Show that the minimum velocity required for a particle to 'loop a loop' while going in a vertical circle of radius r is `sqrt(5gr)`.

Statement I: If a particle of mass m is connected to a light rod and whirled in a vertical circle of radius R, then to complete the circle, the minimum velocity of the particle at the lowest point is sqrt(5gR) . Statement II: Mechanical energy is conserved and for the minimum velocity at the lowest point, the velocity at the highest point will be zero.

Dimensionally prove the expression (mv^(2))/( r) for the centripetal force F acting on a particle of mass m moving with velocity v in a circle of radius r.

Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is 6sqrt3r .

What is the minimum velcoity with which a body of mass m must enter a vertical loop of radius R so that it can complete the loop ?