Considering a body of mass m, radius R rotating with angular speed `omega` about the centre of the mass and with a velocity v of centre of mass, the most appropriate definition of rolling motion will be

Options

A. v = R `omega`
B. the point of contact is stationary
C. the point of contact is stationary with respect to the surface
D. the body purely rotates about the point of contact

Answer

To solve the problem regarding the definition of rolling motion for a body of mass \( m \), radius \( R \), rotating with angular speed \( \omega \) about its center of mass and having a velocity \( v \) of its center of mass, we will follow these steps:

### Step-by-Step Solution:

1. **Understanding Rolling Motion**:
- Rolling motion occurs when a body rolls over a surface without slipping. For pure rolling motion, the point of contact between the body and the surface must have zero velocity relative to the surface.

2. **Identifying Key Parameters**:
- Let the radius of the body be \( R \).
- The angular speed of the body is \( \omega \).
- The translational velocity of the center of mass is \( v \).

3. **Velocity of the Point of Contact**:
- The velocity of the center of mass \( v \) is directed along the positive x-axis.
- The point of contact (let's call it point P) will have a velocity due to the rotation of the body. This velocity can be expressed as \( R\omega \) and will be directed opposite to the direction of the translational motion (i.e., in the negative x-direction).

4. **Condition for Pure Rolling Motion**:
- For pure rolling motion, the velocity of the point of contact with respect to the surface must be zero. Therefore, we can set up the equation:
\[ v - R\omega = 0 \]
- Rearranging gives us:
\[ v = R\omega \]

5. **Conclusion**:
- The condition \( v = R\omega \) indicates that the body is rolling without slipping, meaning the point of contact is stationary with respect to the surface. Thus, the most appropriate definition of rolling motion is that the point of contact is stationary with respect to the surface.

### Final Answer:
The most appropriate definition of rolling motion is that the point of contact is stationary with respect to the surface, which corresponds to the condition \( v = R\omega \).

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