A circular hoop rolls without slipping on a flat horizontal surface. Which one of the following is necessarily true?

Options

A. All points on the rim of the hoop have the same speed.
B. All points on the rim of the hoop have the same velocity
C. Every point on the rim of the wheel has a different velocity
D. All of these

Explanation

To solve the problem, we need to analyze the motion of a circular hoop that rolls without slipping on a flat horizontal surface.

### Step-by-Step Solution:

1. **Understanding Rolling Without Slipping**:
- When a circular hoop rolls without slipping, it means that the point of contact with the ground does not slide. This condition implies a specific relationship between the translational and rotational motion of the hoop.

2. **Defining Motion Parameters**:
- Let the radius of the hoop be \( r \).
- Let the translational speed of the center of the hoop be \( v \).
- Let the angular velocity of the hoop be \( \omega \).

3. **Condition for Pure Rolling**:
- The condition for rolling without slipping can be expressed as:
\[ v = r \omega \]
- This means that the linear speed of the center of the hoop is equal to the tangential speed at the rim due to rotation.

4. **Analyzing Points on the Hoop**:
- Consider different points on the rim of the hoop:
- **Point C** (the point in contact with the ground):
- The tangential speed at this point due to rotation is \( r \omega \) (downward), and the translational speed is \( v \) (forward). Since \( v = r \omega \), the net speed at point C is zero. Hence, point C is momentarily at rest.
- **Point A** (the topmost point):
- The tangential speed is \( r \omega \) (upward), and the translational speed is \( v \) (forward). The net speed at point A is \( v + r \omega = 2v \).
- **Point B** (any point above the center):
- The net speed will be a combination of the translational and rotational speeds, leading to different magnitudes and directions.

5. **Conclusion on Speeds and Velocities**:
- Since point C has a speed of 0, point A has a speed of \( 2v \), and other points have speeds that vary between these two extremes, it is clear that:
- **All points on the rim do not have the same speed**.
- **All points on the rim do not have the same velocity** (as velocity includes direction).
- **Every point on the rim has a different velocity** due to the different magnitudes and directions of motion.

6. **Final Answer**:
- Therefore, the correct statement is that **every point on the rim of the wheel has a different velocity**. The correct answer is option 3.