(Solved) Why does a more center focused mass have a larger angular velocity?

rnewell

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11 years ago

Why does a more center focused mass have a larger angular velocity?

I understand that the angular velocity increases if the radius is decreased because w(omega)=v/r.

I also understand that if the radius is decreased, the moment of inertia decreases because I=mr^2.

I'm a bit confused with this question I had in a review. It asked which wheel would roll down an incline first. A heavy tire/light hub or light tire/heavy hub.The tire with the heavy hub rolls to the bottom first.

Could someone explain to me why this is, preferably with equations to back it up? Also, what does the translational and rotational motion have to do with it?
Thanks everyone for the excellent answers!

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rj957

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11 years ago

you already gave the answer.

inertia is a *resistance* to an applied force. more inertia -> more resistance.

i.e. a larger (moment of) inertia is more resistant to an acceleration by an applied force.

the heavy tire/light rim has a larger moment of inertia than a light tire/heavy rim, simply because more mass is concentrated at lower radii in the latter case. i.e. if the total mass of the tire+rim is the same in both cases, the moment of inertia of the latter case is smaller than the moment of inertia in the former case because a larger fraction of the total mass is concentrated at lower radii.

and as you noted, the moment of inertia involves a factor that goes like m*r^2...

cheers

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lespaire24

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11 years ago

think of the figure skater spinning with her leg out. she is spinning slowly. Then she pulls her leg in and starts spinning faster. In the first case she is a heavy tire light hub, in the second, she is light tire /heavy hub.

if the figure skater doesn't work for you, try this: find an office chair that spins around. sit in it and spin it and keep your legs out. then pull your legs in and feel the increase in angular velocity.

The more mass concentrated in the center, the more omega increases to balance the equation.

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ilovebeaches

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11 years ago

This verified answer contains over 230 words.

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Why does a more center focused mass have a larger angular velocity?

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