A circular ring with area $4\mathrm{.}45$ cm$2$cm$2$ is carrying a current of $13\mathrm{.}5$ AA$.$ The ring, initially at rest, is immersed in a region of uniform magnetic field given by B$=(1\mathrm{.}05102$T$)(12$i$^+3$j$^4$k$^)$B$=(1\mathrm{.}05102$T$)(12$i$^+3$j$^4$k$^).$ The ring is positioned initially such that its magnetic moment orientation is given by $$i$=$$(0\mathrm{.}8$i$^+0\mathrm{.}6$j$^)$$$i$=$$(0\mathrm{.}8$i$^+0\mathrm{.}6$j$^),$ where is the $($positive$)$ magnitude of the magnetic moment.

$($a$)$ Find the initial magnetic torque on the ring.

$($b$)$ The ring $($which is free to rotate around one diameter$)$ is released and turns through an angle of $90\mathrm{.}090\mathrm{.}0,$at which point its magnetic moment orientation is given by $$f$=$k$^$$$f$=$k$^.$ Determine the decrease in potential energy.

$($c$)$ If the moment of inertia of the ring about a diameter is$8\mathrm{.}50107$kg$$m$28\mathrm{.}50107$kg$$m$2,$ determine the angular speed of the ring as it passes through the second position.