Question

A light string is wound several times around a spool of mass $M$ and radius $R$ as shown. The free end of the string is attached to a fixed point and the spool is held so that the part of the string not in contact with it is vertical. If the spool is let go, the tension of the string is $\frac{K M g}{12}$. Find $K$.

Answer 1

Equation of motion is
$M a=M g-T\,\,\,...(i)$
Torque $\tau=T R=I \alpha$
$=\frac{1}{2} M R^{2} \frac{a}{R}\,\,\,...(ii)$
$\therefore T=\frac{1}{2} M a\,\,\,...(iii)$
Solving (i) and (iii)
$a=\frac{2}{3} g $
$T=\frac{M g}{3}$