To find the radius of the circular path of the ball, we need to analyze the forces acting on it. When an object is rotating in a circular path, it experiences centripetal force, which is provided by the tension in the string.
The centripetal force \( F_c \) required to keep the ball moving in a circular path is given by the equation:
\[ F_c = \frac{m v^2}{r} \]
Where:
- \( m \) is the mass of the ball,
- \( v \) is the tangential velocity of the ball,
- \( r \) is the radius of the circular path.
In this case, the tension in the string \( T \) provides the centripetal force. Thus, we can set the tension equal to the centripetal force:
\[ T = \frac{m v^2}{r} \]
To find \( r \), we can rearrange this equation:
\[ r = \frac{m v^2}{T} \]
This equation gives the radius of the circular path of the ball in terms of its mass \( m \), its velocity \( v \), and the tension \( T \) in the string.