When different regular bodies roll down along an inclined plane from rest, then acceleration will be maximum for a body whose -

Options

A. radius of gyration is least
B. mass is least
C. surface area is maximum
D. moment of inertia is maximum

Answer

To determine which body will have the maximum acceleration when rolling down an inclined plane, we need to analyze the relationship between the moment of inertia and the acceleration of the center of mass.

### Step-by-Step Solution:

1. **Understanding Rolling Motion**:
- When a body rolls down an inclined plane, it undergoes both translational and rotational motion. The total acceleration of the center of mass depends on the forces acting on the body and its moment of inertia.

2. **Identify Forces**:
- The gravitational force acting on the body can be resolved into two components: one parallel to the incline (which causes acceleration) and one perpendicular to the incline (which is balanced by the normal force).

3. **Acceleration Formula**:
- The acceleration of the center of mass (a) for a rolling body can be expressed as:
\[ a = \frac{g \sin(\theta)}{1 + \frac{k^2}{r^2}} \]
where:
- \( g \) is the acceleration due to gravity,
- \( \theta \) is the angle of the incline,
- \( k \) is the radius of gyration,
- \( r \) is the radius of the body.

4. **Effect of Moment of Inertia**:
- The moment of inertia \( I \) is related to the radius of gyration \( k \) by the formula \( I = m k^2 \). A larger moment of inertia (higher \( k \)) means that the term \( \frac{k^2}{r^2} \) increases, which in turn decreases the overall acceleration \( a \).

5. **Maximizing Acceleration**:
- To maximize the acceleration \( a \), we need to minimize the term \( \frac{k^2}{r^2} \). This means we should choose a body with the smallest radius of gyration \( k \).

6. **Conclusion**:
- Therefore, the body with the smallest moment of inertia (or smallest \( k \)) will have the maximum acceleration when rolling down the inclined plane.

7. **Final Answer**:
- The body with the least radius of gyration \( k \) will have the maximum acceleration.