Moment of inertia (multiple rotating axis)

[pinsky]

Hello there,

i have a problem with calculating the moment of inertia for the object on the picture. There are two cases I'm observing. In the first case, the obect rotates around the axis located in the center of the big circle, and the little circle can't rotate around the axis located in it's center.

Note that the smaller circle is mounted on the big one, it it not carved into it replacing the part of the big circle in that area. That is why J_big=1/2 MR² R being the radius of the big circle

In this case, we get the whole J by the parallel axis theorem.

J=J_M + J_m

where J_{m= 1/2 mr² + ml²}

How does the equation change if we allow the small circle to rotate around the axis located at its center while the axis of rotation for which we calculate J remains the center axis of the big circle?

What is the physical explanation for that?


tnx

[PLAIN]http://img848.imageshack.us/img848/2126/vztrajnostni.gif

 

[tiny-tim]

hello pinsky!

How does the equation change if we allow the small circle to rotate around the axis located at its center while the axis of rotation for which we calculate J remains the center axis of the big circle?

What is the physical explanation for that?

hmm … awkward, aren't you?

if they can rotate separately, then it's no longer a rigid body, so you need to consider the two discs separately

the angular momentum of the small disc about the main centre will be the "intrinsic" angular momentum about its own centre, plus an extra angular momentum as if all its mass was at its centre, moving with the velocity of the centre (this is the same as the extra parallel axis term, mr²ω = mrv, only in the case where the point about which you're calculating angular momentum is the centre of rotation)

 

[pinsky]

Thanks for the replay.

I didn't understand you quite clearly. Can you please write the equation for the complete moment of inertia from the centre of the big disc when the small discs rotate freely along their axis.

I believe it will be more clear that way.

Tnx

 

[tiny-tim]

Can you please write the equation for the complete moment of inertia from the centre of the big disc …

the moment of inertia from the centre of the big disc isn't helpful

the angular momentum of the small disc about the centre of the big disc is its "intrinsic" angular momentum (about the centre of the small disc), plus an extra angular momentum as if all the mass of the small disc was at its own centre, moving with the velocity of that centre

 

[pinsky]

I must say, I'm still not following. :)

I got a replay on another forum that J from the point of the big discs center is

J = J_big + 2 m_small l² + m_small r_small²

For when small discs aren't rotatable. (l is the distance between the centre of the small disc and the centre of the big disc)

And when the small disc's are rotatable:

J =J_big + 2 m_small l²

with the explanation that in that case we can observe the small discs as point masses at distance l.

 

[tiny-tim]

angular momentum L_P about a point P is not J_Pω unless P is either C, the centre of mass (of the little disc) or the centre of rotation (of the little disc) …

in the general case, L_P = J_Cω + mPC x v_C

 

[pinsky]

So you're saying that the expression i wrote isn't true?

They kind of physically make sense. (never the less, intuition is often wrong :) )

 

[tiny-tim]

i'm saying that J_P isn't relevant if you aren't allowed to use it (because you have to use J_C)

 

[pinsky]

That is a experimental task in which i can measure the angular acceleration together with the force and the lever lenght.

What does the J i get by

J=F l/ \alpha

mean then? How do i get it analytically?

 

[tiny-tim]

that equation only works if P, the point about which you are measuring everything, is either the centre of mass (C) or the centre of rotation

(it usually is, but not in this case)

 

[pinsky]

Oh well, thank you for your time.

I can't say i understand quite, but i suppose time will make it more clear with the study of classical mechanics.