Showing a top spin won't fall, WITHOUT using angular mechanics

[Mad_Eye]

as far as i know angular mechanics contains no axioms in it.
it just a helpful tool to solve problems with a rotating body, but technically if you regard the body as a set of independent particles you should be able to solve any problem without angular momentum, right?

so how can you show a spinning top or a gyroscope won't fall using normal linear mechanics?

i have absolutely no idea how to do that...
thank you very much for helping :D

 

[Andrew Mason]

as far as i know angular mechanics contains no axioms in it.

What does this mean? The rate of change of angular momentum of body is the torque on the body. Is that an axiom?

it just a helpful tool to solve problems with a rotating body, but technically if you regard the body as a set of independent particles you should be able to solve any problem without angular momentum, right?

Angular momentum is real. It is not just a helpful tool.

AM

 

[Mad_Eye]

What does this mean? The rate of change of angular momentum of body is the torque on the body. Is that an axiom?

Angular momentum is real. It is not just a helpful tool.

AM

well, you might say it an axiom yes..
but however, regarding a body as a set of not spinning particles, each with independent velocity and acceleration,
should let you solve it without angular mechanics...
right?

 

[Vanadium 50]

Sure. A top has 10²⁴ atoms. Would you rather solve the equations of motion of each atom using linear motion or use angular motion?

 

[Andrew Mason]

well, you might say it an axiom yes..
but however, regarding a body as a set of not spinning particles, each with independent velocity and acceleration,
should let you solve it without angular mechanics...
right?

You could consider the motion of each atom in the top using its distance from the centre axis and the top's rate of spin: $$\omega = \dot\theta$$

to find its tangential velocity. In fact, this is exactly what is done when one uses moment of inertia to find angular momentum.

You cannot solve the orbit of one planet without resorting to principles of angular momentum (Kepler's laws). So I don't see how you could solve the motion of all the molecules in a top without considering angular momentum.

AM

 

[Cleonis]

so how can you show a spinning top or a gyroscope won't fall using normal linear mechanics?

That can be done, by making maximum use of symmetries.

First simplifying step: consider the case of a wheel where the spokes have negligable mass compared to the rim. If you have handled the case of a ring then the conclusions carry over to the case of a solid disk, since a disk can be thought of as a series of nested rings.

So the case to consider is that of a spinning ring. At each point in time the line elements of that ring are instantaneously in linear motion, and again, there is a lot of symmetry there.

For a mathematical treatment you can integrate the dynamics of the line elements of the ring. That calculation does not use any of the formula's of angular mechanics, yet it reproduces the results that are obtained with angular mechanics.


Earlier discussion (by me) on physicsforums:
A post from november 2010 about https://www.physicsforums.com/showpost.php?p=2992527&postcount=3". Illustrated with images. Just a qualitative discussion.

More detailed discussion (including math) is in the http://www.cleonis.nl/physics/phys256/gyroscope_physics.php" article on my website.

 

[Cleonis]

as far as i know angular mechanics contains no axioms in it.

That is correct.

If you take the set of laws of linear motion as axioms then you can derive conservation of angular momentum as a theorem.

For instance, in the Principia, Newton gives as the first proposition of the first book a derivation of Kepler's law of areas from the (linear) laws of motion. See the thread about https://www.physicsforums.com/showthread.php?t=400562".

 

[Cleonis]

A top has 10²⁴ atoms. Would you rather solve the equations of motion of each atom using linear motion or use angular motion?

Let me discuss this a little further.

As we know, the laws of motion - the axioms - are for the case of point masses. So how do we arrive at expressions for angular mechanics of solids? Well, that's a two stage process.

First, you go from linear dynamics of point masses, to angular dynamics of point masses. (As mentioned earlier, that is covered in the thread about https://www.physicsforums.com/showthread.php?t=400562")

Second, you go from point masses to solids by performing an integration.

For example, what is the moment of inertia of a rod, when that rod is rotating about it's center of mass? (With the rotation axis perpendicular to the rod)
You subdivide the rod in line elements. Each line element has a particular distance to the axis of rotation. To find the overall moment of inertia you integrate the line elements over the length of the rod.Clearly there is no point in considering the atoms of the solid. Ironically, the reason it's unhelpful to consider the constituting atoms is that atoms have a finite size. To perform the integration you treat the solid as a continuum. You take line elements, and for the integration you take the limit of the line elements going to zero.

 

[Andrew Mason]

That is correct.

If you take the set of laws of linear motion as axioms then you can derive conservation of angular momentum as a theorem.

Or, if you started with the conservation of angular momentum you could derive the laws of linear motion (eg. a special case where the radial force is 0). In physics, there are no real fundamental axioms. Everything is derivable from everything else, it seems. What is considered fundamental is a matter of the approach one takes.

AM

 

[Mad_Eye]

That is correct.

If you take the set of laws of linear motion as axioms then you can derive conservation of angular momentum as a theorem.

THANK YOU
that what i wanted to say..

Or, if you started with the conservation of angular momentum you could derive the laws of linear motion (eg. a special case where the radial force is 0). In physics, there are no real fundamental axioms. Everything is derivable from everything else, it seems. What is considered fundamental is a matter of the approach one takes.

AM

no,
that have nothing to do with physics
that's true for almost any set of rules (may be every set of rules, i don't know i need to think about it)
and yet, there fundamentals axioms that should be decided...

for example you can say the shortest distance between two points is a straight line,
setting it as an axiom, and conclude that one side of a triangle is never larger than the sum of the two others.
and you can do the exact opposite.
yet we decided to go for the first one, for aesthetic reasons.
exactly like linear and angular momentum...

 

[Cleonis]

Or, if you started with the conservation of angular momentum you could derive the laws of linear motion (eg. a special case where the radial force is 0). In physics, there are no real fundamental axioms. Everything is derivable from everything else, it seems. What is considered fundamental is a matter of the approach one takes.

In physics, there are no real fundamental axioms.

I'll go one step further, I think there aren't any axioms in the first place.

In physics "axioms" serve a different purpose than in mathematics. In physics the purpose of the axioms is to be evocative.
It's not necessary for any physics discipline to lend itself to axiomatization. There are no axioms for maxwellian electrodynamics; I'm not aware of any axioms of quantum mechanics.

It just so happens that in a couple of physics disciplines it is possible to capture the contents in a small set of principles. There are the laws of thermodynamics. In mechanics there are the laws of motion. The circumstance of being able to formulate a small set of principles, and have pretty much all the bases covered is actually exceptional to the general state of affairs.


For mechanics it does seem to me that linear momentum is more fundamental than angular momentum.

We have the following law of conservation of linear momentum: When two particles exert a force upon each other (repulsing or attracting/colliding) then the state of motion of the center of mass remains the same.

That kind of interaction requires only one dimension of space. Linear momentum is a one-dimensional phenomenon, in the sense that for linear momentum to exist one spatial dimension is sufficient.

Angular momentum is inherently a phenomenon in two spatial dimensions. Two particles can have angular momentum relative to their common center of mass when they are circumnavigating each other. Geometrically angular momentum is proportional to an area, rather than proportional to a vector.

Circumnavigating motion sweeps out a plane, and to specify angular momentum you must specify that plane. Since our space has three spatial dimensions it so happens that we can specify the angular momentum plane with a vector. As we know, the angular momentum vector is defined as the vector that is perpendicular to the angular momentum plane.

Summerizing:
For linear momentum one spatial dimension is sufficient; angular momentum is inherently a phenomenon in two spatial dimensions.

 

[Andrew Mason]

For mechanics it does seem to me that linear momentum is more fundamental than angular momentum.

We have the following law of conservation of linear momentum: When two particles exert a force upon each other (repulsing or attracting/colliding) then the state of motion of the center of mass remains the same.

That kind of interaction requires only one dimension of space.

You would need at least two. Two particles colliding the motion of the centre of mass remains unchanged but the motion of each particle requires two dimensions unless the collision was head-on.

Summarizing:
For linear momentum one spatial dimension is sufficient; angular momentum is inherently a phenomenon in two spatial dimensions.

I think it would be better to say that linear momentum can be defined in one spatial dimension (it requires a line) whereas angular momentum requires two (it requires an angle in a plane).

AM