Question

A sphere is suspended by a thread of length L. What minimum horizontal velocity to has to be imparted to it so that if completes a full vertical revolution?

Answer 1

Tension is the string parties makes angle $\theta$ with vertical, during the circular motion is :

$T=\frac{m{u}^2}{L}-2mg(1-\cos \theta )+mg\cos \theta =\frac{m{u}^2}{L}-mg(2-3\cos \theta )$

Particle completes a circle only with tension in $\le 0$ at highest point $(\theta ={180}^o)$

so,

$\frac{m{u}^2}{L}-mg(2-3\cos ({180}^o\left)\right)\ge 0$

$\Rightarrow \frac{m{u}^2}{L}-5mg\ge 0\Rightarrow u\ge \sqrt{5gL}$

$\therefore$ A minimum velocity of $u=\sqrt{5gL}$ has to be imparted at the lowest point.