A ball tied to the end of the string swings in a verticla plane in a vertical circle under the influence of gravity.

Options

A. When the string makes an angle `90^(@)` with the vertical, the tangential acceleration is zero and radial acceleration is somewhere between minimum and maximum
B. When the string makes an angle `90^(@)` with the vertical, the tangential acceleration is maximum and radial acceleration is somewhere between maximum and minimum
C. At no place in circular motion, tangential acceleration is equal to radial acceleration
D. When radial acceleration has its maximum value, the tangential acceleration is zero

Explanation

To solve the problem regarding a ball tied to a string swinging in a vertical circle, we will analyze the forces acting on the ball at various points in its motion and determine the truth of the given statements.

### Step-by-Step Solution:

1. **Understanding the Forces**:
- When the ball is at any point in the vertical circle, two main forces act on it: the tension (T) in the string and the gravitational force (mg) acting downward.

2. **Identifying Components of Forces**:
- At an angle θ from the vertical, the gravitational force can be resolved into two components:
- Radial (centripetal) component: \( mg \cos \theta \)
- Tangential component: \( mg \sin \theta \)

3. **Centripetal Acceleration**:
- The centripetal acceleration (a_c) required to keep the ball moving in a circle is given by:
\[ a_c = \frac{v^2}{R} \]
- At any point in the circle, the net force providing this centripetal acceleration is the difference between the tension and the radial component of the weight:
\[ T - mg \cos \theta = m \frac{v^2}{R} \]

4. **Tangential Acceleration**:
- The tangential acceleration (a_t) is given by:
\[ a_t = g \sin \theta \]
- This indicates that the tangential acceleration is maximum when θ = 90° (the lowest point in the swing).

5. **Analyzing Specific Angles**:
- **At θ = 0° (top of the swing)**:
- Tangential acceleration: \( a_t = g \sin 0 = 0 \)
- Centripetal acceleration: \( a_c = \frac{v^2}{R} \) (maximum)
- **At θ = 90° (lowest point)**:
- Tangential acceleration: \( a_t = g \sin 90 = g \) (maximum)
- Centripetal acceleration: \( a_c = \frac{T - mg}{R} \) (depends on tension)

6. **Evaluating the Statements**:
- **Statement 1**: When the string makes an angle of 90 degrees with the vertical, tangential acceleration is zero.
- **False**: Tangential acceleration is maximum at this point.
- **Statement 2**: When the string makes an angle of 90 degrees with the vertical, the tangential acceleration is maximum and radial acceleration is somewhere between maximum and minimum.
- **True**: This is correct.
- **Statement 3**: At no place in circular motion is tangential acceleration equal to radial acceleration.
- **False**: They can be equal at certain points, such as at the highest point if conditions allow.
- **Statement 4**: When radial acceleration has its maximum value, tangential acceleration is zero.
- **True**: This is correct, as seen at the top of the swing.

7. **Conclusion**:
- The true statements are: **Statement 2 and Statement 4**.