A thin, flat, uniform disk has mass $8\mathrm{.}0$ kg and radius R $=2\mathrm{.}0$ m$.$ A circular hole of radius $1\mathrm{.}0$ m$,$ centered at a point $1\mathrm{.}0$ m from the disk's center, is then punched in the disk. Find the moment of inertia of the disk with the hole about an axis through the center of the hole, perpendicular to the plane of the disk.

Hint:$$

a$)$ Find the y$-$coordinate of the center of mass.

b$)$ Find the x$-$coordinate of the center of mass taking the following steps.

i$)$ Find the ratio of the masses of the disk punched out $($mP$)$ and the piece left $().($Hint: Because the disk is uniform, mass is proportional to the area of disk.$)$

ii$)$ Find the x$-$cordinate of the center of mass for the disk punched out and the x$-$cordinate of the center of mass for the original disk.

iii$)$ Assume the disk's mass is concentrated as a particle at its center of mass. Find the of the original disk using $,,,$ and $.$

iv$)$ Plug in the x$-$cordinate of the center of mass for the original disk and find $.$

c$)$ Find the moment of inertia of the disk with the hole about an axis through the original center of the disk, perpendicular to the plane of the disk. $($Hint: Find the moment of inertia of the piece punched from the disk.$)$

Question $12$Answer

a$.$

none of these

b$.$

$26\mathrm{.}0$

c$.$

$39\mathrm{.}0$

d$.$

$13\mathrm{.}0$

e$.$

$6\mathrm{.}5$