At any instant, a rolling body may be considered to be in pure rotation about an axis through the point of contact. This axis is translating forward with speed

Options

A. equal to centre of mass
B. zero
C. twice of centre of mass
D. None of these

Explanation

To solve the problem regarding the rolling body and its instantaneous axis of rotation, we can break down the explanation into clear steps:

### Step-by-Step Solution:

1. **Understanding Pure Rolling Motion**:
- A rolling body is one that rolls without slipping. This means that the point of contact with the ground does not slide; it has zero velocity relative to the ground.

2. **Identifying the Instantaneous Axis of Rotation**:
- For a rolling object, at any instant, we can consider the axis of rotation to be at the point of contact with the ground. This is known as the instantaneous axis of rotation.

3. **Velocity of the Center of Mass**:
- The center of mass of the rolling body moves forward with a certain velocity (let’s denote it as \( v_{cm} \)). For pure rolling motion, the velocity of the point of contact (which is at rest relative to the ground) must equal zero.

4. **Condition for Pure Rolling**:
- The condition for pure rolling is that the velocity of the center of mass is equal to the product of the angular velocity (\( \omega \)) and the radius (\( r \)) of the body:
\[ v_{cm} = \omega \cdot r \]
- This means that the point of contact has a velocity of zero when the body rolls without slipping.

5. **Translating Axis of Rotation**:
- The axis of rotation is translating forward with the speed of the center of mass. Thus, if the body rolls forward, the instantaneous axis of rotation moves forward at the same speed as the center of mass.

6. **Conclusion**:
- Therefore, the condition for the body to be in pure rotation about the instantaneous axis through the point of contact is that the velocity of the center of mass must equal the velocity of the point of contact, which is zero at that instant.

### Final Answer:
- The correct option is that the velocity of the point of contact is equal to the velocity of the center of mass when the axis is translating forward with speed.

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