# Is Moment of Inertia, always, independent of angular velocity? [closed]

I once heard in a ted talk that the elementary particles like electron and proton can exist in two different positions at the same time.

Now,I'm trying to understand a rotating rod from this perspective:

Consider a rod rotating about an axis perpendicular to the plane of the rod and passing through the centre. Increase the angular velocity of the rod to very high values (tending to infinity(hypothetical case)).

What we( a frame from ground) would observe or what conclusions can we draw:

• we could say that the rotating rod looks pretty much like a disc(And I'm not sure if it would look like a stationary disc or rotating disc).

• The above point can be understood easily,if we treat rod like a fast moving electron where it has tendency to exist at two positions.Now that the rod is in rotational motion (not something like the motion of an electron ,precisely.) it would have a tendency to exist at different positions (something like drawing two diameters of the circle traced by the rod, and saying that the rod exists at both the diameter)

As rod is moving with very high(almost infinity) angular velocity we could say that the rod exists at all the diameter of the circle.If you put your finger at any point on the circle, traced by the rod, you would feel the rod because (since rod has very high angular velocity) it exists at many different positions at the same time.

I think we can consider this rod (with high angular velocity) to be a solid object,precisely, a disc.

CASE I: I'm assuming that the disc(so formed from observers frame) to be rotating.

So,finally we have a rotating disc from observers frame.

Points we know about a rotating disc and a rotating rod:

• Moment of inertia of a rod about an axis through the centre and perpendicular to the plane=[M(L^2)]/12=P
• Moment of inertia of a disc about an axis through the centre and perpendicular to the plane=[M((L/2)^2)]/2=(3p)/2

How has the moment of inertia of the rod changed like that?

We see that the length of rod hasn't changed(Assuming the rod to be a ,strictly, rigid body) and since moment of inertia has increased, does this mean that mass has increased?

Does this mean that moment of inertia is dependent on angular velocity(if it is high enough).

CASE II:Now,I'm assuming that the disc is a stationary one:

In such a case moment of inertia of the disc(so formed from observers frame) would be zero.

How's this possible?

How can a body lose moment of inertia due to very high(angular velocity)?

IS MY UNDERSTANDING AND CORRELATION CORRECT ?

If wrong please explain where I have misunderstood the concepts.

## closed as unclear what you're asking by sammy gerbil, Qmechanic♦Jul 11 '18 at 18:34

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I once heard in a ted talk that the elementary particles like electron and proton can exist in two different positions at the same time.

Saying that particles can exist in two places at once is just people trying to explain quantum mechanics to the lay person. This is not actually what is happening. According to traditional interpretations, particles like electrons do not have any defined position until it is measured to be at some position.

As rod is moving with very high(almost infinity) angular velocity we could say that the rod exists at all the diameter of the circle.If you put your finger at any point on the circle, traced by the rod, you would feel the rod because (since rod has very high angular velocity) it exists at many different positions at the same time.

Classically, if an object is in motion it does not mean it exists in multiple positions. Even something moving at a high velocity only exists at one position in one instant of time.

I think we can consider this rod (with high angular velocity) to be a solid object,precisely, a disc.

Just because due to our biology a fast rotating rod looks like a disk, it does not mean the spinning rod is actually a disk. It is still just a spinning rod. In other words, a spinning rod is not the same thing as a stationary disk. After this point, your other assumptions and cases do not need to be persued any further.

• So,all those statements given in the ted talks about particles existing in two different positions at the same time not true ? or they mean something else? – Banchin Jul 11 '18 at 14:15
• @Banchin Like I said, they were probably trying to speak at a lighter level so that people could understand what was being talked about. While I wouldn't say it is completely off the mark as far as interpreting quantum mechanics, it for sure is not precisely correct. Ted Talks are intended for general audiences, not experts. If you can supply a link to the talk then maybe I could be more help in interpreting what was said. – Aaron Stevens Jul 11 '18 at 14:18
• Sorry,I couldn't find the talk,it was long ago.But,I surely remember the three phenomena in Quantum mechanics which he explained and one of it was related to our topic.I remember the statement something like this:It is possible for subatomic partciles to exist in two different positions at the same time(and I think He even added a point about quantum state also).Has quantum state got anything to do with our topic? – Banchin Jul 11 '18 at 14:31
• @Banchin I do not think the question of a macroscopic rotating rod needs to be approached through quantum mechanics. I was just addressing this because it seemed like you were basing all of your thoughts on this idea of particles being in two locations at once. The way you applied this idea was in saying that high a high velocity rotating rod exists at multiple locations. The premise is wrong (that particles can exist at two places at once) and the application is also wrong (we cannot put things at multiple places in space anywhere we want to). – Aaron Stevens Jul 11 '18 at 15:13

We can't claim that in a limiting case of fast rotation the rod becomes a disk. It certainly can't be a stationary disk, as a stationary disk would be fairly easy to rotate out of its plane (an axis of rotation that forms a diameter for the disk), but for an "infinitely" fast rotating rod, it would be impossible to rotate out of the plane of the disk. Another thing to note is that moment of inertia is intrinsic to the geometry and mass of the object, and even a stationary object has a moment of inertia (just like a stationary object has mass).

Thinking about the rotating rod as becoming a rotating disk is more interesting, though there is ultimately no clear reason we should be able to think of a fast moving rod as a disk. Classically speaking, the rod is really only takes up one given diameter of the circle at any given time.

There is perhaps a way to think about the rod as a disk though, in the limiting case of very fast rotation we could think of the "disk" as rotating at $\frac{2}{3}$ the rate of the rod. This would maintain the same angular momentum as the "rod picture", so is in some sense a valid way to look at the situation.

Finally, I'm not quite sure what this has to do with QM. While in QM we do speak of particles "being in multiple places at the same time", it is a wholly different effect from things moving very quickly so as to appear to be in multiple places at the same time. A better way to think about it is as a stationary particle having a distribution of probabilities of being in various places when the location of the particle is measure (e.g. by a photon interacting with the particle).

You had a great idea of correlating the motion of electrons and protons with a rigid rod moving at a very high speed. But the two cases have no correlation whatsoever. First of all I would like to point out where you went wrong. You said that humans wont't be able to perceive the very high velocity of the rod and think that it looks like a disk. This observation is completely due to the fact that the response of the human eye or other body parts is not fast enough for the high speed of the rod. Now think it in this way, take a snapshot of the rotating rod (make sure that the camera has a shutter speed $\rightarrow0$). What will you observe? You will surely observe a single rigid rod. Hence you can say that the disk does not exist at two place at one time. All these motions obey Newton's laws unless you are considering speeds close to the speed of light.

Now for the motions of electrons and protons, you cannot apply Newton's laws to calculate their paths. The very question of trajectory of electrons and protons do not make sense anymore. This is because they have a wave nature associated with them, resulting in a wave particle duality. This was first shown by De Broglie, and he said that all particles have a De Broglie wavelength. The wave nature only become significant if the De Broglie wavelength is of the order of the dimensions of the body or more. It is calculated as $\lambda=\frac{h}{mv}$ where $h$ is the Plank's constant. Hence for bodies with large masses, or the bodies which we can see with our naked eyes have no wave nature.

Now there is different world of quantum mechanics, which is guided by the Heisenberg's uncertainty relation which is applicable to the microscopic particles i.e. protons, electrons etc. They do not have a fixed deterministic position, and in fact has a probability distribution as a position function. Hence they have a finite probability of being found at more than one position.

So both of them have different reasons.

Now there is one more interesting case. If you imagine the rod rotating at a speed close to light, the rod will lose its rigidity. This is because the sense of motion of the rigid body, is transferred with the body due to elastic vibration in the material. The elastic vibrations has a speed of the order of speed of sound in that material which is much less than the speed of light. You can imagine this situation as one end of the rod is moving at a high speed, but the other end still does not know that the rod has started moving, because vibrations are slower. Hence the body is no longer rigid in nature. This also has some "funny" consequences.

Finally I want to make the point that moment of inertia is independent of angular velocity in non relativistic speeds, till the point the body remains rigid. If the system is in relativistic domain, then mysterious things happen.

I hope this helps.