Question

State and prove principle of perpendicular axes.

Answer 1

moment of inertia of system about an axis perpendicular upon plane of system is summation of moment of inertia of system about axis lies in plane.

Let a particle of point mass, m is placed in xy plane such that location of it in three dimensional co-ordinate system is given by $\vec{r}=x\hat{i}+y\hat{j}+$

so, moment of inertia of particle about x - axis , $I_x=mx^2$
moment of inertia of particle about y - axis, $I_y=my^2$

now , moment of inertia of particle about z - axis, $I_z=mr^2$
$I_z=m(\sqrt{x^2+y^2})^2$
$I_z=mx^2+my^2$
$I_z=I_x+I_y$ hence proved