A ring (`R`), a disc (`D`), a solid sphere (`S`) and a hollow sphere with thin walls (`H`), all having the same mass but different radii, start together from rest at the top of inclined plane and roll down without slipping. Then

Options

A. all of them will reach the bottom of the incline together
B. the body with the maximum radius will reach bottom first
C. they will reach the bottom in the order `S, D, H` and `R`
D. all of them will have the same kinetic energy at bottom of the incline

Explanation

To solve the problem of determining which object reaches the bottom of the inclined plane first, we need to analyze the motion of each object (ring, disc, solid sphere, and hollow sphere) as they roll down the incline without slipping.

### Step-by-Step Solution:

1. **Understanding Rolling Motion**:
- When an object rolls down an incline without slipping, its acceleration can be expressed as:
\[ a = \frac{g \sin \theta}{1 + \frac{I}{M R^2}} \]
where \( g \) is the acceleration due to gravity, \( \theta \) is the angle of the incline, \( I \) is the moment of inertia of the object, \( M \) is the mass, and \( R \) is the radius.

2. **Moment of Inertia for Each Object**:
- For the ring: \( I_R = MR^2 \)
- For the disc: \( I_D = \frac{1}{2} MR^2 \)
- For the solid sphere: \( I_S = \frac{2}{5} MR^2 \)
- For the hollow sphere: \( I_H = \frac{2}{3} MR^2 \)

3. **Calculating Acceleration for Each Object**:
- **Ring**:
\[ a_R = \frac{g \sin \theta}{1 + 1} = \frac{g \sin \theta}{2} \]
- **Disc**:
\[ a_D = \frac{g \sin \theta}{1 + \frac{1}{2}} = \frac{g \sin \theta}{\frac{3}{2}} = \frac{2g \sin \theta}{3} \]
- **Solid Sphere**:
\[ a_S = \frac{g \sin \theta}{1 + \frac{2}{5}} = \frac{g \sin \theta}{\frac{7}{5}} = \frac{5g \sin \theta}{7} \]
- **Hollow Sphere**:
\[ a_H = \frac{g \sin \theta}{1 + \frac{2}{3}} = \frac{g \sin \theta}{\frac{5}{3}} = \frac{3g \sin \theta}{5} \]

4. **Comparing Accelerations**:
- The accelerations calculated are:
- \( a_R = \frac{g \sin \theta}{2} \)
- \( a_D = \frac{2g \sin \theta}{3} \)
- \( a_S = \frac{5g \sin \theta}{7} \)
- \( a_H = \frac{3g \sin \theta}{5} \)

- To determine which object has the highest acceleration, we can compare the coefficients:
- \( a_S > a_H > a_D > a_R \)

5. **Conclusion on Time Taken**:
- Since acceleration is inversely related to the time taken to reach the bottom (higher acceleration means shorter time), we conclude:
- The solid sphere will take the least time, followed by the hollow sphere, then the disc, and finally the ring will take the longest time.

### Final Order of Time Taken:
1. Solid Sphere (minimum time)
2. Hollow Sphere
3. Disc
4. Ring (maximum time)