To determine the validity of the two statements regarding circular motion, we will analyze each statement step by step.
### Step 1: Analyze Statement A
**Statement A:** In uniform circular motion, \( \vec{\omega} \), \( \vec{v} \), and \( \vec{a} \) are always mutually perpendicular.
1. **Definition of Uniform Circular Motion:** In uniform circular motion, an object moves in a circular path with a constant speed. Although the speed remains constant, the direction of the velocity vector changes continuously.
2. **Velocity Vector (\( \vec{v} \)):** The velocity vector is always tangent to the circular path.
3. **Centripetal Acceleration (\( \vec{a} \)):** In uniform circular motion, the only acceleration present is centripetal acceleration, which is directed towards the center of the circle. This means that the acceleration vector is perpendicular to the velocity vector.
4. **Angular Velocity Vector (\( \vec{\omega} \)):** The angular velocity vector points along the axis of rotation (using the right-hand rule). For motion in the x-y plane, \( \vec{\omega} \) points out of the plane (along the z-axis).
5. **Conclusion for Statement A:** Since both \( \vec{v} \) and \( \vec{a} \) are perpendicular to \( \vec{\omega} \) and \( \vec{a} \) is perpendicular to \( \vec{v} \), all three vectors \( \vec{\omega} \), \( \vec{v} \), and \( \vec{a} \) are mutually perpendicular. Therefore, Statement A is **True**.
### Step 2: Analyze Statement B
**Statement B:** In non-uniform circular motion, \( \vec{\omega} \), \( \vec{v} \), and \( \vec{a} \) are always mutually perpendicular.
1. **Definition of Non-Uniform Circular Motion:** In non-uniform circular motion, an object moves in a circular path but with a changing speed. This means that there are two components of acceleration: centripetal acceleration and tangential acceleration.
2. **Tangential Acceleration (\( \vec{a_t} \)):** This component is responsible for the change in the speed of the object and is directed along the tangent to the circular path.
3. **Centripetal Acceleration (\( \vec{a_c} \)):** This component is directed towards the center of the circular path.
4. **Net Acceleration (\( \vec{a} \)):** The net acceleration in non-uniform circular motion is the vector sum of the tangential and centripetal accelerations. Since these two components are not perpendicular to each other (the tangential acceleration is along the tangent while the centripetal is towards the center), the net acceleration vector will not be perpendicular to the velocity vector.
5. **Conclusion for Statement B:** Therefore, \( \vec{\omega} \), \( \vec{v} \), and \( \vec{a} \) are not mutually perpendicular in non-uniform circular motion. Thus, Statement B is **False**.
### Final Conclusion
- **Statement A:** True
- **Statement B:** False
