Talk:Moment of inertia

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1 Moment of mass
2 Rotational Inertia
3 Ix, Iyz, Ixx ?
4 calculations
5 doubts
6 Newton's second law
7 Product of Inertia
8 Reorganization
9 Moment of inertia# Angular momentum and torque
10 Angular momentum

[edit] Moment of mass

Whats the difference between moment of inertia and moment of mass if there is any? --Light current 03:49, 23 December 2005 (UTC)

[edit] Rotational Inertia

"Rotational Inertia" is the term that has been adopted by the College Board for the Advanced Placement Physics exams, beginning at least in 2002. I use that term in my teaching, because "Moment" of inertia is confusing to my students, since most high school students that I've encountered can't separate 'moment' in this sense from 'moment' of time. jpsi 10:11 March 9 2007

"Rotational inertia" is the term used in a popular sophmore physics textbook by Halliday & Resnick (Physics, 2nd Ed.,Chapter 12) as "a measure of the resistance a body offers to a change in its rotational motion about a given axis." Yes it is the exact same thing as "moment of inertia". To Tokerboy and Cyp below, i would reply that any professor worth his salt would not laugh, but rejoice that a student had made a conceptual break-thru in understanding how "mass" or "translational inertia" relates to rotational motion. Kenny56 12:50, 29 September 2005 (UTC)

No, they are not the same, but can be easily confused. I will explain...

First, let's just think about linear motion. How hard it is to stop an object is related to it's mass and it's velocity. Mass times velocity is inertia. So, you could say that mass, velocity, and inertia are each "a measure of the resistance a body offers to a change in it's linear motion". However, this does NOT mean that mass, velocity, and inertia are all the same thing.

Now, let's look at circular motion. How hard it is to stop an object is related to it's rotational mass (moment of inertia) and it's rotational/angular velocity (spin rate). Moment of inertia times rotational velocity is rotational inertia. So, you could say that moment of inertia, rotational velocity, and rotational inertia are each "a measure of the resistance a body offers to a change in it's rotational motion". However, this does NOT mean that moment of inertia, rotational velocity, and rotational inertia are all the same thing. StuRat 20:52, 1 October 2005 (UTC)

Are you confusing inertia and momentum? Mass times velocity is the definition of momentum, not inertia. Mass is the amount of matter that an object contains; Inertia is a property of matter related to its resistance to change of state, either at rest or in motion, unless acted upon by an external force. Momentum is a measure of the change in velocity that results when an object with mass is acted upon by an external force, and it is also related to the kinetic energy of an object in motion.

Whether in rotation or translation, the mass of the object is the same. But it is the inertia property of the mass that is different in each case, and it is calculated by different formulas to account for either rotation or translation. The "rotational mass" is a confusing term and I am reverting back to Halliday and Resnick's usage quoted above. Kenny56 05:07, 3 October 2005 (UTC)

I think your "mass quantifies the inertia of an object" is more confusing than the phrase "rotational mass". StuRat 23:27, 5 October 2005 (UTC)

Okay, here's another way to consider this concept. Mass is a physical quantity; it measures the amount of matter that an object contains. Inertia is a physical property; it describes the persistant behaviour of matter in uniform motion or at rest.

The strength of this property is related to the quantity of matter, so an object of greater mass has more inertia than one of lesser mass. In this sense the units of measure of mass can be used to quantify inertia by providing numeric values and units for comparing the inertias of different objects.Kenny56 04:26, 6 October 2005 (UTC)

I still find the original intro much clearer:

Moment of inertia is a measure of the resistance of a physical object to increases or decreases in its rate of spin about its axis (angular acceleration). Moment of inertia is to rotational motion as mass is to linear motion. In this sense, the moment of inertia may be thought of as the rotational mass.

According to Keith R. Symon in his book Mechanics, Addison-Wesley, 1971, the analogy between mass and moment of inertia breaks down due to three differences:

1. Moment of inertia is a tensor, but mass is a scalar.

2. The inertia tensor is not constant with respect to axes fixed in space, but changes as the body rotates, whereas mass is constant.

3. There is no symmetrical set of three coordinates analogous to X-Y-Z with which to describe the orientation of a body in space. As a result the solution to the rotational equations of motion requires different methods than those used for linear motion.

I'm a big fan of using intuitive concepts in explanations whenever possible, however in the case of rotational motion I think it would be too misleading to use "mass" in place of "inertia" in light of Symon. Kenny56 03:51, 11 October 2005 (UTC)

1. As far as I can tell, mass and an INDIVIDUAL moment of inertia (for a given axis) are both scalars, which are considered to be the simplest type of tensor.

2. I assume this means the "moment of inertia tensor", not the "inertia tensor", which are different things. If this means there is a different moment of inertia about different axes of rotation, then I agree, that is a difference. It is explained in great detail later in the article, though, so any misconception here will be covered later. I'm not opposed to adding a disclaimer to this effect up top, however, if you think it would help.

3. I'm not sure what this is saying. There certainly seem to be similar methods to me, such as $F = ma\,$ and the rotational equivalent, $T = I{\alpha}\,$ , or linear inertia, $mv\,$ , and rotational equivalent, $I{\omega}\,$ .

Also, the mere fact that he was listing differences means he also thought there were substantial similarites (you wouldn't bother to say things are different if it's obvious to everyone that they are). StuRat 16:11, 11 October 2005 (UTC)

What on earth does "Moment of inertia is the name for rotational inertia" mean? If there is no difference, then say they are synonyms. This makes it seem like the page should be moved to rotational intertia. Tokerboy

Suppose it's because, for some inexplicable lack of reason, rotational inertia isn't (officially) called rotational inertia. Perhaps, so that if a new person asks a bunch of professors about rotational inertia, they can look confused, and say there is no such thing, and later on laugh together behind the new persons back, about how that stupid little new person didn't even know that it's called moment of inertia, not rotational inertia. From then on, all the professors can mock and play practical jokes on the new person, with the new person having no idea why. -(Cyp)

I don't believe they are the same. Think of the case of linear inertia, which equals mass times velocity. Rotational inertia would then equal rotational mass (called the moment of inertia) times the rotational velocity. StuRat 01:49, 23 September 2005 (UTC)

I think the definition of I should only have one integral sign, i.e $I = \int r^2\,dm$ instead of $I = \int\int\int r^2\,dm$ , because there is only one integral variable (dm). Or am I wrong? --Stw 22:03, 25 Jan 2004 (UTC)

I'm quite sure I'm right on this, so I changed it now --Stw 21:49, 1 Feb 2004 (UTC)

User:Stw you are wrong. dm is the mass density over the volume V. The triple integral is the customary notation for a volume integral.

Regarding the formula which follows the phrase "These quantities can be generalized to an object with continuous density":

$\mathbf{I}=\int_V \left[\left( \mathbf{r} \cdot \mathbf{r}\right)\mathbf{\delta} - \mathbf{r} \otimes \mathbf{r}\right]\ dm$

The delta is NOT explained, and in fact should be replaced with the 3x3 Identity matrix.

[edit] Ix, Iyz, Ixx ?

The notation "I_x= I_yz = moment of inertia about the axis parallel to the X-axis or in the plane parallel to the YZ plane" confused me at first - later on in the article, the notation I_yz is used for the non-diagonal elements of the tensor.

Wouldn't it be better to have I_x stand on its own in the initial paragraph, to be consistent? I am in doubt whether it would be useful to include I_x = I_yy + Izz ?

1st: done; 2nd: seems wrong, or again some other notation.--Patrick 00:47, 4 October 2005 (UTC)

I am familiar with $I_{x}\;$ , not the $I_{xx}\;$ notation, for the moment of inertia about an axis parallel to the X-axis, thru the centroid of the object. Are you using $I_{x}\;$ to mean the moment of inertia directly about the X-axis ? If so, this can be simplified by moving the origin to the centroid of the object in question, in which case they are the same thing. StuRat 10:10, 5 October 2005 (UTC)

I used the same notation as later on. If they are the same we can add that.--Patrick 22:07, 5 October 2005 (UTC)

I added the simplified versions. StuRat 23:24, 5 October 2005 (UTC)

[edit] calculations

It might be important to know that the only way of calculating the moment of inertia of an arbatrary polygon is triangulate, calculate moment of each triangle about center of mass of the triangle, use off-axis equation to get it about center of mass of polygon, and add them up. This is complicated to implement however.

[edit] doubts

why moment of inertia is related to rotation.
- Cyborg 18:20 APRIL 01 2006 (UTC)

That's things only thing it's related to. It's not related to much else, except for perhaps mass if you compare linear motion and circular motion. --M1ss1ontom a rs2k4 ^{(T | C | @)} 22:30, 29 July 2006 (UTC)

[edit] Newton's second law

In the torque section the equation F = ma is given, but there is no mention or link provided to Newton's second law.

[edit] Product of Inertia

Product of inertia is a redirect to this page, yet this page does not mention it, at least I don't think it does. Wouldn't it help to have a section about this for those that are following this redirect? I know I would appreciate it, as I know little on the subject and was looking to learn more. Thanks. --Zsinj^Talk 17:24, 13 April 2006 (UTC)

[edit] Reorganization

I have reorganized and consolidated the article essentially under two headers: the scalar and the tensor. Within each the exposition is similar. Very little content was removed or added (except see below); some was changed in a copyedit way and some moved to reflect the reorganization. I also tweaked some math markup a bit.

I think it reads nicer now, and I hope everyone agrees. One section where I anticipate some may wish to revert somewhat is the notational comments about the scalar integration definition. I removed all notations about "infinitesimals", figuring if someone need to elaborate on that, they would consult integration references. But I do understand how the actual definition used, the one in terms of m, mass, may seem to be somewhat ambiguous. So have a go at it and feel free to discuss here. Baccyak4H 16:42, 26 October 2006 (UTC)

Could there be descriptions without calculus, here and in many physics articles?

I'm not sure we should remove calculus when it's pretty integral to the description (*resisting bad pun*). However, I would support alternate descriptions that don't require calculus knowledge. The issue is that many people who understand this subject don't know how to explain it without calc. - EndingPop 10:49, 1 November 2006 (UTC)

There are such descriptions at the top of the article. I am not sure there is a description that allows one to compute

I

that can do any better than calling it a weighted sum, which is merely a special case of an integral anyway. Baccyak4H 15:52, 1 November 2006 (UTC)

[edit] Moment of inertia# Angular momentum and torque

Out of the blue comes $\boldsymbol{\omega}\cdot\mathbf{r}_i=0$ for all i. This is only true in special cases. Is this on purpose or by mistake? --P.wormer 10:10, 31 January 2007 (UTC)

[edit] Angular momentum

Please stop changing the angular momentum definition. Angular momentum is defined with respect to an origin, NOT an axis of rotation. Egendomligt 03:03, 5 March 2007 (UTC)

I second this--P.wormer 15:36, 5 March 2007 (UTC)

I agree that angular momentum is strictly a vector quantity and thus includes the axis in its definition, but beginners often think of it as a scalar quantity about an axis of rotation. This is equivalent for basic problems, and is a much simpler concept for a beginner. Could we not allow both (with a caveat)? Dbfirs 22:16, 14 March 2007 (UTC)

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