I must have missed something about the standard definition of moment of inertia. how did they come up with the basic formula I= the integral of radius^2times dm?

 

Cheers

It's the integral of m*r^2, not r^2 times dm.

 

Torque = F*r

 

If you have a mass dm at r, then

 

F = dm*a

 

but a = r*alpha

 

where alpha is the angular acceleration (rad).

 

Put these together and you get

 

Torque = F*r = dm*r^2*alpha

 

This is analogous to F=m*a, but with torque and alpha replacing m and a, so that means for rotational dynamics mass is replaced by the moment of inertia dm*r^2.

 

To get it for the whole body we integrate over all m.

 

That okay?

yes thank you!