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I'm hoping any gravity or friction can be ignored.

I gather a spinning object is a non-inertial frame. I suppose that's because change of direction is acceleration. Continued acceleration requires continued force.

But apparently the constant rotation can also somehow be described as angular velocity. And the rotation described as having rotational inertia, aka moment of inertia. Inertia being resistance to change in velocity, meaning maintaining velocity unless forced.

Some explanations seem to result in a third explanation, conservation of angular momentum, explained abstractly in accord with a Noether's theorem about symmetry.

Qmechanic
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  • Think of an athlete spinning a hammer before throwing it away. It is only spinning because of the centripetal force. Once released, there is no spin anymore. – Claudio Saspinski Feb 04 '22 at 23:08
  • I think comment shouldn't be used to answer? Plus I think the example only explains the direction not the speed. –  Feb 09 '22 at 10:31

3 Answers3

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There are rotational analogs of Newton's 1st and 2nd laws that apply to rotating objects.

According to Newton's 1st law, objects in motion tend to stay in motion until acted on by a net force. This is attributed to the object's inertia, which resists acceleration. The rotational analog of inertia is moment of inertia, where every differential mass of a rotating object has its own inertia, depending on how far that differential mass is from the axis of rotation.

According to Newton's 2nd law, $F=ma$, or if you prefer, $a=\frac{F}{m}$. This says that an object must experience a net force to experience an acceleration, and the amount of acceleration is inversely proportional to the object's mass. The rotational analog of the 2nd law is $\tau=I\alpha$, where $\tau$ (aka torque) is the rotational equivalent of force, $I$ (aka moment of inertia) is the rotational equivalent of mass (or inertia), and $\alpha$ is the rotational equivalent of acceleration.

This means that rotating objects in frictionless environments continue to rotate due to their rotational inertia.

David White
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  • Ty. Would you accept that moment of inertia can be replaced by moment of momentum aka angular momentum? Moment of momentum seems a bizarre phrase (somehow the same root word took on almost opposite meanings?) but used historically at least. –  Feb 05 '22 at 20:00
  • And would you say that someone who says inertia keeps it spinning, without saying rotational inertia, is wrong, because it's not inertial it's accelerating around? –  Feb 05 '22 at 20:20
  • @Aday, rotational inertia IS inertia for rotating bodies. – David White Feb 05 '22 at 20:26
  • Inertia seems to refer to non-rotational inertia by default. Which is presumably why you said rotational inertia is an analog of inertia (or does analog in physics mean a type of?). I'm not trying to be pedantic, I'm trying to understand the real meaning of inertia since it's defined as resistance to change in velocity, not resistance to change in acceleration. –  Feb 05 '22 at 22:07
  • @Aday, change in velocity is acceleration. By default, if you resist changes in velocity, you resist changes in acceleration. You seem to have a hidden assumption that you have not stated regarding changes in velocity and changes in acceleration. If you can manage to state that assumption, you will probably get the explanation that you are looking for. – David White Feb 06 '22 at 01:23
  • I may have some basic misunderstanding of the distinctions, I've tried to appreciate first that velocity is speed in a direction, then that it's actually defined as rate of displacement. I'm still thinking of rotation as acceleration not velocity, because of the change of direction. I don't see that inertia by default leads to resistance to deceleration, because isn't the whole point that objects continue at constant velocity but don't continue accelerating when force is removed. Yet somehow in rotating frames, resistance to change in 'angular velocity' results in resistance to deceleration. –  Feb 06 '22 at 12:37
  • There does seems to be something about rotating frames that puzzles https://physics.stackexchange.com/questions/239477/how-does-a-spinning-object-know-that-it-is-spinning 'Given that, apparently, the laws of physics are intrinsically invariant with respect to position, orientation and velocity (= the time derivative of position), it's somewhat surprising that they're not invariant with respect to angular velocity' –  Feb 06 '22 at 12:49
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Which is true - objects keep spinning because of inertia; objects keep spinning because of centripetal force

The law of inertia, also called Newton’s first law, postulate in physics that, if a body is at rest or moving at a constant speed in a straight line, it will remain at rest or keep moving in a straight line at constant speed unless it is acted upon by a net force. See https://www.britannica.com/science/law-of-inertia

So since the inertia of an object makes an object want to continue to move in a straight line at constant speed. It takes a net force to prevent that from happening.

Therefore, in the case of a spinning object, that net force is the centripetal force acting towards the center of the circular motion. So to answer your question, objects keep spinning because of the centripetal force, not inertia.

Here's another way to think about it. Think of the inertia of an object as a description of the way an object behaves when there is no net external force acting upon it. Which means, it remains at rest or moves at constant speed in a straight line.

Hope this helps.

Bob D
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  • The link seems to give the spin of the earth (and therefore us on it) as an example of inertia. –  Feb 09 '22 at 10:36
  • I would argue that gravity is the centripetal force that keeps us from flying off the surface of the earth in a straight line due to our inertia. You asked the question and I gave you my opinion. However if you feel it is our inertia that keeps us on the surface, that’s fine. – Bob D Feb 09 '22 at 11:03
  • I didn't mean that's not gravity. I only meant the reason the earth keeps spinning. I don't disagree with your logic about spin being change of direction therefore acceleration therefore requiring force. Yet somehow it seems instead to be defined as change of orientation, with angular velocity, with inertia. But then you get weird puzzles like https://physics.stackexchange.com/questions/239477/how-does-a-spinning-object-know-that-it-is-spinning –  Feb 09 '22 at 21:34
  • @Aday The earth keeps spinning due to conservation of angular momentum. But it wasn't always spinning. As I understand it, during its early formation it was impacted by space rocks (forces) which caused it to spin. My point is, the law of inertia is Newton's first law and the first law talks about constant speed in a straight line. If it applies to an object in circular motion, why the statement "in a straight line"? Why not just say "constant speed"? – Bob D Feb 10 '22 at 17:02
  • That's what bothers me too, I had found the extract of that law and he says rectilinear. But in the next paragraph refers to a spinning top... –  Feb 11 '22 at 19:17
  • @Aday I don't know what else to tell you. Every reference to the law of inertia that I have seen equates it to Newton's first law. In order for an object to remain at rest or move at constant speed in a straight line there can be no net force acting upon it. An object in circular motion moves that way because there is a net force acting upon it- a centripetal force. You'll just have to make up your own mind. – Bob D Feb 11 '22 at 19:31
  • @Aday Here's another way to think about it. The inertia of an object describes the way an object behaves when there is no net external force acting upon it. Which means, it remains at rest or moves at constant speed in a straight line. I've added this to my answer. – Bob D Feb 11 '22 at 19:40
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Definitely inertia (although more accurately, "angular momentum", but definitely not the frames or internal forces). Angular momentum is the important conserved quantity here. (In detail, this is because space is invariant under rotation and invariant motions lead to a conserved momentum.)

The "inertia", in this case, "moment of inertia" for a rotating body, is a scale factor for the angular momentum, describing, for example, how much torque applied over what time is required to stop the object. And, for example, the moment of inertia could change dynamically as when a spinning skater pulls in their arms, but the angular momentum is still preserved.

Angular momentum would be conserved regardless of how the body is constructed. For example, if it was sand that was all of sudden let loose, and flew everywhere, it would fly off in such a way that angular momentum would still be conserved. But if it's a solid body, all of the pieces that would fly off are instead bound, and that binding (centripetal) force will be required for it to act as a solid body, but it's not what creates a conserved quantity. Also, again here, it's interesting to think of the skater: the angular momentum stays constant, while the forces also change.

tom10
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  • I read to keep the concepts of inertia and momentum distinct but the terminology of 'moment of inertia' seems to overlap them more, unless the root 'moment' has different etymology. Your solid answer gives me a lot to process which I will do. I had thought i'd understood that the main distinction in motion in the universe is between inertial vs accelerating (including change of direction). Then got clouded by this rotational stuff and I've just read a comment that 'Conservation of momentum is the most fundamental law in our universe'. –  Feb 05 '22 at 00:49
  • @Aday: For the vocabulary: "Momentum" whether linear or angular (rotational) is very clear and well defined and everyone means the same thing. Eg, the wikipedia pages have a bunch of equations describing what these are. "Inertia" on the other hand has a wider range of uses, more variation to where it's applied, etc. Eg, in wikipedia page there are no equations and things like: In common usage, the term "inertia" may refer to an object's ... depending on the context. It's not a term I use much in my own thinking about things. – tom10 Feb 05 '22 at 01:33
  • I see the WP explanation for conservation of angular momentum is "According to Noether's theorem, if the action (the integral over time of its Lagrangian) of a physical system is invariant under rotation, then angular momentum is conserved." But that leads to this puzzle, right? https://physics.stackexchange.com/questions/239477/how-does-a-spinning-object-know-that-it-is-spinning –  Feb 08 '22 at 11:07
  • @Aday: I don't think SE is a good place for long dorm-room type discussions. What SE is (somewhat) good for is that if you can post a question that's clear enough to get an answer, such that the answer will answer your deeper question, then it might help you resolve deeper questions. 1) Here your question is very clearly stated in your title; and that question has a very clear answer which I think I have posted. 2) Yes, Mach's principle is interesting. But what else is there to say on this? If there is another specific question you can post on it, I encourage you to do so. – tom10 Feb 08 '22 at 16:59