A sphere cannot roll on

Options

A. (a) a smooth inclined surface
B. (b) a smooth horizontal surface
C. (c) a rough inclined surface
D. (d) a rough horizontal surface

Explanation

To determine where a sphere cannot roll, we will analyze each of the given surfaces: smooth inclined surface, smooth horizontal surface, rough inclined surface, and rough horizontal surface.

### Step-by-Step Solution:

1. **Smooth Horizontal Surface:**
- When a sphere rolls on a smooth horizontal surface, the forces acting on it are the weight (downward) and the normal force (upward).
- There are no horizontal forces acting on the sphere, which means the linear acceleration in the horizontal direction is zero.
- Since both the normal force and weight act at the center of the sphere, the torque about the center is zero.
- Therefore, the angular acceleration is also zero, indicating that the sphere can roll with a uniform velocity.
- **Conclusion:** A sphere can roll on a smooth horizontal surface.

2. **Rough Horizontal Surface:**
- On a rough horizontal surface, friction acts on the sphere.
- This frictional force does not act at the center, creating a torque that allows the sphere to roll.
- The friction provides the necessary force for rolling motion.
- **Conclusion:** A sphere can roll on a rough horizontal surface.

3. **Rough Inclined Surface:**
- Similar to the rough horizontal surface, a rough inclined surface also has friction acting on the sphere.
- The friction will again create a torque about the center, allowing the sphere to roll down the incline.
- **Conclusion:** A sphere can roll on a rough inclined surface.

4. **Smooth Inclined Surface:**
- On a smooth inclined surface, there is no friction.
- The weight of the sphere can be resolved into two components: one parallel to the incline (W sin θ) and one perpendicular (W cos θ).
- The absence of friction means there is no torque acting on the sphere, so it cannot roll; it will slide down the incline instead.
- **Conclusion:** A sphere cannot roll on a smooth inclined surface.

### Final Answer:
A sphere cannot roll on a **smooth inclined surface**.