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 need to analyze the acceleration of each object based on their moments of inertia.

### Step-by-Step Solution:

1. **Understand the Problem**:
We have four objects: a solid sphere, a solid disc, a hollow ring, and a hollow sphere (shell). They are rolled down an inclined plane without slipping. We need to find out which one reaches the bottom first.

2. **Identify the Moment of Inertia**:
- For a solid sphere: \( I = \frac{2}{5} m r^2 \)
- For a solid disc: \( I = \frac{1}{2} m r^2 \)
- For a hollow ring: \( I = m r^2 \)
- For a hollow sphere (shell): \( I = \frac{2}{3} m r^2 \)

3. **Use the Formula for Acceleration**:
The acceleration \( a \) of an object rolling down an incline is given by:
\[ a = \frac{g \sin \theta}{1 + \frac{I}{m r^2}} \]
where \( g \) is the acceleration due to gravity, \( \theta \) is the angle of the incline, \( I \) is the moment of inertia, and \( m \) is the mass.

4. **Calculate the Acceleration for Each Object**:
- **Solid Sphere**:
\[ a_s = \frac{g \sin \theta}{1 + \frac{2/5 m r^2}{m r^2}} = \frac{g \sin \theta}{1 + \frac{2}{5}} = \frac{g \sin \theta}{\frac{7}{5}} = \frac{5g \sin \theta}{7} \]
- **Solid Disc**:
\[ a_d = \frac{g \sin \theta}{1 + \frac{1/2 m r^2}{m r^2}} = \frac{g \sin \theta}{1 + \frac{1}{2}} = \frac{g \sin \theta}{\frac{3}{2}} = \frac{2g \sin \theta}{3} \]
- **Hollow Ring**:
\[ a_r = \frac{g \sin \theta}{1 + \frac{m r^2}{m r^2}} = \frac{g \sin \theta}{1 + 1} = \frac{g \sin \theta}{2} \]
- **Hollow Sphere**:
\[ a_h = \frac{g \sin \theta}{1 + \frac{2/3 m r^2}{m r^2}} = \frac{g \sin \theta}{1 + \frac{2}{3}} = \frac{g \sin \theta}{\frac{5}{3}} = \frac{3g \sin \theta}{5} \]

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

To find the order, we can compare the coefficients:
- Solid Sphere: \( \frac{5}{7} \approx 0.714 \)
- Solid Disc: \( \frac{2}{3} \approx 0.667 \)
- Hollow Sphere: \( \frac{3}{5} = 0.6 \)
- Hollow Ring: \( \frac{1}{2} = 0.5 \)

6. **Determine the Order**:
The object with the highest acceleration reaches the bottom first. Thus, the order from fastest to slowest is:
1. Solid Sphere
2. Solid Disc
3. Hollow Sphere
4. Hollow Ring

### Final Order:
- **Solid Sphere** > **Solid Disc** > **Hollow Sphere** > **Hollow Ring**