Calculate the rotational inertia through integration

[jamie.j1989]

hello, I've gone through countless textbooks and website trying to find the deeper relation between the mass and the radius when calculating the rotational inertia of an object, however know one is giving it up. I know how to calculate the rotational inertia through integration, and i know how to use the equation, but why does I=mr^2.

 

[radou]

I = mr^2 is the moment of inertia of a point mass rotating around an axis. It follows almost directly from the definition of the moment of inertia.

 

[Cleonis]

hello, I've gone through countless textbooks and website trying to find the deeper relation between the mass and the radius when calculating the rotational inertia of an object, however know one is giving it up. I know how to calculate the rotational inertia through integration, and i know how to use the equation, but why does I=mr^2.

In circumnavigating motion the following quantity is a conserved quantity: $$\omega r^2$$
(The greek letter $$\omega$$ stands for the angular velocity)
That is: if the centripetal force that sustains the circumnavigating motion has reduced the radial distance by half then the angular velocity has doubled.

Angular momentum is defined as $$m \omega r^2$$
Angular momentum is defined that way to capitalize on the conserved quantity $$\omega r^2$$
In the absence of a force linear momentum is conserved; in the absence of a torque angular momentum is conserved.

So your question translates to: in circumnavigating motion, why is the quantity $$\omega r^2$$ conserved?
To some extent that question is answered by the https://www.physicsforums.com/showpost.php?p=2701193&postcount=1" in a recent physicsforums thread.

 

[sgsawant]

As radou has said it's the definition of moment of inertia. In principle you can't question the definition. Definition is like axiom in Maths.

But perhaps you would better digest it if you see an analogy. Continuing to talk on lines similar to Cleonis, we can go reverse from your understanding of angular momentum ( I hope you find the definition of angular momentum more palatable).

L=r X mv

But v=rw. Therefore L=mr^2w.

Thus you may justify Moment of Inertia (I like to call it angular mass) to be equal to mr^2.

If you are equally confused about why angular momentum is defined so, try going back from kinetic energy of body moving in a circle. It is (1/2)mv^2=(1/2)mr^2w^2=(1/2)Iw^2.

By analogy, the current definition of moment of inertia has the same relationships with angular velocity, as those between normal mass and linear velocity.

Regards,

-sgsawant