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I'm learning about rotational motion and the moment of inertia. Unlike inertia that I learned before, there is a formula to calculate rotational inertia. I'm having trouble understanding why it's possible to calculate inertia for a rotating object, but not a regular moving object. After doing research, not only is there no formula for normal objects, but different interpretations of what inertia is. Is there a fundamental difference between the moment of inertia and the inertia of an object, or am I misunderstanding something?

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    $\begingroup$ Probably the concept of inertia is not what you want for most circumstances. You want the mass. Depending on whether it's non-relativistic or relativistic you then calculate how the object responds to force. $\endgroup$
    – Dan
    Dec 22, 2021 at 4:57
  • $\begingroup$ The idea of moments carries over from math (to characterize functions, and where we get the idea of mean, variance, etc) to physics: en.wikipedia.org/wiki/Moment_(physics) $\endgroup$ Dec 23, 2021 at 5:00

4 Answers 4

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Classically, the inertia of something is just its mass. If you want an analogous equation, just integrate the mass density $\rho$ of the object over the volume of the object:

$$m=\iiint \text dm=\iiint\rho\,\text dV$$

Compare this to what you usually see in introductory physics as $$I=\iiint r^2\,\text dm=\iiint r^2\rho\,\text dV$$

which, for a given axis, is one element of the moment of inertia tensor.

Is there a fundamental difference between the moment of inertia and the inertia of an object?

Yes. The inertia of an object does not depend on where the mass is within the body, only on how much mass there is. The moment of inertia about a given point does depend on how that mass is distributed about the point / axis you are calculating the moment of inertia about.

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    $\begingroup$ As a person never schooled in physics... this is (MoI) comparing a figure skater doing a spin with arms and legs spread about their axis of spin, vs one who's arms and legs are tight to the body (axis)? $\endgroup$
    – CGCampbell
    Dec 22, 2021 at 15:49
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    $\begingroup$ @CGCampbell: Yes, basically. $\endgroup$
    – Vikki
    Dec 22, 2021 at 17:37
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    $\begingroup$ I suppose, by the general definitions of a moment, this makes the MoI the second moment of inertia (with center of mass being the first)? $\endgroup$ Dec 23, 2021 at 4:59
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    $\begingroup$ @onetoinfinity Yes. I think the term is second moment of mass though. $\endgroup$ Dec 23, 2021 at 5:19
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BioPhysicist is right.

Your confusion comes from the fact that the word "inertia" is not a "technical" term. It refers to a notion that can apply to many things, physical or psychological, like some person being slow to act.

"Moment of inertia" is a precise technical term, related to rotational motion. The corresponding technical term for what you call "regular motion" is just "mass".

In fact, your question is deeper than meets the eye. Technically, the analog of "moment of inertia" for translational (your "regular") motion, rather than rotation, is "inertial mass". The tendency of an object to gravitationally attract another one and be attracted by it is its "gravitational mass".

Experimentally, one has always found, since Newton, that "inertial mass" and "gravitational mass" are equal.

But there was no deep reason for this identity.

Postulating that this identity is a fundamental principle of Nature is at the very origin of Einstein's theory of General Relativity.

In fact, there are some theories of gravitation (for instance Tensor-Vector-Scalar Gravity) that differ from General Relativity. For these theories, "inertial mass" and "gravitational mass" are not identical. Of course, if it exists, the difference is so small that no experiment on Earth has been able to measure this difference, but it could explain some difficulties with astronomical observations.

If these theories are correct, then one should, in principle, use the phrase "inertial mass" rather than just "mass" for the analog of "moment of inertia", and "gravitational mass" to describe the way an object is attracted, say, by Earth's gravity.

In practice, it makes no difference, but in principle, it really is a very fundamental distinction.

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I'm assuming this is from the standpoint of first-semester introductory physics, so I don't think it's appropriate at all to talk about tensors or multivariable integration.

Forget about what the complicated functional description of $ I $ looks like for some object for just a moment and call it some constant. Generally when teaching a basic physics class I just like to give the following analogous statements of Newton's 2nd Law in simple cases $$ \vec{F} = m \vec{a}, \\ \vec{\tau} = I \vec{\alpha}, $$ and note that if mass (or intertia, they mean the same thing) is a resistance to a change in linear motion, then moment of intertia is a resistance to a change in rotational motion. Since both $ m $ and $ I $ are a positive constants for an object, they are the constants that control how easy/hard it is to accelerate the translation and rotation of the object, respectively. That's all. The moment of inertia is acting as the thing that impedes rotational acceleration, just as mass impedes linear acceleration. It just so happens that $ I $ changes a lot depending on mass, radius, shape, distribution of mass, etc.

The origin of this strange word moment will become clear at higher levels as you study the inertia tensor, but when first introduced to it in (I presume) intro physics, stick to the basics of the above equations. If it helps, sometimes people say rotational intertia instead of moment of intertia, so you can use the perhaps more illuminating terms $ m = $ translational inertia, $ I = $ rotational inertia.

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Inertia in general, is a term used to describe how easily something changes its speed or direction of movement, when a force is applied to it.

  • Lower inertia = easier for a given force to move/change it.
  • Higher inertia = harder for a given force to move/change it.

In general, for example, we would expect objects with larger "inertia", to move or change velocity less easily than an equivalent body with smaller "inertia".

When we look at movement in a line, we find that the ease with which the object changes speed, seems to be proportional to the weight it has due to gravity. That's an odd way of phrasing it, and worth looking at closely.

Suppose we say a small object weighs 1 kg, or has mass 1 kg. What do we ultimately mean by that? In simple terms, we mean that gravity pulls it equally as it would any other point object that we also label "1 kg". We know 1 kg is 1 kg, because gravity attracts it like it attracts some other object we labelled 1 kg in the past. Because if we put the object on a simple set of scales, it balances if we put another object we called "1 kg" on the other side. That's when we know this object is or isn't 1 kg as well.

(Or it was, while the kilogram was defined by a platinum mass in Paris, until not long ago!)

So there's some deep truths about the universe here. Objects moving in a straight line have inertia - some change speed or direction easier than others when we apply the same force - but this inertia seems to be the same as the ease they change speed or direction due to gravity. So we identify 2 kinds of inertia, and give them the name "mass". In simple terms, inertia when gravity acts on an object is called its gravitational mass. Inertia when another force acts on the object is called its inertial mass. So far, they seem to be the same thing, but that's still something physicists are checking.

But there's other situations where the same force moves one object easier than another. Imagine three playground child's roundabouts. One is empty, one has 500kg of weights at the centre, and one has 500kg weights at the rim. All 3 can turn easily, but spinning them up takes a lot more force for the 3rd than the other two. So this is also a form of inertia.

But it's not just how much mass there is,otherwise the 2nd roundabout would be just as hard to spin. Its something about how the mass is positioned as well.

It turns out that you can do a calculation that starts with "what masses you have where, and where are you spinning it around", and ends with a single number that says how hard it is to change its speed of spinning.

By analogy with other forms of inertia, we name this property, inertia as well. We call it "moment of inertia", because moment is a physics term related to turning, and turning forces. But you can't just take a scale to measure it. You need to calculate it, or measure it compared to some standardised "moment of inertia" objec, like the 1 kg block.

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    $\begingroup$ You have it all wrong. Inertia is not the ease to change speed, on the contrary, it is what reduces the change of speed, the acceleration, which is inversely proportional to the inertial mass When the force is due to gravity, the force is proportional to the gravitational mass. Einstein Strong Equivalence Principle states the two masses are identical, hence all objects have the same acceleration when the only force is gravitation. But a priori the two masses could be different, as in other theories of gravitation. $\endgroup$
    – Alfred
    Dec 22, 2021 at 17:51
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    $\begingroup$ I said it describes how easy it is for a given force to change velocity. Which is correct. Low inertia -> easy to change velocity. High inertia -> difficult to change velocity. Which I said, see 2nd paragraph. But added 2 bullets so its absolutely clear, as.well. $\endgroup$
    – Stilez
    Dec 22, 2021 at 20:07
  • $\begingroup$ Your bullet says "Higher inertia=harder for force to move/change it", which is right. But a bit lower you write that "this inertia seems to be the same as the ease they change speed or direction due to gravity" so inertia here is "ease", which is wrong. $\endgroup$
    – Alfred
    Dec 22, 2021 at 21:47
  • $\begingroup$ You're reading a direction into something, which there's no basis to do. The ease with which something moves under a force, does indeed seem to be the same ease with which they move under gravity. $\endgroup$
    – Stilez
    Dec 23, 2021 at 5:20
  • $\begingroup$ What you are saying in the above comment is that the inertial mass is the same whether the force is due to gravity or to another cause (EM force, for instance) which again, is correct. But the "gravitational mass" is not the inertial mass in case of force of gravitation. The "gravitational mass" is the source of the force of gravitation. And in principle it need not be equal to the inertial mass. It is in practice, as per observations. It is in principle in General Relativity. It is not for other theories of gravitation, but the difference is too small to be detected. $\endgroup$
    – Alfred
    Dec 23, 2021 at 15:21

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