A ring, a solid cylinder,a hollow sphere and a solid sphere are released from rest on an inclined plane from same level.If there is no sliping then which one of these will reach the ground at last?

Options

A. Ring
B. Solid cylinder
C. Hollow sphere
D. Solid sphere

Answer

To solve the problem of which object will reach the ground last when released from rest on an inclined plane, we need to analyze the motion of each object based on their moment of inertia. The objects in question are a ring, a solid cylinder, a hollow sphere, and a solid sphere.

### Step-by-Step Solution:

1. **Understanding the Motion on an Inclined Plane**:
- When an object rolls down an inclined plane without slipping, its acceleration depends on its moment of inertia. The total acceleration can be expressed in terms of the gravitational force acting on it and its moment of inertia.

2. **Using the Formula for Acceleration**:
- The acceleration \( a \) of a rolling object can be given by:
\[ a = \frac{g \sin(\theta)}{1 + \frac{k}{r^2}} \]
where \( g \) is the acceleration due to gravity, \( \theta \) is the angle of the incline, \( k \) is the moment of inertia, and \( r \) is the radius of the object.

3. **Identifying the Moment of Inertia for Each Object**:
- For each object, we need to identify the value of \( k/r^2 \):
- **Ring**: \( k/r^2 = 1 \)
- **Solid Cylinder**: \( k/r^2 = \frac{1}{2} \)
- **Hollow Sphere**: \( k/r^2 = \frac{2}{3} \)
- **Solid Sphere**: \( k/r^2 = \frac{2}{5} \)

4. **Comparing the Values**:
- The larger the value of \( k/r^2 \), the smaller the acceleration \( a \) will be, meaning it will take longer to reach the bottom.
- The values we have are:
- Ring: \( 1 \)
- Solid Cylinder: \( \frac{1}{2} \)
- Hollow Sphere: \( \frac{2}{3} \)
- Solid Sphere: \( \frac{2}{5} \)

5. **Determining Which Object Takes the Longest**:
- The ring has the highest value of \( k/r^2 \) (which is 1), meaning it will have the lowest acceleration and therefore take the longest time to reach the bottom of the incline.

6. **Conclusion**:
- Thus, the object that will reach the ground last is the **ring**.

### Final Answer:
The ring will reach the ground last.