| Topic: |
Science > Physics |
| User: |
"Richard Herring" |
| Date: |
11 Aug 2003 07:39:18 AM |
| Object: |
Re: Angular mechanics in 3 dimensions. |
In message <4aa861fb.0308110143.3b189ff5@posting.google.com>, Starblade
Darksquall <Starblade13@Yahoo.com> writes
What is the most complex three dimensional combination of rotations
you can get before it can be described by a simpler combination of
rotations? IE, how many different kinds of rotations and rotational
accelerations and torques, etc. can you all put together
gyroscopically before you have to describe it in terms of something
simpler?
Addition of infinitesimal rotations commutes. Angular rates of change
are limits involving infinitesimal rotations. So any number of angular
momenta, torques etc. can be summed according to the normal rules of
vector arithmetic.
A better way of putting it is this: What is the most complex rotation
function or class of rotation functions you can make?
A (pseudo) vector.
That is, when
you have the most complex rotation function or class of rotation
functions, and additional rotation function or class of rotation
functions you add to it does not increase its complexity.
To envision this question, imagine I have a gyroscope with dozens of
different circles, or chambers, and I set them in motion, one by one,
and describe the complex rotation by a number of mathematical
function. Exactly how many times can I do this before I don't have to
describe it by any more numbers of mathematical functions? Or is there
a limit at all?
No limit. Its angular momentum is a single vector quantity.
And now the physics part of the question: Is there an angular energy
and angular momentum four vector such that we get an invariant
quantity? Or do we count any energy and momentum gotten from rotations
and spinning as just regular energy and momentum? Meaning that things
that spin, even if they're massless, can not go the speed of light if
they have any energy in their spinning,
Why not?
meaning they also cannot have
momentum? Or if they do have momentum, they must have energy, and
cannot go the speed of light?
Why not?
I'm just wondering this, because it seems to me that this means that
light, which is a number of photons, is NOT the fastest thing in the
universe then, since it has angular momentum, but no net momentum in
the axises orthogonal, or normal, to its motion, yet it still would
have energy, making it have nonzero mass,
Why would having energy give it nonzero mass?
Here's a hint: E = mc^2 is not generally true.
and therefore not going at
the universal top speed.
Or is the spin something that doesn't follow these rules, IE,
something that is entirely nonphysical?
(...Starblade Riven Darksquall...)
Reading too many fantasy novels is bad for you.
--
Richard Herring
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| User: "Michael Moroney" |
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| Title: Re: Angular mechanics in 3 dimensions. |
11 Aug 2003 01:11:48 PM |
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Richard Herring <junk@[127.0.0.1]> writes:
A better way of putting it is this: What is the most complex rotation
function or class of rotation functions you can make?
A (pseudo) vector.
....
No limit. Its angular momentum is a single vector quantity.
This brings up a question: An irregular asteroid (Toutatis?) that was
visited by the Galileo or another spacecraft was described as having
a chaotic spin, or different spins on different axes. If angular momentum
is a simple vector quantity, how is this possible?
--
-Mike
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| User: "Robert Israel" |
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| Title: Re: Angular mechanics in 3 dimensions. |
11 Aug 2003 02:04:52 PM |
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In article <bh8m94$ah$1@pcls4.std.com>,
Michael Moroney <moroney@world.std.spaamtrap.com> wrote:
Richard Herring <junk@[127.0.0.1]> writes:
This brings up a question: An irregular asteroid (Toutatis?) that was
visited by the Galileo or another spacecraft was described as having
a chaotic spin, or different spins on different axes. If angular momentum
is a simple vector quantity, how is this possible?
Yes, Toutatis. See
<http://www.solarviews.com/eng/toutatis.htm>
The same angular momentum can be obtained by different motions.
Robert Israel
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2
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| User: "Bored Huge Krill" |
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| Title: Re: Angular mechanics in 3 dimensions. |
11 Aug 2003 08:45:41 PM |
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"Robert Israel" <> wrote in message
news:bh8pck$gkj$1@nntp.itservices.ubc.ca...
In article <bh8m94$ah$1@pcls4.std.com>,
Michael Moroney <moroney@world.std.spaamtrap.com> wrote:
Richard Herring <junk@[127.0.0.1]> writes:
This brings up a question: An irregular asteroid (Toutatis?) that was
visited by the Galileo or another spacecraft was described as having
a chaotic spin, or different spins on different axes. If angular
momentum
is a simple vector quantity, how is this possible?
Yes, Toutatis. See
<http://www.solarviews.com/eng/toutatis.htm>
The same angular momentum can be obtained by different motions.
Robert Israel
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2
Robert,
thanks for the clarifications.
I have one more question - and apologies, my math is extremely rusty here -
I'm fairly sure I knew this stuff at one point:
Angular velocity about one axis can be described as a vector, which when
referenced to the center of rotation is in the direction of the axis, with
magnitude equal to the angular velocity about that axis in a clockwise
direction (correct so far?).
In the case of an object such as Toutatis, which I understand to be
described as rotating at different angular velocities about each of two
different axes, I don't think it can be described by a single vector. I'm
not sure though.
1. Is this correct? Are two vectors required here?
2. If so, what is the maximum number of vectors required to describe the
arbitrary rotations of a three dimensional object?
I'm thinking the answer is either one or three, but I don't know how to
prove it.
Many thanks in advance for any enlightenment
Regards
Krill
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| User: "Starblade Darksquall" |
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| Title: Re: Angular mechanics in 3 dimensions. |
12 Aug 2003 04:40:57 AM |
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"Bored Huge Krill" <bored_huge_krill@hotmail.com> wrote in message news:<vjgiclfodtva05@corp.supernews.com>...
"Robert Israel" < > wrote in message
news:bh8pck$gkj$1@nntp.itservices.ubc.ca...
In article <bh8m94$ah$1@pcls4.std.com>,
Michael Moroney <moroney@world.std.spaamtrap.com> wrote:
Richard Herring <junk@[127.0.0.1]> writes:
This brings up a question: An irregular asteroid (Toutatis?) that was
visited by the Galileo or another spacecraft was described as having
a chaotic spin, or different spins on different axes. If angular
momentum
is a simple vector quantity, how is this possible?
Yes, Toutatis. See
<http://www.solarviews.com/eng/toutatis.htm>
The same angular momentum can be obtained by different motions.
Robert Israel
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2
Robert,
thanks for the clarifications.
I have one more question - and apologies, my math is extremely rusty here -
I'm fairly sure I knew this stuff at one point:
Angular velocity about one axis can be described as a vector, which when
referenced to the center of rotation is in the direction of the axis, with
magnitude equal to the angular velocity about that axis in a clockwise
direction (correct so far?).
I really don't know... I've looked up the mathematical definition of
angular momentm, and I find that sometimes it doesn't point in a
direction parallel to its axis. Like if it's a spinning sphere, and I
take the cross product between the displacement of the particle from
its origin and the regular momentum vector, but I chose a point on the
top or bottom, it will be pointing diagonally. At least I'm assuming
origin means the actual center rather than simply something having to
do with the curvature, which would make a lot more sense if it was.
In the case of an object such as Toutatis, which I understand to be
described as rotating at different angular velocities about each of two
different axes, I don't think it can be described by a single vector. I'm
not sure though.
I think if you define it one way, it can... but if you define it
another way it can't. I'm not so sure about this though.
1. Is this correct? Are two vectors required here?
2. If so, what is the maximum number of vectors required to describe the
arbitrary rotations of a three dimensional object?
That's what I'm asking!
And I think angular momentum is just one vector, but that the
complexity of rotations is much more than one vector... and may not be
a vector at all!
I'm thinking the answer is either one or three, but I don't know how to
prove it.
Many thanks in advance for any enlightenment
Regards
Krill
(...Starblade Riven Darksquall...)
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| User: "Robert B. Israel" |
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| Title: Re: Angular mechanics in 3 dimensions. |
12 Aug 2003 02:45:45 AM |
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"Bored Huge Krill" <bored_huge_krill@hotmail.com> wrote in message news:<vjgiclfodtva05@corp.supernews.com>...
"Robert Israel" < > wrote in message
news:bh8pck$gkj$1@nntp.itservices.ubc.ca...
In article <bh8m94$ah$1@pcls4.std.com>,
Michael Moroney <moroney@world.std.spaamtrap.com> wrote:
Richard Herring <junk@[127.0.0.1]> writes:
This brings up a question: An irregular asteroid (Toutatis?) that was
visited by the Galileo or another spacecraft was described as having
a chaotic spin, or different spins on different axes. If angular
momentum
is a simple vector quantity, how is this possible?
Yes, Toutatis. See
<http://www.solarviews.com/eng/toutatis.htm>
The same angular momentum can be obtained by different motions.
Robert Israel
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2
Robert,
thanks for the clarifications.
I have one more question - and apologies, my math is extremely rusty here -
I'm fairly sure I knew this stuff at one point:
Angular velocity about one axis can be described as a vector, which when
referenced to the center of rotation is in the direction of the axis, with
magnitude equal to the angular velocity about that axis in a clockwise
direction (correct so far?).
Yes.
In the case of an object such as Toutatis, which I understand to be
described as rotating at different angular velocities about each of two
different axes, I don't think it can be described by a single vector. I'm
not sure though.
1. Is this correct? Are two vectors required here?
Exactly what are you hoping to describe?
The angular velocity at any particular time is a single vector.
However, this changes as time goes on. The angular momentum stays
constant (in the absence
of torque), but if the object is not symmetric the moments of inertia
about different axes are different, so the angular velocity and
angular momentum
don't have to be in the same direction.
Robert Israel
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2
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| User: "Old Man" |
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| Title: Re: Angular mechanics in 3 dimensions. |
12 Aug 2003 05:41:12 PM |
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Michael Moroney <moroney@world.std.spaamtrap.com> wrote in message
news:bh8m94$ah$1@pcls4.std.com...
Richard Herring <junk@[127.0.0.1]> writes:
A better way of putting it is this: What is the most complex rotation
function or class of rotation functions you can make?
A (pseudo) vector.
...
No limit. Its angular momentum is a single vector quantity.
This brings up a question: An irregular asteroid (Toutatis?) that was
visited by the Galileo or another spacecraft was described as having
a chaotic spin, or different spins on different axes. If angular momentum
is a simple vector quantity, how is this possible?
--
-Mike
Suppose that a rigid body possesses three orthogonal angular
moments of inertia such that I1 < I2 < I3. Any component of
rotation about I2 is unstable. [Old Man]
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| User: "Matthew F Funke" |
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| Title: Re: Angular mechanics in 3 dimensions. |
11 Aug 2003 01:49:50 PM |
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Michael Moroney <moroney@world.std.spaamtrap.com> wrote:
Richard Herring <junk@[127.0.0.1]> writes:
No limit. Its angular momentum is a single vector quantity.
This brings up a question: An irregular asteroid (Toutatis?) that was
visited by the Galileo or another spacecraft was described as having
a chaotic spin, or different spins on different axes. If angular momentum
is a simple vector quantity, how is this possible?
Actually, Toutatis has never been visited by a spacecraft; the images
we have of that asteroid were obtained with radar from the Goldstone Deep
Space Communications Complex in the Mojave Desert (California, USA).
As to the main question, consider that any complex rotation can be
broken down into simple rotations about more than one axis. You can add
the angular momentum of each particle around each axis to get a *combined*
angular momentum of that particle. Add all the particles' angular
momentums together and you have one angular momentum vector for the entire
asteroid.
Angular momentum vectors don't *have* to coincide with the axis of
rotation. In the case of a spherical body with one principal axis of
rotation (like the Earth), they just happen to -- but it's not mandatory,
especially in the case of something with more than one principal axis of
rotation.
--
-- With Best Regards,
Matthew Funke (mff@hopper.unh.edu)
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| User: "Brian Quincy Hutchings" |
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| Title: Re: Angular mechanics in 3 dimensions. |
11 Aug 2003 04:56:33 PM |
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any "compound" rotation (initial torque
about different axes) resolve to a single rotation
about a single axis, just as any two do;
these problems can be solved by "purely imaginary" quaternions,
such as pitch/yaw/roll of a craft.
try throwing a bottle in the air with two axes
o'spin.
mff@hypatia.unh.edu (Matthew F Funke) wrote in message news:<bh8oge$ite$1@tabloid.unh.edu>...
As to the main question, consider that any complex rotation can be
broken down into simple rotations about more than one axis. You can add
the angular momentum of each particle around each axis to get a *combined*
angular momentum of that particle. Add all the particles' angular
momentums together and you have one angular momentum vector for the entire
asteroid.
--les ducs d'Enron!
http://members.tripod.com/~american_almanac
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| User: "Robert Israel" |
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| Title: Re: Angular mechanics in 3 dimensions. |
11 Aug 2003 06:49:19 PM |
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In article <bde404c9.0308111356.512cb5f6@posting.google.com>,
Brian Quincy Hutchings <QncyMI@netscape.net> wrote:
any "compound" rotation (initial torque
about different axes) resolve to a single rotation
about a single axis, just as any two do;
these problems can be solved by "purely imaginary" quaternions,
such as pitch/yaw/roll of a craft.
try throwing a bottle in the air with two axes
o'spin.
This is a common mistake. What is true is that for any two
times t1 and t2, the change in orientation of a rigid body from
t1 to t2 can be described by a single rotation about a single
axis. But it's not true that the evolution of the orientation
in time is described by rotations about a fixed axis. Indeed,
this won't happen for a free motion when the body is
non-symmetric (all three moments of inertia distinct) and the motion is
not rotation about one of the principal axes.
Robert Israel
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2
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| User: "Brian Quincy Hutchings" |
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| Title: Re: Angular mechanics in 3 dimensions. |
12 Aug 2003 04:06:30 PM |
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must they be the three orthogonal axes?
israel@math.ubc.ca (Robert Israel) wrote in message news:<bh9a1v$n54$1@nntp.itservices.ubc.ca>...
try throwing a bottle in the air with two axes
o'spin.
This is a common mistake. What is true is that for any two
times t1 and t2, the change in orientation of a rigid body from
t1 to t2 can be described by a single rotation about a single
axis. But it's not true that the evolution of the orientation
in time is described by rotations about a fixed axis. Indeed,
this won't happen for a free motion when the body is
non-symmetric (all three moments of inertia distinct) and the motion is
not rotation about one of the principal axes.
--Dec.2000 'WAND' Chairman Paul O'Neill, reelected to Board. Newsish?
http://www.rand.org/publications/randreview/issues/rr.12.00/
http://members.tripod.com/~american_almanac
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