A hollow sphere and a solid sphere having same mass and same radii are rolled down a rough inclined plane.

Options

A. the hollow sphere reaches the bottom first
B. the solid sphere reaches the bottom with greater speed.
C. the soid sphere reaches the bottom with greater kinetic energy
D. the two spheres will reach the bottom wilth same linear momentum

Answer

To solve the problem of a hollow sphere and a solid sphere rolling down a rough inclined plane, we will analyze the forces acting on each sphere and derive their respective accelerations. We will then compare their speeds, kinetic energies, and linear momenta at the bottom of the incline.

### Step-by-Step Solution

1. **Identify Forces Acting on the Spheres**:
- Each sphere experiences gravitational force \( mg \) acting downwards.
- The gravitational force can be resolved into two components:
- \( mg \sin \theta \) (along the incline)
- \( mg \cos \theta \) (perpendicular to the incline)
- There is also a normal force \( N \) acting perpendicular to the surface and a frictional force \( f \) acting opposite to the direction of motion.

2. **Apply Newton's Second Law**:
- For the motion along the incline, we can write:
\[ mg \sin \theta - f = ma \]
- For rotational motion, the torque \( \tau \) about the center of mass due to friction is given by:
\[ \tau = f \cdot r = I \alpha \]
- Since \( \alpha = \frac{a}{r} \) (for rolling without slipping), we can substitute to get:
\[ f \cdot r = I \frac{a}{r} \]
- Rearranging gives:
\[ f = \frac{I a}{r^2} \]

3. **Substitute Frictional Force into the Linear Motion Equation**:
- Substituting \( f \) into the linear motion equation:
\[ mg \sin \theta - \frac{I a}{r^2} = ma \]
- Rearranging gives:
\[ mg \sin \theta = ma + \frac{I a}{r^2} \]
- Factoring out \( a \):
\[ mg \sin \theta = a \left( m + \frac{I}{r^2} \right) \]
- Thus, the acceleration \( a \) can be expressed as:
\[ a = \frac{mg \sin \theta}{m + \frac{I}{r^2}} \]

4. **Calculate Moment of Inertia for Each Sphere**:
- For the solid sphere:
\[ I_{\text{solid}} = \frac{2}{5} m r^2 \]
- For the hollow sphere:
\[ I_{\text{hollow}} = \frac{2}{3} m r^2 \]

5. **Calculate Acceleration for Each Sphere**:
- For the solid sphere:
\[ a_{\text{solid}} = \frac{mg \sin \theta}{m + \frac{2/5 m r^2}{r^2}} = \frac{mg \sin \theta}{m + \frac{2}{5}m} = \frac{5}{7} g \sin \theta \]
- For the hollow sphere:
\[ a_{\text{hollow}} = \frac{mg \sin \theta}{m + \frac{2/3 m r^2}{r^2}} = \frac{mg \sin \theta}{m + \frac{2}{3}m} = \frac{3}{5} g \sin \theta \]

6. **Compare Accelerations**:
- Since \( \frac{5}{7} g \sin \theta > \frac{3}{5} g \sin \theta \), the solid sphere has a greater acceleration than the hollow sphere.

7. **Determine Which Sphere Reaches the Bottom First**:
- The solid sphere reaches the bottom first due to its greater acceleration.

8. **Kinetic Energy at the Bottom**:
- Both spheres start with the same potential energy \( mgh \) and convert it entirely into kinetic energy at the bottom. Thus, they have the same kinetic energy.

9. **Linear Momentum at the Bottom**:
- The linear momentum \( p \) is given by \( p = mv \). Since the velocities of the two spheres differ, their linear momenta will also differ.

### Conclusion
- The solid sphere reaches the bottom first (Option 2 is correct).
- Both spheres have the same kinetic energy at the bottom (Option 3 is incorrect).
- Both spheres have different linear momenta (Option 4 is incorrect).
- The hollow sphere does not reach the bottom first (Option 1 is incorrect).