An ancient Chinese clock mechanism consisted of a weight hanging from a rope wound around a cylinder. The rope would slowly unwind, driving the hands of the clock.

A student tries to recreate the mechanism using a cylinder of mass $23$ kg and radius $4\mathrm{.}2$ cm and a hanging weight of mass $1\mathrm{.}0$ g$.$

$($i$)$ The moment of inertia of a cylinder of mass M and radius R is I $=1/2$ MR$^2.$

Show that the acceleration of the hanging weight is $8\mathrm{.}5*10^-4$ m$/$s$^2$

$($ii$)$ How long does it take the weight to fall through a distance of $2\mathrm{.}0$ m$?$

$($iii$)$ How many rotations does the cylinder make in this time?