A particle is connected to a string of length l and rotated in a vertical circle, with the string's other end held fixed. To ensure the particle completes the circle, its minimum speed when the string is horizontal must be

Step-by-step solution

Let the point where the string is horizontal be B, and the highest point of the vertical circle be C. For the particle to just complete the vertical circle, the tension at the highest point C must be zero. The minimum speed at the highest point is given by: v_ C = gl Applying the principle of conservation of mechanical energy between point B (horizontal position) and point C (highest position): E_ B = E_ C 1 2 mv_ B ² = 1 2 mv_ C ² + mgh Here, the height difference between B and C is h = l . Substituting v_ C = gl