Question

A mass is tied to a string and rotated in a vertical circle. What should be the minimum velocity of the body at the top such that the string remains taut?

• A. $\sqrt{gr}$
• B. $g/r$
• C. ${\left(\frac{g}{r}\right)}^{3/2}$
• D. $gr$

Answer 1

The correct option is A $\sqrt{gr}$

Let velocity at $A=v_A$ and velocity at $B=v_B$
Applying conservation of energy at $A$ and $B$
$\frac{1}{2}m{v}^2+2gmr=\frac{1}{2}m{v}_B^2[\because (P.E.)=mg(2r\left)\right]$
$v_B^2=v_A^2+4gr......\left(i\right)$
Now as it is moving in circular path it has centripetal force.
At point $A\Rightarrow T+mg=\frac{m{v}_A^2}{r}$
for minimum velocity $T\ge 0$
Or $\frac{m{v}_A^2}{r}\ge mg\Rightarrow v_A^2\ge gr\Rightarrow v_a\ge \sqrt{gr}$