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If I have a system with particles moving with varying accelerations such that it's center of mass is moving with accelerations (there is a non-zero net force on the system (acting on one or all of the particles) where there is a non-zero acceleration of the center of mass? Would the frame centered around the center of mass be therefore non-inertial?

user1007028
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    The only way for the center of mass to be accelerating (in an inertial frame) is for there to be some external force acting on the objects in the system from some source that is not considered as part of the system. – The Photon May 21 '22 at 14:40
  • so we would ignore the COM frame in the case, but yeah this was the idea I had. – user1007028 May 21 '22 at 15:03
  • if there was an external force acting on one part of the system this would also count as acting on it and cause this as well? – user1007028 May 21 '22 at 15:05
  • Yes, the external force only needs to act on one component of the system to accelerate the center of mass of the system. – The Photon May 21 '22 at 15:06

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The momentum of an isolated system is conserved. This means the center of mass of an isolated system has a constant velocity in an inertial frame.

A system is isolated if there are no external forces acting on it. So a rock floating in space far from any mass or other source of force is an isolated system. A rock dropped on earth is not. Gravity accelerates the rock. The center of mass is not an inertial frame. (In classical physics. We are leaving General Relativity out of this.)

Note that the isolated rock might be spinning. The center of mass will still travel at a constant velocity.

You can take this farther. The COM of an isolated firecracker travels at a constant velocity. After it explodes, the COM of the shards travels at the same constant velocity. The forces from the explosion are internal. Shard 1 exerts a force on shard 2. There is an equal and opposite reaction force from shard 2 on shard 1. The two shards have equal and opposite changes in momentum.

It is the same for a rocket, but you have to be careful about what you consider to be the system. A rocket plus its fuel is an isolated system. The rocket alone is not. Here is more on that. Clarification about what makes a system isolated

mmesser314
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