A rod of length `l=100cm` is fixed at `30cm` from both end. If velocity of transverse wave in rod is `v(cms^(-1))`, then chose the correct option(s)

Options

A. Fundamental frequencey (in Hz) of transverse wave in rod is `v/40`
B. Second overtone in rod will `5^(th)` harmonic
C. Frequency of third overtone (in Hz) is `7v//40`
D. Third overtone is`5^(th)` harmonic

Explanation

To solve the problem, we need to analyze the fixed rod and the transverse waves traveling through it. Here's a step-by-step breakdown of the solution:

### Step 1: Understand the Setup
The rod has a total length \( l = 100 \, \text{cm} \) and is fixed at \( 30 \, \text{cm} \) from both ends. This means that the effective length of the rod that can vibrate is:
\[ L = 100 \, \text{cm} - 30 \, \text{cm} - 30 \, \text{cm} = 40 \, \text{cm} \]

### Step 2: Identify the Modes of Vibration
The rod is fixed at both ends, which means it can support standing waves. The fundamental frequency (first harmonic) corresponds to one complete wave fitting in the length of the vibrating section of the rod.

### Step 3: Fundamental Frequency Calculation
For a rod fixed at both ends, the fundamental frequency \( f_1 \) is given by:
\[ f_1 = \frac{v}{\lambda_1} \]
where \( \lambda_1 \) is the wavelength of the fundamental mode. For the fundamental mode, the wavelength is twice the length of the vibrating section:
\[ \lambda_1 = 2L = 2 \times 40 \, \text{cm} = 80 \, \text{cm} \]
Thus, the fundamental frequency is:
\[ f_1 = \frac{v}{80} \]

### Step 4: Determine the Harmonics
The harmonics for a rod fixed at both ends are given by:
\[ f_n = n \cdot f_1 = n \cdot \frac{v}{80} \]
where \( n \) is the harmonic number (1, 2, 3, ...).

### Step 5: Identify Overtones
- The first overtone (second harmonic) corresponds to \( n = 2 \):
\[ f_2 = 2 \cdot \frac{v}{80} = \frac{v}{40} \]
- The second overtone (third harmonic) corresponds to \( n = 3 \):
\[ f_3 = 3 \cdot \frac{v}{80} \]
- The third overtone (fourth harmonic) corresponds to \( n = 4 \):
\[ f_4 = 4 \cdot \frac{v}{80} = \frac{v}{20} \]

### Step 6: Find the Specific Frequencies
- The second overtone is the fifth harmonic, which corresponds to \( n = 5 \):
\[ f_5 = 5 \cdot \frac{v}{80} = \frac{5v}{80} = \frac{v}{16} \]
- The third overtone is the sixth harmonic, which corresponds to \( n = 6 \):
\[ f_6 = 6 \cdot \frac{v}{80} = \frac{6v}{80} = \frac{3v}{40} \]

### Step 7: Final Answers
- The fundamental frequency is \( \frac{v}{80} \).
- The second overtone is the fifth harmonic.
- The frequency of the third overtone is \( \frac{3v}{40} \).

### Conclusion
The correct options based on the calculations are:
1. The fundamental frequency is \( \frac{v}{80} \).
2. The second overtone is the fifth harmonic.
3. The frequency of the third overtone is \( \frac{3v}{40} \).