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) rolls down an inclined plane the fastest, 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 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 inclination, \( I \) is the moment of inertia, \( m \) is the mass, and \( r \) is the radius of the object.

2. **Moments of Inertia**:
We need to find the moment of inertia \( I \) for each object:
- **Ring**: \( I = m r^2 \)
- **Disc**: \( I = \frac{1}{2} m r^2 \)
- **Solid Sphere**: \( I = \frac{2}{5} m r^2 \)
- **Cylinder**: \( I = \frac{1}{2} m r^2 \)

3. **Calculating the Effective Acceleration**:
We can substitute the moments of inertia into the acceleration formula for each object:
- **For the Ring**:
\[ a_{ring} = \frac{g \sin \theta}{1 + \frac{m r^2}{m r^2}} = \frac{g \sin \theta}{2} \]
- **For the Disc**:
\[ a_{disc} = \frac{g \sin \theta}{1 + \frac{\frac{1}{2} m r^2}{m r^2}} = \frac{g \sin \theta}{1.5} = \frac{2g \sin \theta}{3} \]
- **For the Solid Sphere**:
\[ a_{sphere} = \frac{g \sin \theta}{1 + \frac{\frac{2}{5} m r^2}{m r^2}} = \frac{g \sin \theta}{1.4} = \frac{5g \sin \theta}{7} \]
- **For the Cylinder**:
\[ a_{cylinder} = \frac{g \sin \theta}{1 + \frac{\frac{1}{2} m r^2}{m r^2}} = \frac{g \sin \theta}{1.5} = \frac{2g \sin \theta}{3} \]

4. **Comparing Accelerations**:
Now we compare the accelerations:
- \( a_{ring} = \frac{g \sin \theta}{2} \)
- \( a_{disc} = \frac{2g \sin \theta}{3} \)
- \( a_{sphere} = \frac{5g \sin \theta}{7} \)
- \( a_{cylinder} = \frac{2g \sin \theta}{3} \)

To determine which is the largest, we can convert them to a common denominator or simply compare their coefficients:
- Ring: \( 0.5 \)
- Disc: \( 0.6667 \)
- Solid Sphere: \( 0.7143 \)
- Cylinder: \( 0.6667 \)

5. **Conclusion**:
The solid sphere has the highest acceleration, thus it will reach the bottom of the incline first.

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