Statement-1 : A thin uniform rod is undergoing fixed axis rotation about one its end with variable angular accleration. Then acceleration vector of any two moving point on the can not be parallel at an instant of time.
Statement-2: for a rod undergoing fixed axis rotation, the velocity of any two moving points on the rod at different distances from centre of rotation are different.

Options

A. Statement-1 is true, Statement-2: is true, Statement-2 is a correct explanation for Statement-1.
B. Statement-1 is true, Statement-2: is true, Statement-2 is NOT a correct explanation for Statement-1.
C. Statement-1 is true but statement-2 is false
D. Both statement 1 false and statement 2 are true.

Explanation

To solve the problem, we need to analyze the two statements provided.

### Step-by-Step Solution:

1. **Understanding Statement 1**:
- The statement claims that for a thin uniform rod undergoing fixed axis rotation about one of its ends with variable angular acceleration, the acceleration vectors of any two moving points cannot be parallel at an instant of time.
- In rotational motion, the total acceleration \( \vec{a} \) of a point on the rod can be expressed as:
\[ \vec{a} = \vec{a}_{tangential} + \vec{a}_{centripetal} \]
- The tangential acceleration \( \vec{a}_{tangential} \) is given by:
\[ \vec{a}_{tangential} = r \alpha \]
where \( r \) is the distance from the axis of rotation and \( \alpha \) is the angular acceleration.
- The centripetal acceleration \( \vec{a}_{centripetal} \) is given by:
\[ \vec{a}_{centripetal} = \frac{v^2}{r} = r \omega^2 \]
where \( \omega \) is the angular velocity.
- Since both \( \vec{a}_{tangential} \) and \( \vec{a}_{centripetal} \) depend on the distance \( r \) from the axis of rotation, the direction of the total acceleration vector will vary for different points on the rod. Therefore, the acceleration vectors of two points on the rod will not be parallel.

2. **Conclusion for Statement 1**:
- Thus, Statement 1 is **False**.

3. **Understanding Statement 2**:
- This statement asserts that for a rod undergoing fixed-axis rotation, the velocities of any two moving points on the rod at different distances from the center of rotation are different.
- The linear velocity \( v \) of a point on the rod is given by:
\[ v = r \omega \]
- Here, \( r \) is the distance from the axis of rotation. Since different points on the rod have different distances from the axis, their velocities will also differ.
- For example, if point A is at a distance \( r_1 \) and point B is at a distance \( r_2 \) from the axis of rotation, then:
\[ v_A = r_1 \omega \quad \text{and} \quad v_B = r_2 \omega \]
- Since \( r_1 \neq r_2 \), it follows that \( v_A \neq v_B \).

4. **Conclusion for Statement 2**:
- Thus, Statement 2 is **True**.

### Final Answer:
- Statement 1 is False, and Statement 2 is True. Therefore, the correct answer is option D.

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