**Angular Velocity in a Rotating Rigid Body**

In a rotating rigid body, every point on that body has the same angular velocity at any instant of time. This means that all points on the body move with the same rate of rotation.

**Understanding Angular Velocity:**

Angular velocity is a vector quantity that describes the rate of change of angular displacement of an object with respect to time. It measures how quickly an object is rotating or spinning. Mathematically, angular velocity is given by:

Angular Velocity = Change in Angular Displacement / Time

**Rigid Body Rotation:**

A rigid body is an object in which the distance between any two points on the body remains constant. In other words, the shape and size of a rigid body do not change during motion. When a rigid body undergoes rotation, every point on the body moves in a circular path around a central axis.

**Uniform Angular Velocity:**

In a rotating rigid body, the angular velocity is constant and the same for all points on the body. This is known as uniform angular velocity. It means that every point on the body undergoes the same amount of angular displacement in the same amount of time.

**Reasoning:**

When a rigid body rotates, all points on the body travel along circular paths. Since these circular paths have the same radius (distance from the axis of rotation), the linear velocities of the points may vary (as they depend on the distance from the axis), but the angular velocities remain the same.

**Example:**

To illustrate this concept, imagine a rotating disc. Each point on the disc moves in a circular path around the center of the disc. Although the points at the outer edge of the disc travel a greater distance in the same time compared to the points closer to the center, they all complete one full revolution in the same amount of time. Hence, they all have the same angular velocity.

**Conclusion:**

In summary, every point in a rotating rigid body has the same angular velocity at any instant of time. While the linear velocities of the points may differ due to their distances from the axis of rotation, the rate of rotation (angular velocity) is constant and uniform for all points on the body.