A thin, light string is wrapped around the outer rim of a uniform, hollow cylinder with mass $M=5\mathrm{.}00kg$ having inner radius of $r_1=15cm$ and outer radius $r_2=30cm$ as shown in the figure below. The other end of the string is affixed to the ceiling. The cylinder is released from rest.

$($a$)(8$ points$)$ Immediately after being released from rest, what is the tension in the string?

$($b$)(5$ points$)$ How long would it take for the cylinder to fall $1\mathrm{.}00$ m $?$