The condition that a rigid body is rolling without slipping on an inclined plane is

Options

A. it has acceleration less than g.
B. it has rotational and translational K.E. to be equal
C. it has linear velocity equal to radius times angular velocity
D. the plane is fricitionless.

Answer

To determine the condition for a rigid body to roll without slipping on an inclined plane, we need to understand the relationship between translational motion, rotational motion, and friction.

### Step-by-Step Solution:

1. **Understanding Rolling Without Slipping**:
- Rolling without slipping means that the point of contact between the rigid body and the inclined plane does not slide. This implies that there is a relationship between the linear velocity of the center of mass of the body and its angular velocity.

2. **Condition for Rolling Without Slipping**:
- The condition for rolling without slipping is given by the equation:
\[ v = r \omega \]
where \( v \) is the linear velocity of the center of mass, \( r \) is the radius of the body, and \( \omega \) is the angular velocity.

3. **Role of Friction**:
- For a body to roll without slipping, there must be sufficient frictional force. If the inclined plane is frictionless (i.e., the coefficient of friction \( \mu = 0 \)), the body will not roll; it will slide down the plane. Thus, friction is necessary to provide the torque needed for rotational motion.

4. **Acceleration on the Inclined Plane**:
- When a rigid body rolls down an inclined plane, its acceleration is less than \( g \) (the acceleration due to gravity) because part of the gravitational force is converted into rotational motion. The net acceleration can be expressed as:
\[ a = \frac{g \sin(\theta)}{1 + \frac{I}{mr^2}} \]
where \( I \) is the moment of inertia of the body, \( m \) is its mass, and \( \theta \) is the angle of inclination.

5. **Conclusion**:
- Therefore, the condition for a rigid body to roll without slipping on an inclined plane is that there must be sufficient friction to prevent slipping, and the relationship \( v = r \omega \) must hold true.

### Final Answer:
The condition for a rigid body to roll without slipping on an inclined plane is that the linear velocity of the center of mass is equal to the product of its radius and angular velocity (\( v = r \omega \)), and there must be sufficient friction present.