How to calculate kinetic energy of rod which is rotating about an external axis and spinning about its centre of mass?
1) I know how to calculate kinetic energy of a rod if it is not spinning around centre of mass but rotating around external axis
length=l
mass=m
angular speed $\omega=\sqrt{\frac{g}{l}}$
$$KE=KE_{translation}+KE_{rotation}$$
$$KE=\frac{1}{2}mv_{cm}^2+\frac{1}{2}I_{cm}\omega^2$$
$$v_{cm}=\frac{3l\omega}{2}, I_{cm}=\frac{ml^2}{12}$$
$$KE=\frac{7}{6}mgl$$
$$or$$
$$KE=\frac{1}{2}I\omega^2$$
I is about axis of rotation
$$I=\frac73 ml^2$$
$$KE=\frac{1}{2}(\frac73 ml^2)\frac{g}{l}$$
$$KE=\frac{7}{6}mgl$$
But now if rod is spinning around its com as well as external axis
2) A rod is spinning as shown
in this case im getting trouble