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 of descend will be same
D. the speed and time of descend will be different

Answer

To solve the problem of a solid sphere rolling down two different inclined planes of the same height but with different angles of inclination, we can analyze the situation step by step.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have two inclined planes with the same height (h) but different angles of inclination (θ₁ and θ₂). We need to determine how the speed and time of descent of a solid sphere rolling down these planes compare.

2. **Speed of the Sphere**:
The speed (v) of a rolling solid sphere can be calculated using the formula:
\[ v = \sqrt{\frac{2gh}{1 + \frac{k^2}{r^2}}} \]
For a solid sphere, the radius of gyration (k) is constant, and the radius (r) is also constant. Therefore, the term \(\frac{k^2}{r^2}\) remains constant for both inclined planes.

3. **Height is Constant**:
Since both inclined planes have the same height (h), the speed of the sphere when it reaches the bottom of both planes will be the same. Thus, we can conclude:
\[ v_1 = v_2 \]
where \(v_1\) is the speed at the bottom of the first incline and \(v_2\) is the speed at the bottom of the second incline.

4. **Acceleration of the Sphere**:
The acceleration (A) of the rolling sphere down an incline is given by:
\[ A = \frac{g \sin \theta}{1 + \frac{k^2}{r^2}} \]
Here, the acceleration depends on the angle of inclination (θ). As the angle increases, \(\sin \theta\) increases, leading to a greater acceleration.

5. **Time of Descent**:
The time of descent (t) can be derived from the relationship:
\[ v = A \cdot t \]
Rearranging gives:
\[ t = \frac{v}{A} \]
Since the speed \(v\) is the same for both inclined planes, the time of descent will depend on the acceleration (A). As the angle of inclination increases, the acceleration increases, which leads to a decrease in the time of descent.

6. **Conclusion**:
- The speeds of the sphere at the bottom of both inclined planes are the same.
- The times of descent will be different due to the different accelerations caused by the different angles of inclination.

Thus, the correct option is:
**Option B**: The speed will be the same, but the time of descent will be different.