Physics Discussion Thread

[Vayutuvan]

What is the speed of light for an observer travelling in a frame of reference relative to the photon at c ?

It is c.

[sanjaykumar]

So the observer travelling alongside the photon, at c, will see it travelling at c? Or will he now see a photon at rest mass. If the photon really (whatever that means) has rest mass 0, will it disappear?

So Einstein's theory of relativity is seemingly incomplete just as was Newton's.

[Vayutuvan]

Are there Observers who have zero rest Mass? Otherwise they will take finite amount of time to accelerate to c.

[sanjaykumar]

Yes that is the standard argument against these postulates, that acceleration to c is not possible. But apparently that also precludes thought experiments, which is a bit troubling.

[sanjaykumar]

Anyway, photon rest mass of zero may be only a convention, perhaps a convenient fiction or fudge factor.

What is troubling is that a theory that has inbuilt rules for disqualifying certain questions approaches theology. It may of course be internally consistent.

[TSJones]

If your question exceeds the physical limits of the universe, then you would have to say "I don't know". For instance what happens to matter (including photons) when falling below the Schwarzschild Radius of a black hole? Hawking used to say he knew. Now he is not so sure. He used to say all the information was destroyed. Now he says, maybe not. In other words, he doesn't know. He is using math to calculate things that may not be in this physical plane of the universe. Is it religion? Who knows? If my uncle was a woman she would be my aunt.

Are there perpetual motion machines? Not so far as we know today.

If we filled the universe with boiling pots of water, one of them would freeze (all the pots vibrating atoms could potentially miss rubbing with each other). OK, so what?

[sanjaykumar]

TSJ, firstly with your use of idioms and argumentativeness, you are eligble for a passport issued by Government of India.

If your question exceeds the physical limits of the universe, then you would have to say "I don't know".


That is one of the problems I was alluding to above. It is a case of begging the question: We have already decided the physics of the universe thus it is facile (in both senses of the word) to decide what is an inadmissable/incongruent question.


If we filled the universe with boiling pots of water, one of them would freeze (all the pots vibrating atoms could potentially miss rubbing with each other). OK, so what?


I am not sure I follow, Boltzmann Brains are statsitically possible-it's just that the universe is not old enough to have observed one.

Relativity assumes equivalence of graviational and inertial frames of reference.
Let us test this. Supend a bucket of water from a rope. Now spin the bucket, the water will rise at the periphery. From the frame of reference of the water the stars are revolving about a point.
Now in part two, being an Indian, you spin the stars above the bucket instead.

Will the water rise?

[Amber G.]

http://coldfusionnow.org/current-scienc ... t-in-lenr/

http://coldfusionnow.org/special-lenr-i ... lable-now/

Hi, you may already know that, but I did post (in 2009) about PKI ("father of India's cold fusion")'s video about cold fusion and "mega-gauss-bombs"...

This was, of course, when there was heated debate in " Pokhran II not fully successful' dhaga .. and PKI's was name was often mentioned there...

The post can be found here: ..http://forums.bharat-rakshak.com/viewto ... 42#p745742

There was "debate" about cold fusion even earlier in nuke dhaga.. (2008 or so)
(http://forums.bharat-rakshak.com/viewto ... 05#p496005

[Amber G.]

I would say in addition to physics courses, some applied mathematics will be very helpful. Then one doesn't have to depend on "descriptive" physics.

Yes, Good math background is needed..(I say physics is math with purpose ).. it makes other things easier to understand .. and IMO, if there is any opportunity one should take as rigorous math courses as one can take -- specially if taught by good teachers...

[SriKumar]

Cross-posting 'ankar's' post to here. An object (a space 'spanner'?) flipping between two states of unstable (?) equilibrium.

Unstable spin in Zero-Gravity

Seems like the object has 2 points of equilibrium both unstable, and is alternating between the two due to lack of gravitational resistance in space and any sources of dissipation.

[Amber G.]

Cross-posting 'ankar's' post to here. An object (a space 'spanner'?) flipping between two states of unstable (?) equilibrium.

Unstable spin in Zero-Gravity

Seems like the object has 2 points of equilibrium both unstable, and is alternating between the two due to lack of gravitational resistance in space and any sources of dissipation.

To add - this and similar problems, at least to me, are fairly well known -- I have heard it many times in serious discussion. Feynman was known, for example, throwing spinning food trays up in the air, and asking one to explain the "flipping". (he claimed that thinking about math part of these is what inspired him to do noble-prize winning type serious physics)

But in my humble opinion, this kind of behavior is definitely not intuitive, and I can't think of any any good, simple way to describe it (which will predict what will happen in all the situations) without deriving it from the governing equations..(any good physics text book - motion of rigid body - an example given below:
http://farside.ph.utexas.edu/teaching/3 ... ode99.html

***

One of interesting experiment you can do, and it really looks interesting if you have not seen it, is to throw a spinning booK (or a rectangular object such as DVD with a case with sides,say a>b>c) straight up. Spin the book , along a horizontal axis ((say x-axis) and throw it vertically(z-axis). If the horizontal exis is parallel to a or c, the spin will be stable, but if it is b, the book will "flip". (It is hard to describe, but try the experiment - the result looks very strange)

Another you tube similar principle - http://www.youtube.com/watch?v=H0WM_nnMswo

***

Although this was well understood by Newton and Euler etc.. as I said, to many it may not be intuitive..
One famous case was the US's very first satellite (Explorer 1) which along with few other discoveries (like Van Allen radiation belt) also discovered a problem related to stability that caught its designers by surprise.

Explorer 1 was a cylinder that was supposed to spin about its minimum moment of inertia axis -- along its axis of symmetry --, which should be stable according to rigid body analysis. However, it had flexible antennas sticking out that dissipated energy and changed the dynamics such that it was unstable. The result was that the satellite ended up tumbling unpredictably.

( For details see: http://en.wikipedia.org/wiki/Explorer_1#Results

[Amber G.]

^^^ For those who want to look it up to get detailed explanation -- try searching for "Dzhanibekov effect" or "Tennis racket theorem" (You may need some understanding of vectors and tensors) (Or rigid body rotation in physics text book).

The experiment I talked about can be seen here in slow motion in an you tube video -- Object here is a table tennis racket, but any rectangular box (say dvd case, or a book or iPhone) will do. Try it.. enjoy..
Link: https://www.youtube.com/watch?v=4dqCQqI-Gis

***

...due to lack of gravitational resistance in space and any sources of dissipation.

Just a comment - one will see the same kind of rotary motion on earth (except that the object will be failing down so the flip may be hard to observe).. as the above video shows. Since in space, things do not fall down, one has a longer time to observe.

(The object will not flip, even in a space station, if the rotation was along the other two principle axis..)

[Vayutuvan]

Counter intuitive indeed but also interesting .

[Amber G.]

^^^Yes.
(I still remember the first time some one asked me to explain - by a well known physics professor, who threw a spinning book at a party..I was a graduate student and it took much more math to understand and I have not heard any "simple"/intutive way to explain.

Recently I saw Terry Tao (famous mathematician) post about this who explained it in novel way.. (Interesting that a "mathematician" is explaining physics problem by "intution" ) The idea is fun/short
(for detail see https://plus.google.com/+TerenceTao27/posts/e3GLg4Ki4dj) and more intuitive than typical textbook derivation.

But still ... it uses "herpolhode" a concept one is not likely to know unless one has taken graduate level physics/math course ... )

(Still interesting aspect I learned that, these instability axis exists in n (>3) dimensions .. while in 3 dimensions there is only 1 axis.. for general n it is n-2.

Anyway found this video, which shows the book thrown, as I mentioned earlier .. (start looking at 2:00 minute into the this 3+ minute video)
https://www.youtube.com/watch?v=LzVItPwiQyI

(As said before, one does not have to be in a space station to see this, you can just throw the book up and watch this - using a rubber band around the book to hold pages may help )

Enjoy.
(Thanks SriKumar for bringing this fun topic)

[SriKumar]

You are welcome. Actually I posted it here for a discussion/explanation...so thanks for that.
It is funny but the 'book flipping' test was something a middle-school classmate of mine mentioned to me...when he was trying to flip his lunch box (a cubiod shape). Since then, I've flipped a lot of books ......with no new enlightenment dawning on me. The exercise in space is still quite interesting because the book (space spanner) stays in the 'air' long enough to show it reverse the direction of its flipping, which one cannot see it at earth because the book 'lands' quickly.

Question: So it seems like the rotating object (book/box/space spanner) swings through 2 points of unstable equilibrium. If there was dissipation in the system to bleed out energy from the rotating object e.g. due to air, or better yet, if it were done in water; how would the motion change over time? Would it 'finally settle' into a state of stable equilibrium (as assuming there is one, between the 2 unstable states) or would it continue to alternate between the 2 unstable states but with a lower 'amplitude' i.e. it does not swing far enough? Or does the rpm slow down but it continues to hit the points of unstable equilibrium? I am guessing it is the last option.

[Vayutuvan]

My guess is that microgravity is the one of the keys, so are the starting conditions and the non-existence of a perfectly symmetric physically realizable body.

If there is no gravity at all - is there any place in the universe which does not have 0 gravitational field? - and the object is rotated such that the rotational axis is aligned with the axis of symmetry perfectly and there are no imperfections in the object, there are no flips.

Same way, if a top made out of a disk and and a round stick going through the centre of the disk is spun on the ground, no distinction can be made between the handle and the tip.

But the every present gravity (the curvature of space) introduces an asymmetry.

Broadly speaking, the Hamiltonian is (double? I have to think more about this) saddle shaped. That is the connection to JVM and Morgenstern's game theory.

[Amber G.]

For those who are not afraid of "physics talk"

You are welcome. Actually I posted it here for a discussion/explanation...so thanks for that.
It is funny but the 'book flipping' test was something a middle-school classmate of mine mentioned to me...when he was trying to flip his lunch box (a cubiod shape). Since then, I've flipped a lot of books ......with no new enlightenment dawning on me. The exercise in space is still quite interesting because the book (space spanner) stays in the 'air' long enough to show it reverse the direction of its flipping, which one cannot see it at earth because the book 'lands' quickly.

Question: If there was dissipation in the system to bleed out energy from the rotating object e.g. due to air, or better yet, if it were done in water; how would the motion change over time? Would it 'finally settle' into a state of stable equilibrium (as assuming there is one, between the 2 unstable states) or would it continue to alternate between the 2 unstable states but with a lower 'amplitude' i.e. it does not swing far enough? Or does the rpm slow down but it continues to hit the points of unstable equilibrium? I am guessing it is the last option.

SriKumar - I think you are right that throwing books (or lunch boxes) up in the air may have been much more common, but many would not notice/think about this strange phenomena unless pointed out (at least I did not). Funny thing is, it is NOT the longest or the shortest side (as one would intuitively may find reasons for) but the middle side where , even with good practice, it is almost impossible to keep the rotation stable.

The "flipping", btw is quite observable and impressive even in a living room. (I really want people to try the experiment, and see for themselves) it may require a little practice but it is fairly easy to do).
(BTW, I am told iPhone tis he object used in such cases now a days )

So it seems like the rotating object (book/box/space spanner) swings through 2 points of unstable equilibrium.

If I have to describe it, it is really not "swinging" between 2 points - nothing like a pendulum, or regular swing etc..every point (as far angular velocity in a given direction is concerned) there is instability all the time except that there are two points where there is equilibrium which is unstable. IOW it is very hard (without using computer -numerical methods, in practice) to describe/predict the motion except to say it is something like "unpredictable". (It is like, one can easily "study" collision of two particles (their momentum/energy before and after the collision - but it is very hard to calculate what the shape of a car will be after an accident)..(The particular video you posted, there does seem some sort of regular motion (may be due to particular shape of the object), but see some of the other videos, or do the experiment, all one can say is that the tumble is irregular)

If there was dissipation in the system to bleed out energy from the rotating object e.g. due to air, or better yet, if it were done in water; how would the motion change over time?

Loss of energy etc has no importance here. Loss of energy or angular momentum is not relevant here. (Sure, if done under water, motion will slow down, as one thinks but that is hardly the point here). IOW the energy or angular momentum can be considered CONSTANT (negligible loss to air etc). Similarly practically speaking, gravity (or lack of it) plays NO part. (Gravity introduces NO torque and hence produces no changes in the rotation).

To be clear -

- Gravity (or lack of it) essentially plays no part.
(If you through a rigid body, the motion of center of mass of the body can be thought as a point and its motion is rather easier to study. (first year physics deals with it) -- essentially one study "falling objects, or even orbits ). Gravity (on a falling object) produces no torque so angular momentum is not affected.

- The body (once released) is essentially has ZERO torque so angular momentum remains constant.

Study of angular motion, even in a zero torque case, where angular momentum does NOT change, requires complex math (graduate level physics). (Angular momentum (a vector) is inertia (a tensor)*angular velocity (vector).

(One studies motion of earth (elliptical orbit etc) in UG/high-school physics, but precession of equinox requires more complex math)

The "symmetry" (or lack of it) of the object plays NO part. IOW *any* object (symmetric or not) will do. (As long as three eigen values of Inertia tensor are different, one will observe the "flipping"/tumbling type behavior in one of the axis)

If you really want to understand physics (or physics technique used to study the motion) -

- For any rigid object, one can use center of Mass - and use simple Newton's law to calculate it's motion.

- For studying the rotation, easiest method is to use three principle axis (fixed wrt to body). and "Moment of Inertia" along these axis. (Then complicated tensor equation becomes simple - If the torque and initial angular velocity happens to be in one of the principle axes then torque = angular velocity * moment of Inertia (along that axis).

(Note: One has to calculate or measure experimentally, these three axes - these are not just arbitrary.
if there is any symmetry, the one of the principle axis will pass through it)

- Thus, if there is NO external torque, the object will not tumble (keep the angular velocity and direction constant ) if the axis of rotation happens to be one of the principle axis.

Interesting part is, for two of the principle axes one has a stable equilibrium..
(This is why spinning earth does not "flip" small fluctuations around the spin (called nutations, some times, are indeed very small) (There is precession - a 26000 year cycle which rotates the spin axis -is another phenomena - but that is due to torque exerted by sun/moon on equatorial bulge of the earth)

For the third principle axis the equilibrium is unstable. Thus even a small "error" in not aligning exactly with the axis will result in a tumble in a short time.

Hope this helps...

(For those who want some math - the simplified formula for motion, in zero torque is:

rate of change of w_1 = constant * w_2*w_3
(similar formula for w_2, and w_3)

Where w_1,w_2,and w_3 are angular velocity component in three principle axes.
(Let us assume w_2 direction is "middle", and initial value of w_3=0, and w_2 is almost zero, and w_1 is very large compared to w_2)

So we see change of rate for w_1=0 or w_1 does not change. .. IOW if you careful to keep wobble small w_2 and w_3=0 the angular velocity w_1 remains constant.

The fun part is if you calculate "rate of change of the rate of change (double derivative) of say w_2, it comes out to be = constant * w_2.. and the constant is positive

(Important: The constant happens to be negative for the two other case)

Solving the diff eq will yield exponential solution for w_2, so even if started with small value, it will grow - unstable...

Thus the important part here is not gravity/outer space/air resistance/symmetry (or lack of it) ..etc.. It is rather odd behavior which takes place ONLY in just one of the three directions. ..
(It also happens to one of the difficult to "explain without math " like problem)

It is a rather long post, hope it is some what useful.

[Amber G.]

My guess is that microgravity is the one of the keys, so are the starting conditions and the non-existence of a perfectly symmetric physically realizable body.

If there is no gravity at all - is there any place in the universe which does not have 0 gravitational field? - and the object is rotated such that the rotational axis is aligned with the axis of symmetry perfectly and there are no imperfections in the object, there are no flips.

Hello Vayu tuvan -
In my view, as I said in the previous post (please do read, though it is quite long), gravity (or lack of it) or existence (or non-existence) of symmetry (perfect) is not key point here..

The key point is, if one does an experiment, taking special case of iPhone where there are clearly 3 axes of symmetry, the rotation aligned to *any* one one of the axis exactly ( only along one of the three axes) will keep the rotating body uniformly rotated (with constant angular velocity - no tumble etc). (No torque ==> Angular Momentum Constant ==> Angular Velocity along that axis constant).

The Key point is: (sorry to repeat it again but it is the key point) with slight initial perturbation (eg very small angular velocity component in other direction),
For two of the cases - (along longest and the shortest side) there is only slight wobble (self correcting - like a pendulum swinging slightly if not left exactly at the center)

But for the other one case, it is unstable equilibrium - like a pebble on the top of a hill - a slightest push in either direction will make the pebble roll)

Same way, if a top made out of a disk and and a round stick going through the centre of the disk is spun on the ground, no distinction can be made between the handle and the tip.

The "difference" here is simply moment of Inertia which is different, depending on the axis.

Broadly speaking, the Hamiltonian is (double? I have to think more about this) saddle shaped. That is the connection to JVM and Morgenstern's game theory.

I think many people may not even know what is a Hamiltonian .let alone talking about saddle points etc... .. Reminds me of the story about tiger shikar (hunting) in an quantum jungle written by G. Gamow. When a (quantum) tiger (or his wave function) attacks the hunter and the professor riding on an elephant -- the professor shouts

"Don't worry about precise aiming .. (one can't do that against a wave/particle tiger anyway) just raise the Hamiltonian"...)

(Mr. Tompkins is a wonderful book..Hamiltonian, for those who are unfamiliar, Hamiltonian, in physics, is generally just the operator corresponding to the total energy of the system ( spectrum /set of possible outcomes )

[Bade]

The same principle works on an imperfect rotating sphere isn't it, like cricket ball with a seam.

[Vayutuvan]

AmberG: Thanks, I get it now that neither gravity nor kimperfections are important. Each point is in a fixed position with the assumption that the body is rigid.

Looks like only part of my guess is correct that the axis of rotation has to be perfectly aligned with one of the axis of symmetry. Does the vibrations of the spacecraft play a role? IOW, it is impossible impart torque to the body so that the torque (cross product r x F) to align exactly in the positive or negative of the direction vector defined by the axis of symmetry. The minima of the total energy are close together and are separated by a very small energy barrier between pairs of the minima so the flip is only between matched pairs.

One more question - all the atronomical bodies are also exerting gravitational force and are moving in some complicated fashion with respect to the rotating object.

My doubt is that even if one were to align the torque exactly with the axis of symmetry, wouldn't the varying gravitational field on different particles of the rigid body due to the relative moment of all the astronomical bodies in the universe is not going to affect the body (differential gravity results in a changing force) such that it can overcome the small energy barrier separating the two minima and flip between these two states?

Another followup question: Is the curvature of space non-changing over long duration - duration comparable to the age of the universe?

Amber G and Bade and other Phizziks gurus: I hope I am making sense. Most likely not, but one can sure hope

[Vayutuvan]

Another question:

Let us assume that I have a system of point masses connected by mass less rigid rods such that these point masses are all in fixed position with each other. Then we rotate this point mass assembly around an axis that goes through its center of {mass, gravity, momentum} (I am not sure which one to chose here). How many stable and unstable equillibria this assembly has?

AmberG: Why are there only three eigenvalues? The DOFs are six for a rigid body and hence we need six eigenvalues for the rigid body, right? The eigenvalues are {+,-} pairs, i.e. three 2x2 sub spaces which are well separated depending on the three principal axes of the enclosing ellipsoid. These axes need not necessarily be orthogonal.

I have to go through this Dover book on Theoretical Physics by Georg Joos (no pingrezi here - that is the verbatim name of the author). Looks great but is a magnum opus of ~700 pages of small print.

[Amber G.]

Another question:

Let us assume that I have a system of point masses connected by mass less rigid rods such that these point masses are all in fixed position with each other. Then we rotate this point mass assembly around an axis that goes through its center of {mass, gravity, momentum} (I am not sure which one to chose here). How many stable and unstable equillibria this assembly has?

AmberG: Why are there only three eigenvalues? T

We live in 3-dim space and the 3 eigenvalues we mean are of Inertia matrix(tensor).

In the above system, for each point mass of mass 'm' and at (x,y,z) simply write down,
(And later sum for all the masses)
I_11= m(y^2+z^2), I_22=m(z^2+x^2),I_33=m(x^2+y^2)
I_12=I_21=-m(xy), I_23=I_32=-mxz and I_13=I_31=-mxz

(After doing a sum on all particles) You have a 3x3 Matrix, If you diagonalize the matrix, you will find three eigenvectors (the three principle axes) and three eigenvalues (Moment of Inertia in these principle directions).

(I am sure one will notice that in other words - that if you choose your coordinate system carefully along principle axes - specially if there is some sort of symmetry - I_12, etc will be zero and only I_11,I_22,I_33 will have non_zero values)
Added later: perhaps this gives more details :
Moment of Inertia
Eigen Values - priciple axes
see

[Vayutuvan]

I was reading up a little also. For a 3x3 matrix, if you have one real eigenvalue and a complex conjugate pair with the real part opposite in sign of the real eigen value, then there the complex pair makes the two unstable equilobrium configurations.

The real question is this - let us say we have a cylinder whose radius is as small as one can make - say a carbon nano-tube (heck - I don't know whether that is the smallest cylindrical structure one can make but let us say it is) and is rotated along one out of three of the axis of symmetry using a very precise machine/thingamagic setup, would there be still be tumbling?

If there is a prefect sphere (there are again three principle axis orthogonal to each other) will there be tumbling? In case of a sphere it should not matter. All eigenvalues are equal.

[Amber G.]

I was reading up a little also. For a 3x3 matrix, if you have one real eigenvalue and a complex conjugate pair with the real part opposite in sign of the real eigen value, then there the complex pair makes the two unstable equilobrium configurations.

Except that this matrix is symmetric so.. Eigen values are always real...! (As one will guess for this kind of real system)

Also just a note that principle axes are always orthogonal.

[RamaY]

http://agnitantra.com/battery/

evidently ancient India had knowledge about electricity and also rudimentary electrochemistry

the sage agastya had made dry cell...

the dark ages coincided with the abrahamic religions expansion... One surely needs to ponder how much Indians knew and how far had our great saints had reached in science ... gravity , calculus and astronomy were well known..they had concepts of atoms and photons ...

why did we not transition into an Industrial society back in 1000 AD is a mystery our historians should explore...

[Vayutuvan]

I was reading up a little also. For a 3x3 matrix, if you have one real eigenvalue and a complex conjugate pair with the real part opposite in sign of the real eigen value, then there the complex pair makes the two unstable equilobrium configurations.

Except that this matrix is symmetric so.. Eigen values are always real...! (As one will guess for this kind of real system)

AmberG: Yes, I realize the importance of symmetry. If it is positive (negative) definite, all eigenvalues are real and positive (negative). In that case, there is only one global minimum (maximum).

In this case, if there is one positive (negative) eigenvalue, and the other two are repeated negative (positive) eigenvalues, the above condition is trivially satisfied since the repeated eigenvalues can be considered complex conjugate with imaginary zero part.

I have one more question - how come Earth or moon do not flip? Is it because the axis is perfectly aligned due to construction (i..e the formation process of earth and moon etc.)?

Do deformable bodies have different dynamics? Let us say we start operating an extremely high speed centrifuge in
space. The structure is obviously not rigid and deformations would be measurably high at extremely high speeds.
What happens to governor like structures? Say something similar to the following?

Code: Select all

                          O
                         /   \
                       /      \
                      /        \
                    /           \
                 --+--          --+--
----------------------------------------------------------------
----------------------------------------------------------------
                 --+--          --+--
                    \            /
                      \        /
                       \      /
                         \   /
                           O

[Amber G.]

Hope people are okay for us to keep going in detail..

Except that this matrix is symmetric so.. Eigen values are always real...! (As one will guess for this kind of real system)

AmberG: Yes, I realize the importance of symmetry. If it is positive (negative) definite, all eigenvalues are real and positive (negative). In that case, there is only one global minimum (maximum).

Yes the matrix is positive definite symmetric (can be proven) and so, as expected, all values are real and positive..

I have one more question - how come Earth or moon do not flip? Is it because the axis is perfectly aligned due to construction (i..e the formation process of earth and moon etc.)?

I commented on this before (see previous posts). The interesting case is when 3 moment of inertia's are different , and the spinning axis happens to align with the intermediate principle axis.

(Of course, if it is not aligned with any principle axis, the body will tumble in unpredictable (or difficult to calculate) way ... like many asteroids do)

In earth's case, the three values of Inertia are almost equal, in fact it is spinning along the axis with maximum moment of inertia ( Earth is not exactly spherical and it is budged around equator). So the spin is fairly stable. The north pole does wander a small amount amount (around 100 meters or so) but not much. - There is precession ( slow - about a degree in 70 years)- due to torque produced by Sun and Moon system and nutation.. all fun stuff enjoyed by Newton (who was able to calculate these which fit the actual observed values).. BTW, there have been lot of discussion about this precession of Equinoxes in GDF dhaga...(Around Mahabharata's time, earth's spin was pointing in slightly different direction..and hence the position of stars looked different etc..)

[Vayutuvan]

guru ji:

Yes the matrix is positive definite symmetric (can be proven) and so, as expected, all values are real and positive..

If that were to be the case, there is only minimum whichn by definition is the [global] minimum. I am thinking that you cannot have +ve definiteness for this 3x3 matrix.

Can you chose a different origin which is equivalent to shifting the spectrum wholesale (Sylvester's Theorem of Inertia). If one is engaged in numerical calculations on a computing device, one would do a "Sturm Check", number of negative pivots while doing factorization (usually Cholesky which is $LL^T$ or $LDL^T$ or even $LD^{1/2}D^{1/2}L^T$) in a direct solver. Underlying system has to be +ve (-ve) definite or one has to do pivoting.

I am at a loss to see the connection between the Physics and the Linear Algebra.

[Amber G.]

May be this is helpful Positive Definiteness of the Moment of Inertia Tensor

[Amber G.]

TSJ and others who are interested ...

The Surface Of Light ...

The guy is amazing ... A physics grad student .. good voice .. no musical instruments all sounds created by as single person... Enjoy ..



And <click here for same performed live >

[TSJones]

Yes, he has wonderful voice and some great musical ideas, too! Thanks, for posting that video.

Did you know that the stars sing to us? Yes indeed, they are a cosmic choir and sing in all sorts of voices.

I was reading in Astronomy magazine a few months ago about the Kepler space telescope. It was originally built to find planets as they orbit around their stars. Kepler watches closely for the star to "darken" as the planet orbits around the star in front of Kepler's view. Astronomers also noted that Kepler could monitor the stars vibrations at the same time. These vibrations are somewhat similar to seismic waves that scientists study here on Earth. What happens is that as stars burn fuel they eject large fountains of gas that fall back down on the "surface" of the sun and create sound waves that travel around the inner structure of the star. Using these waves or vibrations scientists can determine how old the star is (how much hydrogen has been converted to helium) as well as various other data that also reveals information about the planets that orbit the star. They also note that as a star ages its inner core spins faster and it emits a higher vibration. So we have all sorts of humming and singing going on in the cosmos. One scientist has even converted these vibrations into an acoustical recording. There is a link on the internet to listen to it.

This was a very heavy revelation to me. To think that there was a heavenly choir so-to-speak blasting out their voices to me from 1000's of light years away was overwhelming at the time. I could not put down the magazine all weekend. I carried it with me everywhere I went. I read and re-read all the details so that I could properly understand it (as much as I was able). To be honest, I am still thrilled about it. And oh yes, does our very own star, Sol, sing to us? Yes, indeed! Although it is nothing too "loud". Just a mild mannered "hum" to accompany the other louder members of the choir. He's a very modest fellow, you know. And thank goodness for that!

I also discovered why gold is so rare. Scientists now think that gold is only created when two stars are closely orbiting each other (scientists tell us these stars are real screamers in the choir) and then collide into each other creating the very rare minerals such as gold. That's the theory anyway. Again. thanks for the video and the mixture of art and science.

I think I will now go to youtube and play Mandolin U Srinivas and read Astronomy magazine again.

[ramana]

VT The book Theoretical Physics by George Joos is a classic. We had The bound edition in our home. Has very good derivation of Bessel functions in it.

Where did you find The Dover edition?

TSJ, I happened To hear U Srinivas when he was little boy. He became much famous later. Related to SHQ.

[Vayutuvan]

ramana garu: Dover started reprinting a lot of classics - low price paper backs. Joos is available at amazon for under $20.

[Amber G.]

Okay - Inspired by recent few comments regarding ancient knowledge about electricity, and old classical mechanics type problems.. here are a few old beauties..

All these problems are not new, but IMO really challenging/fun if you have not heard them before. Discuss.. comment or solve/explain....All experiments below are easy to do so one can even verify the answers. (There are no "trick" questions.. all problems below are practical/real life situations so I encourage to do the experiment and share the result and if it matched your guess. One can assume all normal settings.

4 problems:

Motion 1 - There are two identical sized spheres, one solid other hollow but they look the same from outside and made of the same material. The solid one is, naturally heavier. You roll both the balls on an inclined plane. Which ball (if any) will roll faster? (By how much?)

(Extra credit: Does radius have any effect? If so in which way? How about substance (heavier vs lighter) it is made out of?

Motion 2 - Two identical cars (it is easier to use model cars, if an experiment is done) roll down a hill. (Car engines as well as brakes are not working - just rolling down due to gravity. One of the car has a heavy object in the trunk, other's trunk is empty. So the first car -- including the load -- is heavier. Which car (if any) will reach the bottom of the hill first?

Electricity 1 - One has a piece of breakfast cereal floating floating in water. Use a strong magnet (<modern rare earth magnets work best>) and move it near (without touching) and you will see that you can move the cereal. Why? (Perform the experiment)

Electricity 2 - Start a very narrow stream of water
and bring a elector statically charged comb near it. (Instead of comb one can use a plastic cup or a balloon - just rub it against dry hair ) . Why does water bend? (see the picture below)



(Extra credit - bring a strong magnet (described above) near.. does water bend? which direction? Why?)

Enjoy-- Hope there is some discussion...

[SriKumar]

Problem 1:
Mom. Iner_solid_Sphere = (2/5)*M*r^2; M.I._hollow_Sphere = (2/3)*m*r^2.

Applied Torque = Force x radius
Force seeking to rotate the sphere = mass*g*cos(theta) where theta is the angle of inclination.
Forcce equilibium equation in rotational system= Torque = I* alpha (where alpha = omega/t).

For solid sphere: M*g*cos(theta) = (2/5)* M*r^2/2 *alpha; alpha = 2.5*g*cos(theta)/r^2
For hollow sphere: m*g*cos(theta) = (2/3) m*r^2 * alpha; alpha = 1.5 g*cos(theta)/r^2

The solid sphere should reach the bottom faster because it has a higher acceleration (if both start from rest).
Problem 2: I am assuming a sliding mass (i.e. no car, no wheels, no friction, no rolling anything...just two masses sliding down an inclined plane).
They should reach at the same time because force equilibrium eqn is: mass*g*cos(theta) = mass* acc.
Mass cancels out; so acceleration for both is the same (and both start from rest).
PRoblem 3:
No idea why they move, I did check my cereal box for ingredients, and it contains some metals- a lot of Fe, some Zn, Cu, Mg..... Which might be ferro (or dia) magnetic. Just remembered that water is diamagnetic....need to think this one out (there's the irregular shape of the cereal, surface tension of water and the force exerted by magnet on the water...)


Problem 4:
Pure guess here....tap water has all sorts of charged ions and they are being affected by the statically-charged comb. I have no idea whether the charge in the comb is positive or negative (which would actually tell us the nature of the charged ions in the tap water).

[Amber G.]

^^^Good analysis. Let us see if there are more comments/discussion specially interesting will be to see people (those who have not heard of the problems before) doing (or watching) actual experiments and seeing if their guesses/calculations were correct..

(Next best thing is searching/googleing and seeing the demo on a you-tube , )

For example:

I am assuming a sliding mass (i.e. no car, no wheels, no friction, no rolling anything...just two masses sliding down an inclined plane).

..

...But in real life we do have cars with wheels and friction So, in practice will that matter significantly or will have negligible effect?..

[Sidd]

Problem 3: The cereal must be iron fortified i.e. small quantity of fine iron powder added to the cereal.

[SriKumar]

^^^Good analysis. Let us see if there are more comments/discussion specially interesting will be to see people (those who have not heard of the problems before) doing (or watching) actual experiments and seeing if their guesses/calculations were correct..

(Next best thing is searching/googleing and seeing the demo on a you-tube , )

For example:

I am assuming a sliding mass (i.e. no car, no wheels, no friction, no rolling anything...just two masses sliding down an inclined plane).

..

...But in real life we do have cars with wheels and friction So, in practice will that matter significantly or will have negligible effect?..

Couple of things....my above eqn. should be m*g*sin(theta) and not cos(theta) for the pulling force (the _reaction_ force is m*g*cos(theta))....been a while since I did this.

About having friction...if friction between wheels and the inclined plane is included, then frictional force has to be included as an opposing force. Frictional force = mu*R where R=m*g*cos(theta). Which car gets down first should depend on the values of the extra mass 'm1' being added to the original mass of the car 'm', and the coefficient of friction mu. The eqn I think should be: Is [m*g*sin(theta)-mu*m*g*cos(theta) ] > or < [(m+m1)*g*sin(theta)- mu*(m+m1)*g*cos(theta)]. This one has 2 unknowns: mu and m1. So, not sure how to take it further......

But I think if the above equations are moved around and sorted out (if I did my simplification correctly), the mass term cancels out and you get: mu = sin(theta)/cos(theta). Meaning, if mu is equal to tan(theta) or greater, the cars donot move. If mu less than tan(theta), the heavier car gets down first...i solved the equation assuming mu= 0.5*tan(theta) and saw that the overall pulling force on any car is 0.5*mass*g*sin(theta). Its been a while since I did all this...so there might be mistakes.

[Amber G.]

^^^..So when all is said and done..in practice (if one does an experiment in real life with a real car - or model car(s)) will one see a difference? (Hint: easy to perform an experiment and easy to see )
(Again: all problems above are interesting in a way that one can easily perform an actual experiment).

[member_28108]

So the observer travelling alongside the photon, at c, will see it travelling at c? Or will he now see a photon at rest mass. If the photon really (whatever that means) has rest mass 0, will it disappear?

So Einstein's theory of relativity is seemingly incomplete just as was Newton's.

No it doesn't. the whole premise of Einsteins work was the constancy of the speed of light in all directions for a given observer.

The error is mixing Newtonian and relativistic equations.The rest mass of a Photon is "Zero"
http://math.ucr.edu/home/baez/physics/P ... _mass.html