus, the general expression for the frequency of the longitudinal vibrations of the rod is:
\[ f_n = \frac{(2n-1)}{2L} \sqrt{\frac{Y}{\rho}} \]