If a body A is revolving about an axis with moment of inertia (MOI) $I_1$ with angular velocity $\omega_1$, and rotating about its centre of mass with MOI as $I_c$ with angular velocity $\omega_2$, how would we calculate the kinetic energy of the body?

E.g., a disc with radius R is revolving about an axis $3R$ from its centre, and is simultaneously rotating about a parallel axis through its centre of mass (let's say perpendicular to the plane of the disc for simplicity) with angular velocities $\omega_1$ and $\omega_2$ respectively.

  • $\begingroup$ Or like the Earth orbiting around the sun while also rotating on its axis? $\endgroup$ – BioPhysicist Mar 25 '20 at 15:19
  • $\begingroup$ physics.stackexchange.com/questions/250834/… I feel this explains it better, but an answer on this question says that the cylinder is TRANSLATING about the axis with angular speed $ohm$. However, if we look at the cylinder here, different points on cylinder will have different velocities, as their distances differ from the axis in question. How can we thus say it is translating? $\endgroup$ – K. Chopra Mar 25 '20 at 19:03
  • $\begingroup$ The COM of the cylinder is translating (ie orbiting in a circle). The motion of individual particles within the cylinder is taken into account in the spinning motion about the COM. Generally total KE = KE of COM + KE of body about an axis through the COM. $\endgroup$ – sammy gerbil Mar 25 '20 at 20:41
  • $\begingroup$ The definition of translation is that all particles of the body should have the same velocity in a particular direction. However, this will not be the case considering that all particles of the disc will have differing perpendicular distance from I1. How then can we call the motion translational? $\endgroup$ – K. Chopra Mar 26 '20 at 18:14