$[16$ pts$.]$ The string wrapped around the cylinder $(I=\frac{1}{2}M{R}^2)$ is held by a hand that is accelerated upward so that the cylinder does not move up or down. Assume the mass of the cylinder is $0\mathrm{.}25$ kg $,$ and the radius is $4\mathrm{.}2$

cm$.$ Find:

$($A$)$ The tension in the string.

$($B$)$ The angular acceleration of the cylinder and

$($C$)$ The acceleration of the hand and string $($they are the same$)$

Hint: Follow these steps to get partial credit $(1)$ draw a force diagram for the yo$-$yo $(2)$ create the force equation from the force diagram $(3)$ draw the torque diagram for the yo$-$yo $(4)$ create the torque equation from the torque diagram.