In which situation a body can roll

Options

A. a smooth horizontal surface
B. a moving smooth horizontal surface
C. a smooth inclined surface
D. All of them

Answer

To determine the situations in which a body can roll, we need to analyze the conditions required for rolling motion. Rolling occurs when a body rotates about its axis while simultaneously translating along a surface without slipping. Here’s a step-by-step solution:

### Step 1: Understand the Conditions for Rolling
For a body to roll without slipping, the velocity of the point of contact with the surface must be zero. This means that the linear velocity \( v \) of the center of mass and the angular velocity \( \omega \) must satisfy the relationship:
\[ v = \omega r \]
where \( r \) is the radius of the body.

### Step 2: Analyze the First Situation - Smooth Horizontal Surface
In the first case, if a body is placed on a smooth horizontal surface and given an initial linear velocity \( v \) and angular velocity \( \omega \), we can check if it rolls:
- The point of contact \( P \) must have zero velocity relative to the surface.
- If \( v = \omega r \), then the body rolls without slipping.

**Conclusion for Situation 1:** A body can roll on a smooth horizontal surface if the condition \( v = \omega r \) is satisfied.

### Step 3: Analyze the Second Situation - Moving Smooth Horizontal Surface
In the second case, if the body is on a moving horizontal surface with velocity \( v' \):
- The body can still roll if we adjust its velocity such that the point of contact \( P \) has the same velocity as the moving surface.
- This means we need \( v - \omega r = v' \) or rearranging gives \( v = \omega r + v' \).

**Conclusion for Situation 2:** A body can roll on a moving smooth horizontal surface if the velocities are adjusted accordingly.

### Step 4: Analyze the Third Situation - Smooth Inclined Surface
In the third case, we consider a smooth inclined surface:
- The incline causes an acceleration due to gravity, specifically \( g \sin \theta \).
- The point of contact will also have an acceleration due to the incline, but since the incline is smooth, there is no friction to provide the necessary condition for rolling.
- The acceleration of the center of mass is not equal to the acceleration of the point of contact, which means the condition for rolling is not satisfied.

**Conclusion for Situation 3:** A body cannot roll on a smooth inclined surface because the accelerations of the center of mass and the point of contact do not match.

### Final Conclusion
From the analysis:
- A body can roll on a smooth horizontal surface (Situation 1).
- A body can roll on a moving smooth horizontal surface if velocities are adjusted (Situation 2).
- A body cannot roll on a smooth inclined surface (Situation 3).