y In particular we can write, (Ix)o=/(x2+2)dm=/(v+y)2+(ag+2)2)dm (y2+2)+2/ydm+2c/2dm+(+a)/dm Ix+m(+2) Here, we have use the fact that y and 2 are the coordinates relative to the center of mass and therefore heir integrals over the body are equal to zero. Similarly, we can write (lv)O=l+m(x+2),(12)o=l2+m(x+v) (Ixy)o=(lyx)o=Ixg+mcv,(2)o=(I2x)o=Ixz+mxc,(ly=)o=(1xy)o=lyz+myec Rotation of axes In some situations, we will know the tensor of inertia with respect to some axes zyz and, we will be intereste in calculating the tensor of inertia with respect to another set of axis a'y z. We denote by i, i and k the unit vectors along the direction of zyz axes, and by i', j and k the unit vectors along the direction of r'y'z' yIn particular we can write, (Ixx)O = Z m (y 2 + z 2 ) dm = Z m ((yG + y ′ ) 2 + (zG + z ′ ) 2 ) dm = Z m (y ′2 + z ′2 ) + 2yG Z m y ′ dm + 2zG Z m z ′ dm + (y 2 G + z 2 G) Z m dm = Ixx + m(y 2 G + z 2 G) . Here, we have use the fact that y ′ and z ′ are the coordinates relative to the center of mass and therefore their integrals over the body are equal to zero. Similarly, we can write, (Iyy)O = Iyy + m(x 2 G + z 2 G), (Izz)O = Izz + m(x 2 G + y 2 G), and, (Ixy)O = (Iyx)O = Ixy + mxGyG, (Ixz)O = (Izx)O = Ixz + mxGzG, (Iyz)O = (Izy)O = Iyz + myGzG . Rotation of Axes In some situations, we will know the tensor of inertia with respect to some axes xyz and, we will be interested in calculating the tensor of inertia with respect to another set of axis x ′y ′ z ′ . We denote by i, j and k the unit vectors along the direction of xyz axes, and by i ′ , j ′ and k ′ the unit vectors along the direction of x ′y ′ z ′ axes. 4