A ball of mass `m` is attached to a string whose other end is fixed. The system is free to rotate in vertical plane. When the ball swings, no work is done on the ball by the tension in the string. Which of the following statement is the correct explanation?

Options

A. Tension is always perpendicular to the direction of motion of the ball.
B. Tension is an internal force for the ball.
C. Tension is an external force for the ball.
D. Since the ball is swinging about the axis, total displacement is zero.

Explanation

To solve the question regarding why no work is done on the ball by the tension in the string when it swings in a vertical plane, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Forces**:
- The ball of mass `m` is attached to a string and swings in a vertical plane. The forces acting on the ball are tension (T) from the string and gravitational force (mg) acting downward.

2. **Direction of Tension and Displacement**:
- When the ball swings, it moves along a circular path. At any point in this path, the tension in the string acts radially inward (toward the center of the circular path), while the velocity of the ball is tangential to the circle.

3. **Perpendicular Forces**:
- Since the tension force is directed toward the center of the circle and the displacement (which is tangential to the circle) is perpendicular to the tension force, we can conclude that the angle between the tension force and the displacement is 90 degrees.

4. **Work Done Calculation**:
- The work done (W) by a force is given by the equation:
\[ W = F \cdot d \cdot \cos(\theta) \]
where \(F\) is the force, \(d\) is the displacement, and \(\theta\) is the angle between the force and displacement vectors.
- In this case, since \(\theta = 90^\circ\), we have:
\[ W = T \cdot d \cdot \cos(90^\circ) = T \cdot d \cdot 0 = 0 \]
- Therefore, the work done by the tension force on the ball is zero.

5. **Conclusion**:
- The correct explanation for why no work is done on the ball by the tension in the string is that the tension force is always perpendicular to the direction of motion of the ball.