A rod of length l is given two velocities `v_(1)` and `v_(2)` in opposite directions at its two ends at right angles to the length. The distance of the instantaneous axis of rotation from `v_(1)` is

Options

A. zero
B. `(v_(1))/(v_(1)+v_(2))l`
C. `(v_(2)l)/(v_(1)+v_(2))`
D. `(l)/(2)`

Answer

To solve the problem of finding the distance of the instantaneous axis of rotation from \( v_1 \) for a rod of length \( l \ given two velocities \( v_1 \) and \( v_2 \) at its ends in opposite directions, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Setup**:
- We have a rod of length \( l \).
- The left end of the rod has a velocity \( v_1 \) and the right end has a velocity \( v_2 \).
- The velocities are directed perpendicular to the length of the rod and in opposite directions.

2. **Define the Instantaneous Axis of Rotation**:
- Let the distance from the point where \( v_1 \) is applied to the instantaneous axis of rotation be \( x \).
- Consequently, the distance from the point where \( v_2 \) is applied to the instantaneous axis of rotation will be \( l - x \).

3. **Relate Angular Velocities**:
- The angular velocity \( \omega \) about the instantaneous axis of rotation can be expressed for both ends of the rod:
- For the end with velocity \( v_1 \):
\[ \omega = \frac{v_1}{x} \]
- For the end with velocity \( v_2 \):
\[ \omega = \frac{v_2}{l - x} \]

4. **Set the Angular Velocities Equal**:
- Since both expressions represent the same angular velocity \( \omega \), we can set them equal to each other:
\[ \frac{v_1}{x} = \frac{v_2}{l - x} \]

5. **Cross-Multiply to Solve for \( x \)**:
- Cross-multiplying gives:
\[ v_1 (l - x) = v_2 x \]
- Expanding the left side:
\[ v_1 l - v_1 x = v_2 x \]

6. **Combine Like Terms**:
- Rearranging the equation:
\[ v_1 l = v_1 x + v_2 x \]
- Factoring out \( x \):
\[ v_1 l = x (v_1 + v_2) \]

7. **Solve for \( x \)**:
- Finally, solve for \( x \):
\[ x = \frac{v_1 l}{v_1 + v_2} \]

### Conclusion:
The distance of the instantaneous axis of rotation from \( v_1 \) is given by:
\[ x = \frac{v_1 l}{v_1 + v_2} \]