If a ring, a disc, a solid sphere and a cyclinder of same radius roll down an inclined plane, the first one to reach the bottom will be:

Options

A. disc
B. ring
C. solid sphere
D. cylinder

Answer

To determine which object (a ring, a disc, a solid sphere, or a cylinder) will reach the bottom of an inclined plane first when rolling down, we need to analyze their accelerations based on their moments of inertia.

### Step-by-Step Solution:

1. **Understanding Rolling Motion**:
When an object rolls down an inclined plane, it experiences both translational and rotational motion. The acceleration of the object can be derived from the equation:
\[ a = \frac{g \sin \theta}{1 + \frac{k^2}{r^2}} \]
where \( g \) is the acceleration due to gravity, \( \theta \) is the angle of the incline, \( k \) is the radius of gyration, and \( r \) is the radius of the object.

2. **Finding the Radius of Gyration for Each Object**:
- **Disc**:
- Moment of Inertia \( I = \frac{1}{2} m r^2 \)
- Comparing with \( I = m k^2 \), we find:
\[ k^2 = \frac{r^2}{2} \]
- **Solid Sphere**:
- Moment of Inertia \( I = \frac{2}{5} m r^2 \)
- Comparing with \( I = m k^2 \), we find:
\[ k^2 = \frac{2}{5} r^2 \]
- **Ring**:
- Moment of Inertia \( I = m r^2 \)
- Comparing with \( I = m k^2 \), we find:
\[ k^2 = r^2 \]
- **Cylinder**:
- Moment of Inertia \( I = \frac{1}{2} m r^2 \)
- Comparing with \( I = m k^2 \), we find:
\[ k^2 = \frac{r^2}{2} \]

3. **Comparing the Values of \( k^2 \)**:
- Disc: \( k^2 = \frac{r^2}{2} \)
- Solid Sphere: \( k^2 = \frac{2}{5} r^2 \)
- Ring: \( k^2 = r^2 \)
- Cylinder: \( k^2 = \frac{r^2}{2} \)

From the above, we can see that the solid sphere has the smallest value of \( k^2 \).

4. **Determining the Acceleration**:
Since the acceleration \( a \) is inversely proportional to \( 1 + \frac{k^2}{r^2} \), the object with the smallest \( k^2 \) will have the highest acceleration.

- For the solid sphere:
\[ a = \frac{g \sin \theta}{1 + \frac{2/5}{1}} = \frac{g \sin \theta}{1 + 0.4} = \frac{g \sin \theta}{1.4} \]
- For the disc and cylinder, the acceleration will be lower than that of the solid sphere, and the ring will have the lowest acceleration.

5. **Conclusion**:
The solid sphere has the highest acceleration and will reach the bottom of the inclined plane first.

### Final Answer:
The first one to reach the bottom will be the **solid sphere**.

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