On an inclined plane, two spheres of similar size, one solid while the other hollow, start rolling from the position of rest. The time taken by the hollow sphere, in order to cover the same distance, in comparison with solid one, will be:

Options

A. more
B. less
C. equal
D. data is incomplete

Answer

To solve the problem of comparing the time taken by a solid sphere and a hollow sphere to roll down an inclined plane, we can follow these steps:

### Step 1: Understand the Problem
We have two spheres: one solid and one hollow, both of the same size, rolling down an inclined plane from rest. We need to determine how the time taken by the hollow sphere compares to that of the solid sphere when they cover the same distance.

### Step 2: Identify the Forces Acting on the Spheres
When the spheres roll down the incline, the gravitational force acting on them can be resolved into two components:
- A component acting parallel to the incline: \( F_{\parallel} = mg \sin(\theta) \)
- A component acting perpendicular to the incline: \( F_{\perpendicular} = mg \cos(\theta) \)

### Step 3: Calculate the Acceleration of Each Sphere
The acceleration of each sphere can be derived from Newton's second law. The net force acting on the sphere along the incline is equal to the mass times its acceleration. For rolling objects, we also need to account for rotational inertia.

The equation for the acceleration \( a \) of the center of mass of a rolling sphere is given by:
\[ a = \frac{F_{\parallel}}{m + \frac{I}{R^2}} \]
where \( I \) is the moment of inertia and \( R \) is the radius of the sphere.

- For a solid sphere, \( I = \frac{2}{5} m R^2 \)
- For a hollow sphere, \( I = \frac{2}{3} m R^2 \)

### Step 4: Substitute the Moments of Inertia
1. **Solid Sphere:**
\[ a_{\text{solid}} = \frac{mg \sin(\theta)}{m + \frac{2}{5} m} = \frac{5g \sin(\theta)}{7} \]

2. **Hollow Sphere:**
\[ a_{\text{hollow}} = \frac{mg \sin(\theta)}{m + \frac{2}{3} m} = \frac{3g \sin(\theta)}{5} \]

### Step 5: Compare the Accelerations
From the above calculations, we can see that:
- \( a_{\text{solid}} = \frac{5g \sin(\theta)}{7} \)
- \( a_{\text{hollow}} = \frac{3g \sin(\theta)}{5} \)

To compare these accelerations, we can see that:
\[ \frac{5g \sin(\theta)}{7} > \frac{3g \sin(\theta)}{5} \]
This indicates that the solid sphere has a greater acceleration than the hollow sphere.

### Step 6: Determine the Time Taken
Since both spheres start from rest and roll down the same distance \( s \), we can use the kinematic equation:
\[ s = ut + \frac{1}{2} a t^2 \]
Given that \( u = 0 \):
\[ s = \frac{1}{2} a t^2 \implies t = \sqrt{\frac{2s}{a}} \]
Since the solid sphere has a greater acceleration, it will take less time to cover the same distance compared to the hollow sphere.

### Conclusion
Thus, the time taken by the hollow sphere to cover the same distance will be greater than that taken by the solid sphere.

**Final Answer:** The hollow sphere takes more time than the solid sphere.

---