Two solid spheres of different mass, radii and denisty roll down a rough inclined plane under indentical situation. Their time to come down is independent o their :-

Options

A. Mass
B. Radius
C. Density
D. All of these

Answer

To solve the problem, we need to analyze the motion of two solid spheres rolling down a rough inclined plane. The goal is to determine which factors the time taken to roll down the incline is independent of.

### Step-by-Step Solution:

1. **Identify the Parameters**:
- Let the mass of the first sphere be \( m_1 \), its radius \( r_1 \), and its density \( \rho_1 \).
- Let the mass of the second sphere be \( m_2 \), its radius \( r_2 \), and its density \( \rho_2 \).
- The angle of inclination of the plane is \( \theta \) and the height of the incline is \( h \).

2. **Determine the Acceleration**:
- The acceleration \( a \) of a rolling sphere down the incline can be expressed as:
\[ a = \frac{g \sin \theta}{1 + \frac{I}{m r^2}} \]
- Here, \( I \) is the moment of inertia of the sphere. For a solid sphere, \( I = \frac{2}{5} m r^2 \).

3. **Substituting the Moment of Inertia**:
- Substitute the moment of inertia into the acceleration formula:
\[ a = \frac{g \sin \theta}{1 + \frac{\frac{2}{5} m r^2}{m r^2}} = \frac{g \sin \theta}{1 + \frac{2}{5}} = \frac{g \sin \theta}{\frac{7}{5}} = \frac{5g \sin \theta}{7} \]

4. **Calculate the Distance**:
- The distance \( s \) that the sphere rolls down the incline can be related to the height \( h \) and the angle \( \theta \):
\[ s = \frac{h}{\sin \theta} \]

5. **Using Kinematic Equation to Find Time**:
- The time \( t \) taken to roll down the incline can be derived from the equation of motion:
\[ s = ut + \frac{1}{2} a t^2 \]
- Since the initial velocity \( u = 0 \), we simplify to:
\[ s = \frac{1}{2} a t^2 \implies t^2 = \frac{2s}{a} \implies t = \sqrt{\frac{2s}{a}} \]

6. **Substituting for Time**:
- Substitute \( s \) and \( a \) into the time equation:
\[ t = \sqrt{\frac{2 \cdot \frac{h}{\sin \theta}}{\frac{5g \sin \theta}{7}}} = \sqrt{\frac{14h}{5g \sin^2 \theta}} \]

7. **Independence of Mass, Radius, and Density**:
- Notice that the final expression for time \( t \) does not contain \( m \), \( r \), or \( \rho \). This indicates that the time taken to roll down the incline is independent of the mass, radius, and density of the spheres.

### Conclusion:
Therefore, the time taken for the spheres to roll down the inclined plane is independent of their mass, radius, and density.