question_answer

A mass is tied to a string and rotated in a vertical circle. The minimum velocity of the body at the top is

A) \[\sqrt{gr}\]       

B)        \[g/r\]

C) \[{{\left( \frac{g}{r} \right)}^{3/2}}\]     

D)        \[gr\]

Correct Answer: A

Solution :

[a] Let velocity at A \[={{v}_{A}}\] and velocity at B\[={{v}_{B}}\] Applying conservation of energy at A & B \[\frac{1}{2}mv{}_{A}^{2}+2gmr=\frac{1}{2}mv{}_{B}^{2}\] \[v{}_{B}^{2}=v{}_{A}^{2}+4gr\].........(i) Now as it is moving in circular path it has centripetal force. At point A \[\Rightarrow \]\[T+mg=\frac{mv{}_{A}^{2}}{r}\] for minimum velocity \[T\ge 0\] or \[\frac{mv{}_{A}^{2}}{r}\ge mg\]\[\Rightarrow \]\[v_{A}^{1}\ge gr\]\[\Rightarrow \]\[{{v}_{A}}\ge \sqrt{gr}\]