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Oct
18
comment Why is the angular average of the direction of the momentum squared a third instead of one?
Betohaku: "but where does that $\text{cos}^2 \theta$ come from?" -- That's the big, fair question which I don't know how to seriously answer either. "Somehow a factor $\text{cos} \theta$ for each $\hat p$ ?" -- And correspondingly interpreting "the dot product" $\hat p \cdot \hat p$ as plain square of this factor. Well, perhaps this has something to do with "direction cosines". Perhaps the case can be made that this was intended in the article, and (moreover) is also correct. But I cannot make that case, as a serious answer.
Oct
18
comment Instantaneous Centre of Rotation
ja72: "Also in the answer I say the position relative to A is $r_{icr}$" -- I hadn't read that (sorry!) because I didn't look in any detail at the part of your answer dealing with coordinates. (And I'm not entirely happy with the equation I proposed in my above comment either.) So, after all, I suppose we do agree on: $$\vec r_O[~ICR~] := \vec r_O[~A~] + \frac{\vec \omega \times \vec v_O[~A~]}{\|\vec \omega\|^2}$$ for any member $O$ of an inertial system through which $A$ has been moving. "Do you see that $\vec v_{icr}$ is zero" -- I see that's the case you treated; but why should it?
Oct
18
comment Why is the angular average of the direction of the momentum squared a third instead of one?
Betohaku: With your added guidance (+1) I also find explicitly (on p. 20, following eq. (6.1)) that "{...} $\hat p$ denotes a vector of unit magnitude in the direction of $\vec p$". Now: I can't help but notice that $$\frac{1}{4 \pi}~\int_0^{2 \pi} {\rm d}\phi \int_0^{\pi} \text{Sin}[~\theta~]~{\rm d \theta}~(\text{Cos}[~\theta~])^2 = \frac{1}{4 \pi}~\int_0^{2 \pi} {\rm d}\phi~\frac{2}{3} = \frac{1}{3}.$$ (But I have far too little experience with these sort of calculations to claim that this was actually intended. ...)
Oct
18
comment Instantaneous Centre of Rotation
ja72: "$\vec r_{ICR}$ should be zero by definition." -- Is, or could, "the inst. center of rotation $\vec r_{ICR}$" not be defined such that $$\vec r_{ICR} := \frac{\vec \omega \times \vec v_A}{\| \vec \omega \|^2} + \vec r_A,$$ for all participants $A$ described (this way) by the same parameters $\vec \omega$ and $\vec r_{ICR}$? "Note [...] to use the vector triple product identity [...]" -- a.k.a. the Graßmann identity, sure. p.s. Since you didn't address me in replying I hope I didn't miss other replies of yours.
Oct
18
comment Does the force between two point charges change when inertial reference frame changes?
LCFactorization: If $r$ (the separation between $A$ and $B$) is changing while $v$ (the speed at which both $A$ and $B$ are moving wrt. a suitable reference system) remains constant then that's ... difficult for me to comprehend. On the larger, implied question whether geometric relations can be derived from "laws" of (electro-)dynamics: Formally, perhaps, yes; AFAIU. However, approaching the problem experimentally, we always need to determine geometric relations first, and separately, in order to find out in the first place e.g. whether $A$'s and $B$'s charges were equal (as you presumed).
Oct
18
revised Why don't we consider the electromagnetic field in the equations of Bohr's theory of atoms?
correction to my previous attempts at editing.
Oct
18
comment Why is the angular average of the direction of the momentum squared a third instead of one?
Betohaku: "In the article nano.northwestern.edu/intranet/pubdocs/quasiclassical.pdf it is stated that" $$\langle \hat p \cdot \hat p \rangle_{\hat p} = \frac{1}{3}.$$ -- Would you please point out exactly which number this equation has in the article mentioned? (I'm more familiar with the factor $1/3$ arising in certain averages of $\vec p \cdot \vec p$, rather than $\hat p \cdot \hat p$ ...)
Oct
18
reviewed Edit Why does using the modified Kepler's 3rd law and using the gravitational parameter yields different results?
Oct
18
revised Why does using the modified Kepler's 3rd law and using the gravitational parameter yields different results?
Rewrite mathematical expressions in Tex
Oct
18
comment Negative mass? How it works and can it travel the speed of light?
I consider this response very appropriate and helpful, even though it appears more as a meta-answer (to the meta-question why the OP cannot easily be answered, as it stands), than a direct answer to the OP. Therefore, from me: +1. (YMMV, @Danu.)
Oct
18
comment Time dilation and reference frame in an orbiting context
p.s. I'm very sorry for my mistake in formatting the first comment I had submitted here, which has meanwhile been removed (by others, because I wasn't able to do that myself; mainly due to the effects of the aforementioned mistake in formatting. To those who accomplished that: Thanks for removing my misformatted comment!). I'm especially sorry that this seems to have caused comments from other users having been removed, too; and perhaps even more consequences which I neither intended, not forsaw.
Oct
18
comment Time dilation and reference frame in an orbiting context
John Doe: "A is moving around B from B's perspective. And B is moving around A from A's perspective." -- Oh, like Calcutta is moving around the Easter Islands "from the Easter Islands's perspective", and vice versa? "They can't tell which one is the "inertial system" of reference." -- It seems that you are yourself not quite sure how to tell that; which eventually makes your question problematic, as it stands. Because one way to say that A is (as good as) a member of an inertial system is to quantify B's motion with respect to A (and those at rest wrt. A) by a speed value $v$.
Oct
17
comment Time dilation and reference frame in an orbiting context
Timaeus: "I think I was clear and addressed the case where one is inertial and that there must be an asymmetry in that case." -- Of course the word "inertial" doesn't appear in the OP; but "speed $v$" does. "You are welcome to write your own answers or ask your own questions" -- Right. I'm hesitatant to put an answer to this OP because the notion of "inertial system", and hence of "speed $v$" etc., is by itself so difficult. But yes: I plan to ask about "(the, or a, coordinate-free notion of) speed wrt. members of a non-IS" ... tomorrow.
Oct
17
comment Time dilation and reference frame in an orbiting context
Timaeus: "can describe kinematics in a non inertial frame." -- Sure: determining who met whom, in which order; and who saw whose signal indications in coincidence, or in which order. From this, geometric relations between the participants may be derived. The notion of quasi-distance is helpful ... "If you assign positions and times to events" -- You mean: coordinates?? "you can compute velocities and hence speeds." -- coordinate velocities, or coordinate speeds. But surely, the OP doesn't know better, either.
Oct
17
comment Time dilation and reference frame in an orbiting context
Timaeus: "[...] So we conclude that at most one is moving inertially." -- What do you suggest is meant by "speed $v$" which is prescribed in the OP? Is "speed of $B$ with respect to $A$" defined if $A$ is not a member of an inertial system?
Oct
17
comment Time dilation and reference frame in an orbiting context
John Doe: "From $P_a$'s perspective, $P_b$'s is moving inertially, and from $P_b$'s perspective, $P_a$ is moving inertially." -- Being a member of an inertial system is not a matter of someone else's perspective, but to be determined intrinsically, properly. And by the definition how to do that, and how to determine who is "orbiting" it follows that the setup description is inconsistent. p.s. The first version of my first comment happened to be formatted so unfortunately that I cannot even delete it anymore. Would someone please take care of that.
Oct
17
comment Time dilation and reference frame in an orbiting context
John Doe: "From $P_a$'s reference, $P_b$ is orbiting around him at speed $v$." -- Is it therefore understood that $P_a$ is a member of an inertial system (whose members are all at rest to each other, and who determine fixed distance relations between each other, allowing them to actually determine the value of $P_b$'s speed with respect to $P_a$ and all members of this inertial system)? "But from $P_b$'s reference it is the other way around" -- If $P_a$ is a member of an inertial system then $P_b$ cannot be likewise supposed to be a member of an inertial system while orbiting $P_a$.
Oct
17
comment Instantaneous Centre of Rotation
ja72: $$\vec r_{ICR} := \frac{\vec \omega \times \vec v_A}{\| \vec \omega \|^2}.$$ -- I wonder whether/how this fits with $$\vec v_A = \vec \omega \times (\vec r_A - \vec r_{ICR}) + \vec v_{ICR} $$ ...
Oct
17
revised Simpler derivation of Sackur-Tetrode equation
corrected spelling the surname of O. Sackur in the title; and some copy-editing
Oct
17
revised Relativity and “light years”
added tag [tag:metric-space], since the question is foremost about meaning and determination of distances