Question

A small body of mass m hangs at one end of a string of length a, the other end of which is fixed. it is given a horizontal velocity u at its lowest position so that the string would just becomes slack,

Answer 1

Length of string,a = L.

Mass = m.

1. At the point of projection:

Speed =$u$. Tension = $T_1$.

$T_1$= $mg+\frac{m{u}^2}{a}$

2. At the point the string becomes slack..

Angle with the vertical = $\theta$.

Tension = $T_2$. Speed =$v$.

Balance forces and energy.

$T_2+mgcos\theta =\frac{m{v}^2}{a}$

$\frac{m{u}^2}{2}=\frac{m{v}^2}{2}+mga(1+cos\theta )$

Solving these $T_2=\frac{m{u}^2}{a}-mg\ast (2+3cos\theta ).$

If the string becomes slack then $T_2$ = 0.

$u^2=g\ast a\ast (2+3cos\theta )$

If $\theta =60°$ then, $u^2=\frac{7ga}{2}.$

Finally, tension $T_1=\frac{9mg}{2}.$