A rigid body is in pure rotation, that is, undergoing fixed axis rotation. Then which of the following statement(s) are true?

Options

A. a. . You can find two points in the body in a plane perpendicular to the axis of A. a. rotation having the same velocity.
B. b. You can find two points in the body in a plane perpendicular to the axis of rotation having the same acceleration.
C. c. Speed of all the particles lying on the curved surface of a cylinder whose axis coincides with the axis of rotation is the same.
D. d. Angular speed of the body is the same as seen from any point in the body.

Explanation

To determine which statements are true regarding a rigid body in pure rotation (fixed axis rotation), we can analyze the properties of such a system step by step.

### Step-by-Step Solution:

1. **Understanding Pure Rotation**: In pure rotation, all points in the rigid body move in circular paths around a fixed axis. The angular velocity (ω) is constant for all points in the body.

2. **Velocity of Points in the Body**: The linear velocity (v) of any point in the rigid body can be expressed as:
\[ v = \omega r \]
where \( r \) is the distance from the axis of rotation to the point. Since \( \omega \) is constant for all points, the velocity depends only on \( r \).

3. **Identifying Points with Same Velocity**: For two points in the body to have the same velocity, they must be at the same distance from the axis of rotation (same \( r \)). Thus, it is not possible to find two points at different distances from the axis that have the same velocity. Therefore, the statement about finding two points in a plane perpendicular to the axis with the same velocity is **false**.

4. **Identifying Points with Same Angular Speed**: The angular speed (ω) is the same for all points in the rigid body during pure rotation. Therefore, if the speed is the same, the angular speed must also be the same. This statement is **true**.

5. **Velocity on a Curved Surface**: For a rigid body rotating about a fixed axis, all particles lying on a curved surface (like the surface of a cylinder) will have the same angular velocity but different linear velocities depending on their distance from the axis. However, the magnitude of the linear velocity will be the same for points at the same radius. Thus, this statement is **true**.

### Conclusion:
After analyzing the properties of rigid body dynamics in pure rotation, we conclude that:
- Statement regarding finding two points in a plane with the same velocity is **false**.
- Statement regarding finding two points with the same angular speed when speeds are the same is **true**.
- Statement about the speed of all particles on a curved surface being the same is **true**.

Therefore, the correct answers are **C and D**.