A solid thin disk of radius $R$ is rotating with an angular velocity $$ about the $z$

axis as shown in the figure below.

The suggested stress state, in polar $($cylindrical$)$ coordinates is given by:

${}_{rr}=\frac{3+v}{8}{}^2(R^2-r^2)$

${}_{}=\frac{{}^2}{8}[(3+v)R^2-(1+3v)r^2]$

${}_r=0$

where $$ is the material density of the disk.

$($a$)$ Show that internal equilibrium is satisfied $($note that due to rotation of the body, there will be body force in this problem$).$

$($b$)$ Determine from the stress state the tractions on r $=$ R$.$

$($c$)$ Determine the principal stresses and directions at r $=0$