Momentof inertia and center of mass

[The Trice]

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 !)

 

[The Trice]

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!

 

[Doc Al]

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(x_i*m_i) ≠ Σ(x_i)*Σ(m_i), except in special cases.

 

[The Trice]

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!