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mfb

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The prefactor for the second term looks wrong. That would correspond to a ring instead of a disk.

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- #4

mfb

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But you're sure my formula is for a ring ?

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mfb

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Have you a link where I can find formulas for different shapes ?

- #8

mfb

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First google hit for "moment of inertia" -> List of moments of inertia

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I would like to find the demonstration of this formula to be sure it is correct at least for a ring.

- #10

jbriggs444

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A demonstration I find convincing is to imagine that one is applying the work neccessary to spin this arrangement up.

Start with the rod locked in place and spin up the disk to a rotation rate of ##\omega_a - \omega_b##. How much energy does that take? It should be ##\frac{I_b(\omega_a - \omega_b)^2}{2}##. Where ##I_b## is the moment of inertia of a disk or ring of radius r. (Which you can look up on Google)

Now unlock the rod and spin it up to a rotation rate of ##\omega_a##. Because the pivot on which the disk/ring is mounted is frictionless and mounted at the center of mass of the disk/ring, the energy required to do this is independent of the size, shape or rotation rate of the object on the end of the rod. It depends only on the disk/ring's mass. How much energy is this? It should be ##\frac{I_a\omega_a^2}{2}## where ##I_a## is the moment of inertia of a mass at a distance d. By definition, that's ##md^2##.

Having done this, the relative rotation rate of the disk or ring with respect to the rod will be ##-\omega_b## and we are in the correct final configuration. The energy in the configuration is the total of the energy that went into spinning it up.

There is no requirement that ##\omega_a \gt \omega_b##. The formula works regardless.

- #11

mfb

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There is also no formula for tons of other cases. All the discussion here was about a specific part of the formula. And there is a formula for the moment of inertia of a spinning disk. And somewhere on that page or linked pages you can also find the formula for the energy of a spinning disk. Put both together with the angular velocity you found and you'll see the second term of your formula has a wrong prefactor.Ok, but I don't find the formula: 1/2md²Wa²+1/2mr²(Wa−Wb)²

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