Momentof inertia and center of mass

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Discussion Overview

The discussion revolves around the concepts of moment of inertia and center of mass as presented in a physics textbook. Participants explore the mathematical formulations and implications of these concepts, questioning the validity and interpretation of integrals used in the derivations.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions the transition from discrete mass elements to continuous mass distributions in the context of moment of inertia, suggesting that the integration leads back to the original equation I = mR².
  • Another participant expresses confusion over the integration of the center of mass formula, arguing that it seems to imply a misunderstanding of mass constancy and the role of integration.
  • Concerns are raised about the treatment of the variable r in the moment of inertia calculation, with one participant suggesting that r should not be assumed constant for all mass elements.
  • A later reply clarifies that the summation of products in the center of mass calculation does not equate to the product of sums, indicating a misunderstanding of the mathematical principles involved.
  • One participant proposes that r is a function of variable mass, drawing an analogy to impulse and suggesting that changes in mass affect r.

Areas of Agreement / Disagreement

Participants express differing views on the interpretations of the integrals and the assumptions made in the derivations. There is no consensus on the validity of the approaches or the implications of the equations discussed.

Contextual Notes

Participants highlight potential misunderstandings regarding the assumptions of constancy in mass and distance in the context of rigid bodies, as well as the implications of integrating over variable quantities.

The Trice
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momentof inertia and center of mass!

in book of serway : he says moment of inertia of a body is m(r^2).---->(1)
is mass of the body and r is the distance of the body.
but for a rigid body we will divide into particles of very small masses so i = E(Mi * (Ri)^2)
E() is submission function to number if i Th particles.
and he says if we decrease this mi or delta m i to very small amount
so I = lim(mi->0) mi*(ri)^2 which will equal integration( r^2 dm).
SO WHAT DID HE DO IF I SOLVE THE INTEGRATION IT WILL BE I=MR^2 the same as the first equation(1) !?
and also i have the same problem in center of mass
he says if we want to find x coordinate of center of mass so:
x=E(mi*xi)/M and again if we deal with a rigid body of infinite of particles it will be :
x=lim(mi->0) (mi*xi)/M = integration( x dm)/M
so what! if we solve this integration it will be xcm= (x* M)/M
so xcm=x (and u don't have x because x is xcm that u want to find so what did he do with this stupid integration !)
 
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and also why did he make it integration( x dm)/M it seems tome as as mass in changing, but mass is constant for this particles!and even don't we get the constants out of integrtion soit would be xcm=x* integration(dm)/M so it would be 1=1!
and also in I=intefration(r^2 dm) why did he only change mi to dm and not also r^2 i mean he made the r is equal for all mass particles!
 
Last edited:


The Trice said:
in book of serway : he says moment of inertia of a body is m(r^2).---->(1)
is mass of the body and r is the distance of the body.
I = mr2 is true for a point mass, where r is the distance from the axis of rotation.
but for a rigid body we will divide into particles of very small masses so i = E(Mi * (Ri)^2)
E() is submission function to number if i Th particles.
and he says if we decrease this mi or delta m i to very small amount
so I = lim(mi->0) mi*(ri)^2 which will equal integration( r^2 dm).
SO WHAT DID HE DO IF I SOLVE THE INTEGRATION IT WILL BE I=MR^2 the same as the first equation(1) !?
No. In general it's not true that I = mR2 for a rigid body. It would be true if all the mass is at the same distance from the axis.
and also i have the same problem in center of mass
he says if we want to find x coordinate of center of mass so:
x=E(mi*xi)/M and again if we deal with a rigid body of infinite of particles it will be :
x=lim(mi->0) (mi*xi)/M = integration( x dm)/M
so what! if we solve this integration it will be xcm= (x* M)/M
so xcm=x (and u don't have x because x is xcm that u want to find so what did he do with this stupid integration !)
Careful. Σ(xi*mi) ≠ Σ(xi)*Σ(mi), except in special cases.
 


so r is a function in terms of variable mass right!?
like when i say impulse is integration(f dt) so that means as time chnages the force changes ,soin inertia it means as mass changes!? the r changes!
 

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