A solid sphere rolls down two different inclined planes of the same height but of different inclinations

Options

A. the speed and time of descend will be same.
B. the speed will be same but time of descend will be different.
C. the speed will be different but time are descend will be same.
D. speed and time of descend both are different..

Explanation

To solve the problem of a solid sphere rolling down two different inclined planes of the same height but of different inclinations, we will analyze the motion of the sphere using the principles of energy conservation and rotational motion.

### Step-by-Step Solution:

1. **Understanding the System**:
We have a solid sphere of mass \( M \) and radius \( r \) rolling down two inclined planes from the same height \( h \). The sphere rolls without slipping.

2. **Energy Conservation**:
The potential energy (PE) at the top of the incline will convert into translational kinetic energy (TKE) and rotational kinetic energy (RKE) at the bottom of the incline.
\[ PE = TKE + RKE \]
The potential energy at the height \( h \) is given by:
\[ PE = Mgh \]

3. **Kinetic Energy Components**:
The translational kinetic energy when the sphere reaches the bottom is:
\[ TKE = \frac{1}{2} M v^2 \]
The rotational kinetic energy is given by:
\[ RKE = \frac{1}{2} I \omega^2 \]
For a solid sphere, the moment of inertia \( I \) about its center of mass is:
\[ I = \frac{2}{5} M r^2 \]
Since the sphere rolls without slipping, we have the relationship:
\[ v = r \omega \implies \omega = \frac{v}{r} \]
Substituting \( \omega \) into the RKE equation:
\[ RKE = \frac{1}{2} \left(\frac{2}{5} M r^2\right) \left(\frac{v}{r}\right)^2 = \frac{1}{5} M v^2 \]

4. **Total Kinetic Energy**:
The total kinetic energy at the bottom of the incline is:
\[ KE_{total} = TKE + RKE = \frac{1}{2} M v^2 + \frac{1}{5} M v^2 = \frac{7}{10} M v^2 \]

5. **Setting Up the Energy Equation**:
Setting the potential energy equal to the total kinetic energy gives:
\[ Mgh = \frac{7}{10} M v^2 \]
Simplifying this, we find:
\[ gh = \frac{7}{10} v^2 \implies v^2 = \frac{10gh}{7} \]

6. **Finding the Time of Descent**:
The time taken to roll down the incline can be derived from the kinematic equations, but since both inclines have the same height, the time will depend on the angle of inclination. The steeper the incline, the less time it will take to reach the bottom.

7. **Conclusion**:
The solid sphere will reach the bottom of the steeper incline first due to the greater component of gravitational force acting along the incline, leading to a higher acceleration.