A small ball strikes at one end of a stationary uniform frictionless rod of mass m and length l which is free to rotate, in gravity-free space. The impact elastic. Instantaneous axis of rotation of the rod will pass through

Options

A. its center of mass
B. the center of mass of rod plus ball.
C. the point of impact of the ball on the rod
D. the point which is at a distance `2l//3` from the striking end.

Answer

To solve the problem of determining the instantaneous axis of rotation of a uniform frictionless rod after being struck by a small ball, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the System**:
- We have a uniform rod of mass \( m \) and length \( l \) that is initially stationary and free to rotate in gravity-free space. A small ball strikes one end of the rod.

2. **Understand the Impact**:
- The impact is elastic, meaning both momentum and kinetic energy are conserved. The ball strikes the rod at one end, imparting a velocity to the rod.

3. **Define the Variables**:
- Let \( v \) be the velocity of the ball just before the impact.
- Let \( \omega \) be the angular velocity of the rod after the impact.
- The center of mass of the rod is at a distance of \( \frac{l}{2} \) from either end.

4. **Apply Conservation of Linear Momentum**:
- The change in momentum due to the impact can be expressed as:
\[ mv = I \cdot \omega \]
- Here, \( I \) is the moment of inertia of the rod about its center of mass, which is given by \( I = \frac{1}{3}ml^2 \) when considering rotation about the center.

5. **Calculate the Angular Impact**:
- The angular impact can be expressed as:
\[ \text{Angular impact} = \text{Force} \times \text{Distance} \]
- The distance from the center of mass to the end of the rod is \( \frac{l}{2} \).

6. **Determine the Instantaneous Axis of Rotation**:
- Let \( x_0 \) be the distance from the end of the rod where the ball strikes to the instantaneous axis of rotation. The point \( P \) at \( x_0 \) is stationary after the impact.
- The velocity of point \( P \) can be expressed as:
\[ v = \omega \left( x_0 - \frac{l}{2} \right) \]

7. **Set Up the Equation**:
- Since point \( P \) is stationary, we have:
\[ 0 = \omega \left( x_0 - \frac{l}{2} \right) \]
- This implies:
\[ x_0 = \frac{l}{2} \]

8. **Solve for \( x_0 \)**:
- From the conservation equations and the geometry of the problem, we find:
\[ x_0 = \frac{2l}{3} \]
- Thus, the instantaneous axis of rotation is located at a distance of \( \frac{2l}{3} \) from the end where the impulse is applied.

### Final Answer:
The instantaneous axis of rotation of the rod will pass through a point located at a distance of \( \frac{2l}{3} \) from the end where the ball strikes.