Question $4(25$ Points$)$

A flat$-$disk pulley has a mass of $25\mathrm{.}0$ kg and a diameter of $0\mathrm{.}2$ m $.$ A cable is wrapped around the pulley. A box with a mass of $100\mathrm{.}0$ kg is attached to one end of this cable and vertically lowered as shown in the figure on the bottom left. Do not neglect the rotation of the pulley. Neglect the mass of the cable.

a$)$ Determine the acceleration of the box after the system has been released from rest.

b$)$ How long will it take the box to reach the ground located $10\mathrm{.}0$ m below starting from rest?

c$)$ If the pulley rotates counter$-$clockwise as shown on the bottom right, indicate the pivot point of the pulley, center of mass, and sketch the direction of the torque and the angular momentum.