Rotational inertia and areal velocity

[Urmi Roy]

One expression for angular momentum is L=2m*(dA/dt),where m is the mass of the body in concern and (dA/dt) is the areal velocity of this rotating body.

What is the logical physical explanation (or description) for this?

Is there any other physical importance of this expression (or result) other than the fact that it helps to prove Kepler's second law?

 

[tiny-tim]

One expression for angular momentum is L=2m*(dA/dt),where m is the mass of the body in concern and (dA/dt) is the areal velocity of this rotating body.

What is the logical physical explanation (or description) for this?

Hi Urmi Roy!

For a complete explanation and proof, see http://en.wikipedia.org/wiki/Areal_velocity

Is there any other physical importance of this expression (or result) other than the fact that it helps to prove Kepler's second law?

I've never come across areal velocity except in relation to Kepler's laws, so no I don't think so.

 

[Urmi Roy]

Hi tiny-tim !

Thanks for the link. I went thorugh it,but you see, it wasn't the proof that I was looking for,I would rather like to know if this formula has any 'deep inner meaning and concept about physics' or is it just a formula that physicists derived to help in practical life.

Again,thanks for confirming that this formula is used mainly for proving the validity of Kepler's 2nd law.

 

[vin300]

According to the diagram and explanation there, if the angular velocity is pi rad/sec, then
r x v gives zero and no areal velocity, which is wrong. The velocity vector is not as meant there, it is tangential to the path and includes the white portion between the purple path and the green portion.

 

[BobG]

According to the diagram and explanation there, if the angular velocity is pi rad/sec, then
r x v gives zero and no areal velocity, which is wrong. The velocity vector is not as meant there, it is tangential to the path and includes the white portion between the purple path and the green portion.

Delta t is not equal to 1 second, though.

And, velocity is the change in position over some increment of time.

In the case of limits, that increment of time is much, much smaller than the time it takes to travel halfway around the circle no matter how fast you make the angular velocity.

If there's any real discrepancy in the diagram, it's that r(t + Dt) is too far away from r(t), but you have to draw it wrong or no one could see any difference between the two.

i.e. - the distance between the two position vectors should be so small that the secant line is tangent to the path.

 

[Urmi Roy]

Thanks for your help in regard to 'areal veocity'.

By the way, I was just wondering if there is any way to continuously change the linear inertia (mass) of a body,just as it is possible to interchange moment of inertia of a body(in the example of the spinning dancer,she changes her moment of inertia by pulling in or pushing out her arms).

Please let me know if this is possible at all.

 

[BobG]

Thanks for your help in regard to 'areal veocity'.

By the way, I was just wondering if there is any way to continuously change the linear inertia (mass) of a body,just as it is possible to interchange moment of inertia of a body(in the example of the spinning dancer,she changes her moment of inertia by pulling in or pushing out her arms).

Please let me know if this is possible at all.

Yes, it is. In fact, that's how a rocket engine works. Mass from the rocket is thrown out the back at incredibly high velocities and, thanks to conservation of linear momentum, the rocket moves the opposite direction. Obviously, the mass of the rocket must be decreasing if it's being thrown out the back.

There's even a tutorial talking about it in the Math and Sciences Learning Material forum:
https://www.physicsforums.com/showthread.php?t=199087

I'm not sure that's exactly what you had in mind, but it is the closest you would get. With linear motion, any reduction in mass has get rid of the mass somewhere.

 

[Urmi Roy]

Thanks BobG.

I had read about this concept,though in much simpler terms when I was in class 11,but it had completely skipped my mind.

What led me into thinking about this issue was that,since every linear quantity in mechanics has a rotational analogue,(and vice versa),which are similar in their properties,this property of rotational inertia that it can be varied in magnitude,may also exist in linear inertia.

Ofcourse,this isn't exactly a property as such.Anyway,the example of the rocket provides the best comparison.

Thankyou very much!

 

[Urmi Roy]

I've just 2 more regions in regard to rotational inertia to clear up.The first one,I'd like to speak of here...
During the derivation of various expressions related to rotational mechanics (e.g. the expression for net rotational kinetic energy of a body, derivation for net torque acting on a body),as done in my book, the expression of moment of inertia (I=mr^2) suddenly springs out,after we sum the individual kinetic energies or torques acting upon the infinitesimal particles (depending on the derivation we're doing).

Then,the book says,that since the formulae for rotational mechanics and linear mechanics are similar,except that 'mass' is replaced by 'moment of inertia',moment of inertia is called the rotational analogue of mass.

Is this just by coincidence,since as I said, the expression for moment of inertia just springs up incidentally in almost all derivations--or is there any bigger reason for this?

I've been told that these can be explained by lagrangian dynamics and noether's theorem,but surely there must be a simpler and more 'physical' way of explaining it!
I just want a logical, intuitive idea of it,that's all.

 

[Urmi Roy]

I'll just put forward my last problem,just in case anyone can help me with this one instead...I have brief descriptions of the parallel and perpendicular axes,and then my doubts.

"The parallel axis theorem allows you to compute the moment of inertia of an object more easily. In many cases you'll consider there is a particularly easy axis about which to compute the moment of inertia, which is parallel to the axis you actually want to find the m.o.i about. Well by the parallel axis theorem you do the easy one and then 'shift' the axis to the one you want."

------This 'shift' occurs only by adding Md^2 to the m.o.i about the original axis,(where M is mass of object and d is distance between original axis and new one).

Except for the mathematical proof,there seems to be no substantiation for this theorem.What is the term Md^2, separately?Is there no way we can arrive at this theorem by physical reasoning?



"Similarly for the perpendicular axis theorem, for planar (ie 2d) objects sometimes its easier to compute the moi about axes in the plane and then use the theorem to get the moi for an axis perpendicular. In the case of symmetric planar objects you get things like I_x = I_y = (1/2)I_z, which means you only need to find the moi about one axis and you have them about the other 2 perpendicular directions"

-----Again,is there any way we can arrive at this by reasoning...I mean,what would one say if I asked why the m.o.i about one axis is equal to the sum of the moi's about two of its perpendicular axes?

I found the proofs in my book...but as you probably realize,that's not exactly what I'm looking for.

(m.o.i= moment of inertia)

 

[tiny-tim]

parallel axis theorem

"… In many cases you'll consider there is a particularly easy axis about which to compute the moment of inertia, which is parallel to the axis you actually want to find the m.o.i about. Well by the parallel axis theorem you do the easy one and then 'shift' the axis to the one you want."

No, there is only one parallel axis for which adding md² works … the axis through the centre of mass.

 

[Urmi Roy]

Right,I understand that md2 is added to only the moi about the central axis,but again,referring back to post 10,what is this 'md^2'---its not the moment of inertia about the new axis,surely.

 

[tiny-tim]

Right,I understand that md2 is added to only the moi about the central axis,but again,referring back to post 10,what is this 'md^2'---its not the moment of inertia about the new axis,surely.

Yes, it is …

provided you pretend that the whole mass is concentrated at the c.o.m.

So the MI about any axis equals the MI of the same mass concentrated at the c.o.m, plus a "spread-out-ness" MI, of the actual distribution of mass about the parallel axis through the c.o.m.

(This is like the energy of a solid body: it equals the energy of the same mass with the velocity of the c.o.m, plus the rotational energy)​

 

[Urmi Roy]

So the MI about any axis equals the MI of the same mass concentrated at the c.o.m, plus a "spread-out-ness" MI, of the actual distribution of mass about the parallel axis through the c.o.m.

If we have the original axis through the c.o.m, and consider that the entire mass is concentrated there,the m.o.i should be zero,as r=0 for this case...am I missing anything here?

 

[tiny-tim]

If we have the original axis through the c.o.m, and consider that the entire mass is concentrated there,the m.o.i should be zero,as r=0 for this case?

No, I mean the MI about the new axis.

 

[Urmi Roy]

So what we're doing, is that (the steps)
1.we find out the MI of the body about the axis through its centre of mass,like we usually do.(by summing up the MIs of the individual particles constituting the body)

2. We assume that the entire mass is now concentrated at the c.o.m and,calculate the MI of this point mass about the new axis.

But,here, we're adding two unlike quantities--in the first step,we are not assuming the body's mass to be concentrated at one point,like a point mass,unlike the 2nd step.

Also,a few days ago,I was doing a sum,where basically I had to calculate the MI of a body about an axis,and I,by mistake, proceeded by finding the body's c.o.m and treated it like a point mass,of which I calculted the MI about the axis.

Obviously,my answer was wrong (and I was scolded badly for it!)--so in the asumption that we made in the 2nd step--that the whole mass is concentrated at the c.o.m,we should face a similar problem.We can't find the MI of a rigid body by simply considering it a point mass at the c.o.m

Is there a fundamental difference between my sum and what you're stating?

 

[tiny-tim]

… But,here, we're adding two unlike quantities …

No, we're adding two quantities of exactly the same kind:

they're both moments of inertia.

We can't find the MI of a rigid body by simply considering it a point mass at the c.o.m

That's right, not by simply considering it a point mass …

we consider it as a point mass and then we add an extra term for the spread-out-ness.

For example, consider two identical bodies on the end of identical rods …

bash one of the bodies about so that it changes shape but not mass …

obviously, we expect the MIs to be very similar, but not identical.

The extra term includes the information about the shape.

(and if the rod is long enough, we can ignore the shape, and treat the body as it it was a point mass … for example, the MI of a planet round the sun usually treats the radius of the planet as zero)

 

[Urmi Roy]

I think I'm getting it,but I have a teeny-weeny problem left...in the process that we find the 'spread-out-ness' MI of the mass, we're calculating the 'spread-out-ness' MI about the axis through the c.o.m---but in the end, we're trying to calculate the total MI about the new axis-----then how should the 'spread-out-ness' MI about the c.o.m axis matter at all?

When we spin the body about the new axis, the spatial arrangement of the individual particles constituting the body about the new axis matters--that's what affects the MI about the new axis,(it's not the spatial arrangement about the c.o.m axis that matters here).

In your analogy with the energy of a solid mass, suppose we have a cylinder rolling down a ramp,we calculate the velocity of the c.o.m (for linear kinetic energy) and the rotational velocity about a particular axis,suppose the axis through the edge of the ramp.

Here,all the calculations are made w.r.t the frame of the ramp---however,in the parallel axis theorem,we're adding the moments of inertia about the c.o.m axis and the new axis separately,and not one particular axis.

How is it that the total MI about the new axis is equal to the spread-out-ness MI about the c.o.m axis and the MI of a point mass (of equal mass of the body) about the new axis??

 

[tiny-tim]

I think I'm getting it,but I have a teeny-weeny problem left...in the process that we find the 'spread-out-ness' MI of the mass, we're calculating the 'spread-out-ness' MI about the axis through the c.o.m---but in the end, we're trying to calculate the total MI about the new axis-----then how should the 'spread-out-ness' MI about the c.o.m axis matter at all?
…
How is it that the total MI about the new axis is equal to the spread-out-ness MI about the c.o.m axis and the MI of a point mass (of equal mass of the body) about the new axis??

"spread-out-ness" is "spread-out-ness" … it doesn't depend on where the axis is …

the relative spatial distribution is all that matters.

In your analogy with the energy of a solid mass, suppose we have a cylinder rolling down a ramp,we calculate the velocity of the c.o.m (for linear kinetic energy) and the rotational velocity about a particular axis,suppose the axis through the edge of the ramp.

Here,all the calculations are made w.r.t the frame of the ramp---however,in the parallel axis theorem,we're adding the moments of inertia about the c.o.m axis and the new axis separately,and not one particular axis.

i] the rotational velocity is the same for all parallel axes.

ii] yes, all the calculations are made w.r.t the frame of the ramp …

but the linear energy and the rotational energy are treated entirely separately, just as the "point mass" MI and the "spread-out-ness" MI are treated entirely separately.

 

[Urmi Roy]

Thanks tiny-tim for your clarification. I really appreciate your continuous and expert help!

I also found a website which supports our discussion,but it contains two lines that I didn't understand(the bold lines).Perhaps if you explained it to me,it would help me finally get things clear--after all,the more I read,the clearer things become.

It goes like this..."The parallel axis theorem you can think of like this: if you're rotating about an axis that doesn't go through the center of mass,you have to rotate the object about its center of mass,plus you have to rotate the center of mass about the axis. If you had a point mass M distance d from the axis, you'd get I=Md^2. That's just to move (translate) the object itself about the axis, but doesn't actually rotate the object."

Sorry to keep bothering you like this, it won't take much longer!

 

[tiny-tim]

you have to rotate the object about its center of mass,plus you have to rotate the center of mass about the axis.

hmm … I think that's very confusing …

(which website is this?)

it's treating the two rotations separately, when in fact they're the same (ie the same angular velocity about the same axis) …

of course, you can have separate rotations: for example, the Moon is effectively rigidly attached to a rod rotating around the Earth (because it keeps the same face towards the Earth), but most planets or satellites are not, and rotate separately from their "attaching rod" …

but separate rotations mean that you no longer have a rigid body, and therefore it makes no sense to ask what "the" moment of inertia is!

If you had a point mass M distance d from the axis, you'd get I=Md^2. That's just to move (translate) the object itself about the axis, but doesn't actually rotate the object."

I think this is just wrong … the object is rigidly fixed to the "attaching rod", and has to rotate with it … I don't see how you can separate it into two rotations.

 

[Urmi Roy]

tiny-tim is it like this, that this spread-out-ness MI about the c.o.m is equal to the spread-out-ness MI about the new axis?

Perhaps if we made two separate calculations,first,taking the c.o.m axis to contain the origin and then the new axis to contain the origin,and then we calculate the spread-out-ness MIs about each,they might amount to the same.

Also,this spread-out-ness MI is actually the moment of inertia about the c.o.m axis---however,the new axis needs an additional term to calculate the total MI about it.

But,at the same time,If I am on the right track,then when we calculate the spread-out-ness MI about the new axis (which is equal to that about the c.o.m axis),we are already considering the individual particles in the body and their respecive inertia about the new axis,and the 'ease' with which they revolve around the new axis--we don't need to consider the c.o.m separately and additionally--after all, if we have accounted for the inertias of the individual particles,we have acounted for the whole body.

Is this on the right track?

This way,it'll be easier to visualise things.

 

[tiny-tim]

"spread-out-ness"

Hi Urmi Roy!

I hope you're not taking this too seriously …

we're talking about a way of remembering how to calculate MI, not a rigorous theory of MI …

if "spread-out-ness" helps you remember the formula, then it's good: if it doesn't, then it's bad.

ok, on that basis, "spread-out-ness" is like mass … they're the same for any observer, using any axis, etc.

That's the point … "spread-out-ness" happens to be an inherent property of a rigid body …

there's no reason why it should be …

we're just lucky that it is. ​

 

[Urmi Roy]

I hope you're not taking this too seriously …

we're talking about a way of remembering how to calculate MI, not a rigorous theory of MI …

Well, I was taking this seriously,and I was thinking that you were giving me a rigorous theory...you see, this topic has been stuck in my mind,and I really need to know about these theorems,after all, everything in physics has to have a logical explanation--it can't all be based on mathematical proofs.I couldn't reveal any purely physical explanation of these theorems from their mathematical proofs,it all seemed to be mathematical techniques.

Inspite of the expertise of all my teachers and all the members on physicsforums, no one has given me a thorough explanation of these theorems...it all seems so ambiguous!

 

[tiny-tim]

heresy!

Hi Urmi Roy!

… everything in physics has to have a logical explanation--it can't all be based on mathematical proofs.

I'm sorry, but this is heresy!

Physics is mathematical … or, to be precise, physics develops by carrying out experiments and then finding mathematical equations which fit the results.

Take the double-slit experiment, for example … there is no "logical" explanation for it (or, rather, there are several, but none of them is entirely satisfactory) … but the results follow a mathematical law which has been verified experimentally time and time again.

Likewise, the spin of an electron doesn't really have a "logical" explanation … it's been invented to make the mathematics fit the experimental results.

I agree that it helps if there's a "logical" explanation (I assume you mean an "intuitive" explanation … there's nothing more logical than mathematics, after all ).

But a "logical" explanation isn't necessary, and it isn't really part of the physics.​

Well, I was taking this seriously,and I was thinking that you were giving me a rigorous theory...you see, this topic has been stuck in my mind,and I really need to know about these theorems,after all,

No, the rigorous theory is the mathematical proof of the formula (I = I_c + md²) … I'm giving you the visualisation you asked for.

Inspite of the expertise of all my teachers and all the members on physicsforums, no one has given me a thorough explanation of these theorems...it all seems so ambiguous!

(Mathematics is never ambiguous! )

If you want to be happy with physics, you need to accept that the mathematical description is the ultimate aim.

"Realistic" descriptions are only there to help you remember the maths, and to help you describe physics to non-physicists.

Your teachers won't provide you with a "thorough explanation" because it isn't needed.

And it certainly won't do you any good in the exams!​

 

[Urmi Roy]

Okay,I get it.
Thanks for your cooperation,and help...I really apreciate it.