A solid sphere is roilling down on inclined plane from rest and rectangular block of same mass is also slipping down simultaneously from rest on the same plane. Then:

Options

A. both of them will reach the bottom simultaneously
B. the sphere will reach the bottom first
C. the rectangular block will reach the bottom first.
D. depends on density of material

Answer

To solve the problem of a solid sphere rolling down an inclined plane and a rectangular block slipping down simultaneously, we need to analyze the motion of both objects and compare their accelerations.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have two objects: a solid sphere and a rectangular block, both of the same mass, starting from rest at the same height on an inclined plane.

2. **Identify the Forces**:
- For the rectangular block, the only force acting along the incline is its weight component, which is \( mg \sin \theta \), where \( m \) is the mass of the block, \( g \) is the acceleration due to gravity, and \( \theta \) is the angle of the incline.
- For the solid sphere, it not only translates down the incline but also rotates.

3. **Acceleration of the Rectangular Block**:
- The acceleration \( a_b \) of the rectangular block can be derived from Newton's second law:
\[ a_b = g \sin \theta \]

4. **Acceleration of the Solid Sphere**:
- The acceleration \( a_s \) of the solid sphere can be calculated using the formula for rolling objects:
\[ a_s = \frac{g \sin \theta}{1 + \frac{I}{m r^2}} \]
- For a solid sphere, the moment of inertia \( I \) is \( \frac{2}{5} m r^2 \). Plugging this into the formula gives:
\[ a_s = \frac{g \sin \theta}{1 + \frac{2}{5}} = \frac{g \sin \theta}{\frac{7}{5}} = \frac{5}{7} g \sin \theta \]

5. **Comparing Accelerations**:
- Now we have:
- Acceleration of the block: \( a_b = g \sin \theta \)
- Acceleration of the sphere: \( a_s = \frac{5}{7} g \sin \theta \)
- Since \( g \sin \theta > \frac{5}{7} g \sin \theta \), we conclude:
\[ a_b > a_s \]

6. **Conclusion**:
- The rectangular block has a greater acceleration than the solid sphere, which means it will reach the bottom of the incline first.

### Final Answer:
The rectangular block will reach the bottom first.