One end of massless inextensible string of length `l` is fixed and other end is tied to a small ball of mass `m`. The ball is performing a circular motion in vertical plane. At the lowest position, speed of ball is `sqrt(20gl)`. Neglect any other forces on the ball except tension and gravitational force. Acceleration due to gravity is `g`.
Motion of ball is in nature of

Options

A. circular motion with constant speed
B. circular motion with variable speed
C. circular motion with constant angular acceleration about centre of the circle.
D. none of these

Explanation

To determine the nature of the motion of the ball tied to a massless inextensible string performing circular motion in a vertical plane, we can analyze the situation step by step.

### Step 1: Understanding the System
The ball is tied to a string of length \( l \) and is performing circular motion in a vertical plane. At the lowest point of the motion, the speed of the ball is given as \( v = \sqrt{20gl} \).

### Step 2: Analyzing Forces at the Lowest Point
At the lowest point of the circular path, two forces act on the ball:
1. The gravitational force \( mg \) acting downward.
2. The tension \( T \) in the string acting upward.

The net force acting towards the center of the circular path is the difference between the tension and the weight of the ball:
\[ F_{\text{net}} = T - mg \]

### Step 3: Applying Newton's Second Law
According to Newton's second law, the net force is also equal to the mass times the centripetal acceleration:
\[ F_{\text{net}} = m \cdot a_c \]
where \( a_c = \frac{v^2}{l} \) is the centripetal acceleration.

Setting the two expressions for the net force equal gives:
\[ T - mg = m \cdot \frac{v^2}{l} \]

### Step 4: Substituting the Given Speed
Substituting \( v = \sqrt{20gl} \) into the equation:
\[ T - mg = m \cdot \frac{(\sqrt{20gl})^2}{l} \]
\[ T - mg = m \cdot \frac{20gl}{l} \]
\[ T - mg = 20mg \]
\[ T = 21mg \]

### Step 5: Analyzing the Motion
Since the speed of the ball changes as it moves along the circular path (it is maximum at the lowest point and decreases as it rises), we can conclude that:
1. The speed of the ball is not constant; it varies as the ball moves through the vertical circle.
2. The ball is undergoing circular motion with variable speed.

### Step 6: Considering Angular Acceleration
The ball is also experiencing a change in direction, which implies that there is angular acceleration involved. However, since the tangential speed is changing, we can conclude that there is a non-zero tangential acceleration.

### Conclusion
The nature of the motion of the ball is:
- Circular motion with variable speed.
- Circular motion with constant angular acceleration about the center of the circle.