Question

A small body of mass m hangs at one end of a string given a horizontal velocity u at its lowest

Answer 1

The forces acting on the body are the tension T along the string and the weight mg. As the body moves along a circular path, the forces are balanced along the radius.

$T=mv²/a+mgcosФ$

$=mu²/a-2mg(1-cosФ)+mgcosФ$

$T=mu²/a-mg(2-3cosФ)$ ------- (2)

If the string becomes slack, then $T=0$and later negative. So at $Ф=60°$

$T=0$

$=>u²/a=g(2-3cos60°)=g/2$

$=>u=\surd \left(ag/2\right)$

Tension at initial point of projection $Ф=0°$, (using equation (2)

$T=mg/2-mg(2-3)=3mg/2$