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I'm a bit confused about angular vs linear momentum conservation, plus the relationship to centripetal force. Below is my attempt to convey the confusion.

1. Consider a uniform disc (e.g., a CD disc) rotating at a constant angular speed on an axis through its center. With no external TORQUE, the angular momentum is conserved -- and hence it will continue to rotate forever (ignoring any friction etc).

Above, I think the total linear momentum is zero (as each point on the disk is offset by a diametrically opposite point of mass). And, thus linear momentum is conserved.

Now -- where is the centripetal force? I.e., what is keeping the each point in a circular motion. I presume this is subsumed in the rigidity of the disk, or is coming from an external point of contact (likely from the axle around which it is rotating).

2. Now, let's say the disk is comprised of just one little point of mass while the rest of the disk is massless. Then, there is certainly linear momentum as well as angular momentum. Is my understanding correct that (i) Linear momentum is not conserved, due to centripetal force from somewhere, (ii) Angular momentum is still conserved, since the centripetal force (being radial) contributes to zero torque?

Thanks.

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  • $\begingroup$ . . . . then how is this [centripetal force not changing the linear momentum? The force is at right anglers to the velocity so only the direction of the velocity and hence the linear mome4ntum of a particle is changing. $\endgroup$
    – Farcher
    Commented Feb 19, 2023 at 23:43

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If your system is the entire disk, then there is no motion and no centripetal force.

If your system is a small portion of the disk (say a small element at the edge), then the centripetal force to allow that portion to move in a circle comes from nearby elements. If the forces holding the disk together were to disappear, each element would no longer move in a circle.

Due to symmetry, the disk will spin without any forces from the axle.

let's say the disk is comprised of just one little point of mass while the rest of the disk is massless.

Then the forces on that point would no longer come from other elements of the disk rotating as well, but from the axle. If the disk were disconnected from the axle, it would not rotate about the center (it would rotate around the single point of mass).

[angular momentum] is just a convenient way to represent the aggregate linear momentum of all the points that make a rotating object?

I wouldn't say it represents aggregate linear momentum, but is instead a conserved quantity of it's own.

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  • $\begingroup$ Regarding your last comment about angular momentum being a conserved quality of its own: I know that "Torque = (moment of inertia) x (angular acceleration)" can be derived purely from "work done by force = change in KE" ----> which suggests that if there is no torque, the angular momentum should be conserved. Is the conservation of angular momentum --- coming from something different from Newton's law and energy conservation? $\endgroup$
    – Brian
    Commented Feb 20, 2023 at 0:57

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