A ball moves on a frictionless inclined table without slipping. The work done by the surface on the ball is :

Options

A. Positive
B. Negative
C. Zero
D. None of these

Answer

To solve the problem of determining the work done by the surface on a ball moving on a frictionless inclined table without slipping, we can follow these steps:

### Step 1: Understand the Forces Acting on the Ball
The ball experiences two main forces while on the inclined table:
- The gravitational force acting downward (weight of the ball, \( mg \)).
- The normal force exerted by the surface of the inclined table, which acts perpendicular to the surface.

### Step 2: Analyze the Motion
Since the table is frictionless, there is no frictional force acting on the ball. The ball rolls without slipping, which means that the point of contact between the ball and the surface does not slide.

### Step 3: Determine the Work Done by the Normal Force
The work done by a force is given by the equation:
\[ W = \int F \cdot dx \]
Where \( F \) is the force and \( dx \) is the displacement. However, in this case, the normal force acts perpendicular to the displacement of the ball along the inclined surface.

### Step 4: Calculate the Dot Product
Since the normal force is perpendicular to the displacement of the ball, the angle \( \theta \) between the normal force and the displacement vector is 90 degrees. Therefore, the work done by the normal force can be expressed as:
\[ W = F \cdot d \cdot \cos(\theta) \]
Where \( \theta = 90^\circ \). The cosine of 90 degrees is zero:
\[ W = F \cdot d \cdot \cos(90^\circ) = F \cdot d \cdot 0 = 0 \]

### Step 5: Conclusion
Thus, the work done by the surface on the ball is zero:
\[ \text{Work done by the surface on the ball} = 0 \]