Moment of inertia (multiple rotating axis)

In summary, when small discs rotate around their centre, the equation for the moment of inertia changes. Jbig is still the total moment of inertia, but the addition of the msmall l2 and msmall rsmall2 terms changes the equation to J=Jbig+2mL2+ml2rsmall. This equation is only valid if P, the point about which you are measuring everything, is either the centre of mass (C) or the centre of rotation.
  • #1
pinsky
96
0
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 Jbig=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=JM + Jm

where Jm= 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 [Broken]
 
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  • #2
hello pinsky! :smile:
pinsky said:
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? :biggrin:

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) :wink:
 
  • #3
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
 
  • #4
pinsky said:
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 :redface:

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
 
  • #5
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 = Jbig + 2 msmall l2 + msmall rsmall2

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 =Jbig + 2 msmall l2

with the explanation that in that case we can observe the small discs as point masses at distance l.
 
  • #6
angular momentum LP about a point P is not JPω 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, LP = JCω + mPC x vC
 
  • #7
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 :) )
 
  • #8
i'm saying that JP isn't relevant if you aren't allowed to use it (because you have to use JC) :wink:
 
  • #9
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

[tex] J=F l/ \alpha[/tex]

mean then? How do i get it analytically?
 
  • #10
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)
 
  • #11
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.
 

1. What is moment of inertia and how is it related to multiple rotating axis?

Moment of inertia is a measure of an object's resistance to changes in its rotational motion. When an object is rotating about multiple axes, the moment of inertia is the sum of the individual moments of inertia about each axis. This is due to the principle of superposition, which states that the effects of multiple forces on an object can be calculated by summing the individual effects of each force.

2. How is the moment of inertia calculated for an object with multiple rotating axis?

The moment of inertia for an object with multiple rotating axis can be calculated by using the parallel axis theorem. This theorem states that the moment of inertia about a particular axis is equal to the moment of inertia about a parallel axis passing through the object's center of mass, plus the product of the object's mass and the square of the distance between the two axes.

3. What factors affect the moment of inertia for an object with multiple rotating axis?

The moment of inertia for an object with multiple rotating axis is affected by several factors, including the mass distribution of the object, the distance between the axes, and the orientation of the axes relative to the object's shape. Objects with more mass concentrated towards the axis of rotation will have a smaller moment of inertia compared to objects with more mass distributed further from the axis.

4. How does moment of inertia affect an object's rotational motion?

Moment of inertia plays a crucial role in an object's rotational motion. Objects with a larger moment of inertia will require more torque or force to achieve the same rotational acceleration as an object with a smaller moment of inertia. This is why objects with a larger mass or mass distributed farther from the axis of rotation tend to rotate slower compared to objects with a smaller mass or mass closer to the axis of rotation.

5. Can the moment of inertia be changed for an object with multiple rotating axis?

Yes, the moment of inertia for an object with multiple rotating axis can be changed by altering its mass distribution or the distance between the axes of rotation. For example, a figure skater can change their moment of inertia by bringing their arms closer to their body, reducing their moment of inertia and increasing their rotational speed. Similarly, a diver can change their moment of inertia by changing their body position in the air, allowing them to perform different types of dives with varying degrees of difficulty.

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