A sphere cannot roll on

Options

A. a smooth horizontal surface
B. a smooth inclined surface
C. a rough horizontal surface
D. a rough inclined surface

Answer

To solve the question "A sphere cannot roll on", we need to analyze the conditions under which a sphere can or cannot roll on different surfaces. Let's break it down step by step.

### Step-by-Step Solution:

1. **Identify the Surfaces**: The question refers to various surfaces: smooth horizontal, smooth inclined, rough horizontal, and rough inclined surfaces.

2. **Understand Pure Rolling**: Pure rolling occurs when there is no slipping between the surface and the sphere. For pure rolling, the relationship between the linear velocity (V) and the angular velocity (ω) of the sphere is given by:
\[ V = \omega r \]
where \( r \) is the radius of the sphere.

3. **Smooth Horizontal Surface**: On a smooth horizontal surface, if the sphere is given an initial velocity and angular velocity such that the above relationship holds, it can roll without slipping. Thus, it can roll on a smooth horizontal surface.

4. **Smooth Inclined Surface**: On a smooth inclined surface, the sphere experiences gravitational force acting down the slope. However, since the surface is smooth, there is no friction to provide the necessary torque for rolling. Therefore, the sphere will slide down without rolling. Thus, it cannot roll on a smooth inclined surface.

5. **Rough Horizontal Surface**: On a rough horizontal surface, friction is present. This friction allows the sphere to roll without slipping, as it can adjust to maintain the rolling condition. Therefore, the sphere can roll on a rough horizontal surface.

6. **Rough Inclined Surface**: Similarly, on a rough inclined surface, friction will also be present. This frictional force will enable the sphere to roll down the incline without slipping. Thus, it can roll on a rough inclined surface.

### Conclusion:
From the analysis, we conclude that the sphere cannot roll on a **smooth inclined surface**. Therefore, the answer to the question is that a sphere cannot roll on a smooth inclined surface.