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

 

[PJC01]

Hello,

I understand your frustration in trying to find a deeper understanding of the relationship between mass and radius when calculating rotational inertia. The equation I=mr^2 is a simplified version of the more comprehensive equation for rotational inertia, which takes into account the distribution of mass within the object.

To calculate rotational inertia through integration, you must take into account the mass of each infinitesimal element of the object and its distance from the axis of rotation. By integrating these elements over the entire object, you can find the total rotational inertia.

The simplified equation I=mr^2 assumes that the mass is evenly distributed throughout the object, resulting in a constant value for the distance from the axis of rotation (r). This simplification is often used in introductory physics courses to make the concept more approachable.

However, in reality, the distribution of mass within an object can greatly affect its rotational inertia. For example, a thin hoop and a solid disk with the same mass and radius will have different rotational inertias due to their different mass distributions.

In conclusion, while the equation I=mr^2 is a useful simplification, it is important to understand that rotational inertia is a more complex concept that takes into account the distribution of mass within an object. I hope this helps to clarify the relationship between mass and radius in calculating rotational inertia. Keep exploring and asking questions - that's what being a scientist is all about!