In circular motion, which of the following statements are valid?

Options

A. The angular velocity of a particle in uniform circular motion relative to different points may be different
B. The linear velocity of a particle in uniform circular motion relative to different fixed points must be equal
C. `|(dvecv)/(dt)|gt(d|vecv|)/(dt)` for non – uniform circular motion
D. Uniform circular motion is periodic

Answer

To determine which statements regarding circular motion are valid, we will analyze each option step by step.

### Step-by-Step Solution:

1. **Understanding Angular Velocity in Circular Motion**:
- Angular velocity (ω) is defined as the rate of change of angular displacement with respect to time.
- In uniform circular motion, the angular velocity remains constant for a particle moving along a circular path.
- **Conclusion**: The angular velocity of a particle in uniform circular motion is the same for all points on the circular path. Thus, the first statement is **incorrect**.

2. **Analyzing Linear Velocity**:
- Linear velocity (v) is the tangential speed of the particle moving along the circular path.
- In circular motion, the direction of linear velocity changes continuously, even if the magnitude remains constant.
- However, the linear velocity is not the same at all points on the circular path because it depends on the radius and the position on the circle.
- **Conclusion**: Since the linear velocity varies in direction and can vary in magnitude (in non-uniform circular motion), the second statement is also **incorrect**.

3. **Examining the Equation for Acceleration**:
- The total acceleration of a particle in circular motion can be expressed as:
\[ \vec{a} = \vec{a}_t + \vec{a}_r \]
where \(\vec{a}_t\) is the tangential acceleration and \(\vec{a}_r\) is the radial (centripetal) acceleration.
- The statement provided suggests a relationship involving the derivative of velocity, which is indeed influenced by both types of acceleration.
- **Conclusion**: The third statement is **correct** as it accurately describes the relationship between the change in velocity and the total acceleration.

4. **Periodic Nature of Uniform Circular Motion**:
- Uniform circular motion is characterized by a constant speed along a circular path, and it is periodic, meaning it repeats its motion after a fixed time interval (the period).
- The period (T) can be calculated as:
\[ T = \frac{2\pi}{\omega} \]
- **Conclusion**: The fourth statement is **correct** as uniform circular motion is indeed periodic.

### Final Validity of Statements:
- **First Statement**: Incorrect
- **Second Statement**: Incorrect
- **Third Statement**: Correct
- **Fourth Statement**: Correct

### Summary:
The valid statements regarding circular motion are the third and fourth options.

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