Question

The moment of inertia of the pulley system as shown in the figure is 3 kg - $m^2$. The radii of bigger and smaller pulleys are 2m and 1m respectively. As the system is released from rest, find the angular acceleration of the pulley system (Assume that there is no slipping between string & pulley and string is light) [Take g = $10m/s^2$]

Answer 1

Writing force equations,

$6g-T_1=6\times 2\alpha ...........\left(i\right)$

$T_2-3g=3\times \alpha ...........\left(ii\right)$

Writing torque equation,

$T_1{r}_1-T_2{r}_2=I\alpha$

$2T_1-T_2=3\alpha .........\left(iii\right)$

Substituting $T_1$ from (i) and $T_2$ from (ii) in (iii), we get:

$12g-24\alpha -3g-3\alpha =3\alpha$

$\alpha =\frac{9g}{30}=3rad/s^2$