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How does the conservation of angular momentum work radially with respect to a rotating body?

For example, if a rotating body, such as a spinning top, were to lose an outer layer or shell, would it continue rotating at the same rate, or would it speed up or slow down?

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  • $\begingroup$ Does the outer layer itself keep spinning or is it brought to rest somehow? This seems reminiscent of spinning stars losing outer layers. $\endgroup$ – jacob1729 Mar 4 at 12:18
  • $\begingroup$ Calculate the total angular momentum of the system before the outer layer is lost. Then assert that the total angular momentum of the system (spinning top plus outer layer) is the same immediately after they lose contact with each other. What happens to the top is whatever is necessary to hold that assertion and, therefore, depends on exactly how the outer layer is lost $\endgroup$ – Jim Mar 4 at 14:08
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Angular momentum is the rotational analogue of linear momentum with the formula: $$ L=I\omega$$ It can also be further derived as: $$ L=(mr^2)\left(\frac vr \right)=mvr$$

$r$ being the radius from the center of the mass to the rim, then we can logically deduce that angular momentum is directly proportional to angular velocity and angular velocity is inversely proportional to the radius. So if the angular momentum is conserved, an increase or decrease in radius translates to a decrease or increase in angular velocity respectively.

For the other questions, if by loosing the outer shell you mean the shell collapses into the rest of the mass to form a denser mass with less radius, the top speeds up. If the shell just vanishes off the mass and cease to exist (hypothetically), the top will start spinning a lot lot more, depending on how much mass and radius is lost. Also if the shell comes off the mass by breaking or ripping apart, the mass keeps spinning at the same rate.

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  • $\begingroup$ "If the shell spins off the mass, the top will start spinning a lot more, depending on how much mass and radius is lost." This seems wrong $\endgroup$ – Tom B. Mar 7 at 15:48
  • $\begingroup$ "If the shell spins off the mass, the top will start spinning a lot more, depending on how much mass and radius is lost." This seems wrong. If the shell simply flies off, this implies that it breaks into at least two pieces. Each piece would continue in a direction tangential to that which it had immediately before release. Since these directions are not co-linear, the pieces would have angular momentum between them. They would also maintain their own rotation, so would have some angular momentum from rotation. I believe that the rotation rate of what's left of the top would not change. $\endgroup$ – Tom B. Mar 7 at 15:57
  • $\begingroup$ I think it'll be better if I say vanish right. I meant the outer shell just cease to exist, according to the law of conservation of energy and momentum, all the energy isn't destroyed but collapses in back into the residual mass. $\endgroup$ – TechDroid Mar 7 at 16:02

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