In uniform circular motion

Options

A. both the angular velocity and the angular momentum vary
B. the velocity varies but the momentum remains constant
C. magnitude of both the velocity and the momentum stay constant
D. the momentum varies but the velocity remains constant

Answer

### Step-by-Step Solution:

1. **Understanding Uniform Circular Motion**:
- In uniform circular motion, an object moves in a circular path with a constant speed. This means that while the speed (magnitude of velocity) remains constant, the direction of the velocity vector changes continuously.

2. **Angular Velocity**:
- Angular velocity (ω) is defined as the rate of change of angular displacement (θ) with respect to time (t). Mathematically, it is given by:
\[ \omega = \frac{d\theta}{dt} \]
- In uniform circular motion, since the object moves at a constant speed, the angular displacement changes at a constant rate, which means angular velocity remains constant.

3. **Angular Momentum**:
- Angular momentum (L) for a particle moving in a circle is given by:
\[ L = \mathbf{r} \times \mathbf{p} = \mathbf{r} \times (m\mathbf{v}) \]
- Here, \( \mathbf{r} \) is the position vector from the center of the circle to the object, \( m \) is the mass, and \( \mathbf{v} \) is the velocity vector.
- In uniform circular motion, both the radius (r) and the speed (magnitude of v) are constant, which means that the angular momentum also remains constant in magnitude and direction.

4. **Evaluating the Options**:
- **Option 1**: "Both angular velocity and angular momentum vary."
- This is incorrect because both angular velocity and angular momentum remain constant in uniform circular motion.

- **Option 2**: "The velocities vary but the momentum remains constant."
- This is misleading. While the direction of the velocity vector changes, the magnitude (speed) remains constant. Therefore, the momentum (which depends on velocity) also remains constant in magnitude.

- **Option 3**: "The magnitude of both velocity and momentum stay constant."
- This is correct. The speed (magnitude of velocity) is constant, and since momentum is mass times velocity, the magnitude of momentum also remains constant.

- **Option 4**: "The momentum varies but the velocity remains constant."
- This is incorrect because the velocity's magnitude remains constant, and since momentum depends on velocity, the momentum does not vary in magnitude.

5. **Conclusion**:
- The correct answer is **Option 3**: "The magnitude of both velocity and momentum stay constant."