A metal rod of length L is clamped at a distance L/4 from one end. It is set into longitudinal vibrations . The wavlength for fundamental mode of vibrations will be

Options

A. a. L/4
B. b. 3L/4
C. c. L
D. d. L/2

Answer

To solve the problem, we need to determine the wavelength for the fundamental mode of vibrations of a metal rod that is clamped at a distance of L/4 from one end. Here’s the step-by-step solution:

### Step 1: Understand the setup
The metal rod has a total length \( L \) and is clamped at a point that is \( \frac{L}{4} \) from one end. This means that the remaining length of the rod that can vibrate freely is \( L - \frac{L}{4} = \frac{3L}{4} \).

### Step 2: Identify the nodes and antinodes
In longitudinal vibrations, the clamped end of the rod acts as a node (where there is no displacement), and the free end acts as an antinode (where the maximum displacement occurs).

### Step 3: Determine the length of the vibrating section
Since the rod is clamped at \( \frac{L}{4} \), the effective length of the vibrating section is \( \frac{3L}{4} \).

### Step 4: Relate the length to the wavelength
For the fundamental mode of vibrations in a rod, the length of the vibrating section corresponds to half the wavelength (\( \lambda/2 \)). Therefore, we can set up the equation:
\[ \frac{3L}{4} = \frac{\lambda}{2} \]

### Step 5: Solve for the wavelength
To find the wavelength \( \lambda \), we can rearrange the equation:
\[ \lambda = 2 \times \frac{3L}{4} \]
\[ \lambda = \frac{3L}{2} \]

### Conclusion
Thus, the wavelength for the fundamental mode of vibrations in the metal rod is:
\[ \lambda = \frac{3L}{2} \]