Normally, if a body is attached to a string & is rotated in a vertical circle, then to loop the whole circle, the rope must not slack at the topmost point & there should be a velocity for having centripetal force. For this, the minimum initial velocity required is $\sqrt{5gr}$, $r$ being the radius of the vertical circle.
However there is another theory , which has baffled my sense & is quoted :
If a particle of mass $m$ is connected to a light rod & is whirled in a vertical circle of radius $r$, then to complete the circle, the minimum velocity of the particle at the bottommost point is not $\sqrt{5gr}$. This is because, the velocity of the particle can be zero also. Using conservation of mechanical energy, we get the minimum value as $2\sqrt{rg}$.
Ok, it is understandable that the condition that the rope must not slack is not required here. However, if the velocity at the topmost point is zero, there must be no centripetal force, which would make it rotate. So, isn't this theory wrong? Or, am I mistaking? If so, Where am I mistaken?
Best Answer
The topmost point would be in unstable equilibrium, so the speed at the top can be arbitrarily close to zero, and the mass will start to fall. At the top the potential energy is 2mgr (relative to the bottom of the circle). QED