# Derivation of rotational body interia from newton's laws

We know that rigid body will keep rotating without external force. This answer explains

Rotational inertia differs from ordinary “linear” inertia in that it is a derived principle: it can be derived mathematically from Newton’s laws of motion, so in that sense it has (in my opinion) a slightly less fundamental status among the laws of physics. Rigid bodies don’t “want” to keep rotating in the same fundamental sense that particles “want” to keep moving in a straight line with a fixed velocity - they do end up rotating but it’s because of a process we understand well and can analyze mathematically

Can someone present the mathematical derivation referred above, I am not able to find this anywhere? For me individual mass particles in the body seems to be violating newton's first law as they are changing velocity direction without any force being applied.

• You want to derive the equations of rotating bodies? I guess any introductory physics textbook would include that. See for example: en.wikipedia.org/wiki/Moment_of_inertia Commented Dec 17, 2021 at 9:44
• These textbooks start out with definitions which assume concepts of torque and rotational inertia. Whereas, we want to apply newton's laws on the particles/forces in the rigid body to understand why a rotating body without external force keeps rotating Commented Dec 17, 2021 at 18:14
• Imagine a rigid body composed of 2 particles with fixed distance between each other, you can derive all the rotational physics from forces acting on them. Apply the same principle to any body, just sum/integrate over differentials of particles. Commented Dec 17, 2021 at 18:19
• Yes that is what I am looking for. If you can show forces acting inside a rigid body and show integration over differential of particles I can accept that answer Commented Dec 17, 2021 at 18:54