A mass $m=0\mathrm{.}5kg$ is suspended by a string and attached to a pulley which is pictured below. The pulley wheel is massive and constructed as follows: The inner disk is a solid cylinder of radius $r=0\mathrm{.}1m$ and mass $0\mathrm{.}05$ kg $.$ The outer ring has inner diameter of $0\mathrm{.}25$ m and outer diameter of $0\mathrm{.}3$ m and is made of the same material as the inside disk. Inbetween these circular masses are three solid rods which connect between the inner and outer disks which each have mass $0\mathrm{.}01$ kg $.$

a$)$ Find the mass of the outer disk.

b$)$ Calculate the moment of inertia of the pulley.

c$)$ Write out Newtons second laws for the system.

d$)$ Calculate the tension felt in the string.

e$)$ Calculate the acceleration of the falling mass.

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