A solid sphere, a ring and a disc all having same mass and radius are placed at the top of an incline and released. The friction coefficient between the objects and the incline are same but not sufficient to allow pure rolling. Least time will be taken in reaching the bottom by

Options

A. the solid sphere
B. the hollow sphere
C. the disc
D. all will take same time

Answer

To solve the problem, we need to analyze the motion of a solid sphere, a ring, and a disc as they slide down an incline under the influence of gravity and friction. Since the coefficient of friction is the same for all three objects and is not sufficient for pure rolling, we can derive the equations of motion for each object.

### Step-by-Step Solution:

1. **Identify the Forces Acting on the Objects**:
- Each object experiences gravitational force \( mg \) acting downwards.
- The normal force \( N \) acts perpendicular to the incline.
- Kinetic friction \( f_k \) acts opposite to the direction of motion, which is given by \( f_k = \mu_k N \).

2. **Resolve the Gravitational Force**:
- The gravitational force can be resolved into two components:
- Perpendicular to the incline: \( mg \cos \theta \)
- Parallel to the incline: \( mg \sin \theta \)

3. **Determine the Normal Force**:
- The normal force \( N \) is equal to the perpendicular component of the gravitational force:
\[ N = mg \cos \theta \]

4. **Calculate the Kinetic Friction**:
- The kinetic friction force is:
\[ f_k = \mu_k N = \mu_k (mg \cos \theta) \]

5. **Net Force Acting Along the Incline**:
- The net force \( F_{\text{net}} \) acting on each object along the incline is given by:
\[ F_{\text{net}} = mg \sin \theta - f_k = mg \sin \theta - \mu_k (mg \cos \theta) \]
- Factoring out \( mg \):
\[ F_{\text{net}} = mg \left( \sin \theta - \mu_k \cos \theta \right) \]

6. **Acceleration of Each Object**:
- Using Newton's second law, \( F = ma \), we can find the acceleration \( a \) of each object:
\[ ma = mg \left( \sin \theta - \mu_k \cos \theta \right) \]
- Dividing both sides by \( m \):
\[ a = g \left( \sin \theta - \mu_k \cos \theta \right) \]

7. **Conclusion**:
- Since the net force and hence the acceleration \( a \) is the same for all three objects (solid sphere, ring, and disc), they will all slide down the incline with the same acceleration.
- Therefore, all three objects will reach the bottom of the incline at the same time.

### Final Answer:
All three objects (solid sphere, ring, and disc) will take the same time to reach the bottom of the incline.