Question $7.$

A mechanically operated system involves a pulley $-$ diameter $15$ cm $,$ mass $0\mathrm{.}6$ kg and a radius of gyration $6\mathrm{.}25$ cm $.$ The system is required to lift a container of mass $30$ kg with an acceleration of $1\mathrm{.}25\frac{m}{s^2}$ from rest. It can be assumed that the frictional torque in the pulley is $12$ Nm and the mass of the cable can be ignored.

$($i$)$ Sketch the individual free body diagrams for the pulley and the container.

$[5]$

$($ii$)$ Calculate the tension in the cable when the container is in motion and the torque required by the pulley system to control the lifting of the container.

$[5]$