A solid sphere, a hollow sphere and a disc, all having the same mass and radius, are placed at the top of an incline and released. The friction coefficients between the objects and the incline are same and not sufficient to allow pure rolling. The 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 the solid sphere, hollow sphere, and disc as they slide down the incline. Since the friction is not sufficient for pure rolling, all three objects will slide down the incline.

### Step-by-Step Solution:

1. **Identify Forces Acting on the Bodies:**
- The gravitational force \( mg \) acts downward.
- The normal force \( N \) acts perpendicular to the incline.
- The frictional force \( f \) acts opposite to the direction of motion (up the incline).

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

3. **Determine the Net Force Acting on Each Body:**
- Since the friction is not sufficient for rolling, the net force acting down the incline for all three bodies is:
\[ F_{\text{net}} = mg \sin \theta - f \]
- However, since they are sliding and not rolling, we can ignore the rotational inertia for the purpose of calculating acceleration.

4. **Calculate the Acceleration:**
- The acceleration \( a \) of each body can be derived from Newton's second law:
\[ F_{\text{net}} = ma \implies mg \sin \theta - f = ma \]
- Since the frictional force \( f \) does not change the net acceleration down the incline for all three bodies, we can conclude that:
\[ a = g \sin \theta \]
- This acceleration is the same for all three objects.

5. **Use Kinematic Equations to Find Time:**
- The distance \( L \) traveled down the incline can be related to time \( t \) using the kinematic equation:
\[ L = ut + \frac{1}{2} a t^2 \]
- Since the initial velocity \( u = 0 \):
\[ L = \frac{1}{2} a t^2 \implies t^2 = \frac{2L}{a} \implies t = \sqrt{\frac{2L}{a}} \]
- Substituting \( a = g \sin \theta \):
\[ t = \sqrt{\frac{2L}{g \sin \theta}} \]

6. **Conclusion:**
- Since the distance \( L \) and the acceleration \( g \sin \theta \) are the same for all three bodies, the time \( t \) taken to reach the bottom of the incline will also be the same for the solid sphere, hollow sphere, and disc.

### Final Answer:
All three bodies will take the same time to reach the bottom of the incline.