From an inclined plane a sphere, a disc, a ring and a shell are rolled without slipping. The order of their reaching at the base will be

Options

A. Ring, shell, disc, sphere
B. Shell, sphere, disc, ring
C. Sphere, disc, shell, ring
D. Ring, sphere, disc, shell

Answer

To determine the order in which a sphere, a disc, a ring, and a shell reach the base of an inclined plane when rolled without slipping, we will analyze their accelerations and the time taken to roll down the incline.

### Step-by-Step Solution:

1. **Identify the Bodies and Their Moments of Inertia**:
- Sphere (solid): \( I = \frac{2}{5} m r^2 \)
- Disc (solid): \( I = \frac{1}{2} m r^2 \)
- Ring (hollow): \( I = m r^2 \)
- Shell (hollow sphere): \( I = \frac{2}{3} m r^2 \)

2. **Determine the Acceleration**:
For a body rolling down an incline, the linear acceleration \( a \) can be derived from the forces acting on it. The equation for linear acceleration is given by:
\[ a = \frac{g \sin \theta}{1 + \frac{k^2}{r^2}} \]
where \( k^2/r^2 \) is derived from the moment of inertia.

3. **Calculate \( \frac{k^2}{r^2} \) for Each Body**:
- For the sphere: \( \frac{k^2}{r^2} = \frac{2}{5} \)
- For the disc: \( \frac{k^2}{r^2} = \frac{1}{2} \)
- For the ring: \( \frac{k^2}{r^2} = 1 \)
- For the shell: \( \frac{k^2}{r^2} = \frac{2}{3} \)

4. **Substitute into the Acceleration Formula**:
- Sphere:
\[ a_{\text{sphere}} = \frac{g \sin \theta}{1 + \frac{2}{5}} = \frac{g \sin \theta}{\frac{7}{5}} = \frac{5g \sin \theta}{7} \]
- Disc:
\[ a_{\text{disc}} = \frac{g \sin \theta}{1 + \frac{1}{2}} = \frac{g \sin \theta}{\frac{3}{2}} = \frac{2g \sin \theta}{3} \]
- Ring:
\[ a_{\text{ring}} = \frac{g \sin \theta}{1 + 1} = \frac{g \sin \theta}{2} \]
- Shell:
\[ a_{\text{shell}} = \frac{g \sin \theta}{1 + \frac{2}{3}} = \frac{g \sin \theta}{\frac{5}{3}} = \frac{3g \sin \theta}{5} \]

5. **Compare Accelerations**:
- Sphere: \( \frac{5g \sin \theta}{7} \)
- Disc: \( \frac{2g \sin \theta}{3} \)
- Ring: \( \frac{g \sin \theta}{2} \)
- Shell: \( \frac{3g \sin \theta}{5} \)

To determine the order, we need to compare these values. The greater the acceleration, the less time it takes to reach the bottom.

6. **Order of Accelerations**:
- The highest acceleration corresponds to the lowest moment of inertia ratio:
- Shell: \( \frac{3}{5} \) (highest acceleration)
- Sphere: \( \frac{5}{7} \)
- Disc: \( \frac{2}{3} \)
- Ring: \( 1 \) (lowest acceleration)

7. **Final Order of Reaching the Base**:
Therefore, the order in which they reach the base from fastest to slowest is:
1. Shell
2. Sphere
3. Disc
4. Ring

### Conclusion:
The order of the bodies reaching the base of the inclined plane will be: **Shell > Sphere > Disc > Ring**.