Talk:Harmonic oscillator/Archive 1

list
I agree that todo lists are handy, but they belong here.

Driven harmonic oscillator: a few note about what the response of the circuit to different AC frequencies.

Damped harmonic oscillator: Note well underdamped, critically damped

Damped, driven harmonic oscillator: Notes for above apply, transient vs steady state response, and quality factor.

Mat-C 18:33, 8 January 2005 (UTC)

Random elaborations
What about adding a bit about coupled oscillators? To take a famous and interesting example, two equal masses on three springs in linear area (spring constants a,b,a respectively) exhibit two modes, fast and slow.

What about adding more about quasiharmonic and nonlinear oscillators versus harmonic oscillators? What about links with the pendulum equation $$\ddot{x} + \sin(x) = 0$$?

What about symmetry breaking? Example (might better go in article on pendulum equation): take a pendulum swinging from a sliding bead on a frictionless rod, with bob and bead having same mass. Compute symmetry group. Now provide the sliding bead with a restoring force (spring with spring constant a). This breaks some symmetry (new point symmetry group subgroup of previous). ---CH 23:31, 15 March 2006 (UTC)

GA failed
For these reasons :
 * If we define [FORMULA], then the equation can be written as follows ... where does this equation comes from? ... is it angular velocity?
 * It is a bit harsh to be in an encyclopedia with half the variables not defined. It would be nice for a physics book but it is tough to understand for neophytes.
 * Needs wikilinks for words like nonhomogeneous, second order, inductor-capacitor, etc.
 * Not enough references too.
 * Too many formulas for the amount of prose. Lincher 15:11, 23 June 2006 (UTC)

Impedance vs linear response function
Reference is made to impedance, an article which does not exist yet. If just written an article on linear response function and added a corresponding link here. What about deleting the link impedance? --Benjamin.friedrich 15:35, 13 December 2006 (UTC)


 * There is now an article on Electrical impedance ... also fixed link in Benjamin.friedrich's comment because it pointed to a disambiguation page which causes it to show up on a database dump for things to fix :-) --mlewis000 18:59, 21 December 2006 (UTC)

Q
I'd like to see at least a brief mention of the Q of a harmonic oscillator. In addition to extensive use in engineering applications, it is run into regularly in physics. It's one of the most common characterizing statistics of underdamped resonant systems (Why is an atomic clock a better timekeeper than a quartz crystal? It has a higher Q). -- Chetvorno TALK 09:03, 25 November 2007 (UTC)

Fourier transform method is missing
There should be a section showing how to deal with the oscillator equations by using Fourier transforms. It is an extremely powerful tecnique and actually makes the physics easier to understand.

Agreed.... Try a general solution as a fourier series of harmonic vibrators:  Σ Aneiω n t. Transforming to the frequency domain not only makes life a lot easier, but can account for loss (imaginary component). BrintCorb 04:02, 14 August 2006 (UTC)

Do we have a young Einstein here on Wikipedia?


 * Before trying to add more and more complex equations, it's needed first to explain these motions with a more accessible language. For example, you can obtain the X=A cos (ωt) formula first with geometrical methods, then by solving the differential equation and finally with the advanced mathematical methods like Fourier or Laplace transforms. What do you think? --Twicemost (talk) 23:03, 3 February 2008 (UTC)

Merge the two Simple Harmonic Oscillator sections?
There are two distinct section "simple armonic oscillator" that say more or less the same things in different ways, someone should merge them.--Pokipsy76 08:09, 28 October 2006 (UTC)

There are two sections named "Simple harmonic oscillator", of which I found the second one is somewhat redundant as it looks similar to the first one. Also, the second one is not so related to the above paragraph and the following one. Therefore, I suggest the second one should be merged into the first one.--Kris Huang 05:57, 22 July 2007 (UTC)

I concur. --Chetvorno 22:52, 13 August 2007 (UTC)

I agree. The second section is very confusing and redundant. --Tweenk (talk) 11:04, 14 December 2007 (UTC)


 * I now deleted it. /Pieter Kuiper (talk) 16:54, 15 September 2008 (UTC)

Hooke's law
The reference to Hooke's law in the opening sentence is misleading. Hooke's law is about elasticity of materials. As such, it is a special case. There are many examples of classical harmonic oscillators that don't involve elastic materials - even some examples of these are already given in the article (small amplitude pendulum, LC circuit). These have a relation of the form $$d^2x/dt^2 \propto - x$$ but you can't really call it Hooke's law even though it is analogous. —Preceding unsigned comment added by 192.100.78.57 (talk) 11:49, 3 November 2009 (UTC)

question
I have a really stupid question. Why does the potential energy equation given here look like the equation for gravitational _kinetic_ energy? and what does the kinetic energy equation for a harmonic oscillator look like? thanks--165.247.80.72 01:51, 4 December 2005 (UTC)

Effect on a classical simple harmonic oscillator if the value of w is decreased? Effect on a quantum mechanical simple harmonic oscillator if the value of w is decreased? —Preceding unsigned comment added by 129.12.213.255 (talk) 20:31, 22 November 2009 (UTC)

Merge with Simple harmonic motion?
It seems that both articles (Simple harmonic motion and Harmonic oscillator, have the same topic and scope. As I am not involved in the editing of any of these articles I leave it to any of the editors of these pages to do so. If you guys think this merging does not make sense then feel free to delete the merging tag. Any comments? - Sanpaz (talk) 19:57, 15 September 2008 (UTC)


 * Although they are closely related as now written, my feeling is that Simple harmonic motion is an important introductory concept widely used in elementary physics applications, as well as being needed to understand more complex mathematical topics such as Harmonic oscillator, so it merits its own article. Differential equations are not necessary to understand simple harmonic motion, while they are necessary to Harmonic oscillator, so nonmathematical readers will be scared off by the mathematics in the merged article.  Also, Harmonic oscillator is big enough as it is; explanations of simple harmonic motion will get lost in the article, IMHO. -- Chetvorno TALK 05:59, 16 September 2008 (UTC)


 * As an alternative to merger, I'd rather see the overlapping material about harmonic oscillators removed from simple harmonic motion. Just have one harmonic oscillator example, and beyond that stick to describing the characteristics of sinusoidal motion. -- Chetvorno TALK 06:05, 16 September 2008 (UTC)


 * Where I come from with this merge is that I've seen many articles in wikipedia that start without any context and they evolve without knowing that other articles are already talking about the same thing. I am trying to make the editors of these articles to be aware of it. And ask them if they see the possibility of merging some of them, so there is a more compact structure of articles without eliminating content. For example these articles are related and each one is on its own, but each one covers somewhat the same thing: Simple harmonic motion, Vibration, Harmonic oscillator, Damping, Damping ratio. Perhaps the Classical mechanics navigation box is sufficient to link them and put them in context, i.e. mechanics. Comments? Sanpaz (talk) 23:45, 21 September 2008 (UTC)

I would support the merger, seeing as an SHO is just an object the displays SHM, the overlap is (for all purposes that I can see) total. Elocute (talk) 17:02, 7 May 2009 (UTC)


 * There is something to be said for the fact the the harmonic oscillator article includes driven and damped motion, as well as SHM, but I think that this is not enough to keep the articles seperate. I could easily see SHM, damped HM and driven HM as sections in a larger Harmonic Oscillators article. None of them are particularly complex concepts and I don't believe that there is enough to be said about any of them that would make ever resplitting the article necessary. Philc 17:06, 7 May 2009 (UTC)


 * As you said the harmonic oscillator article includes driven, damped, and driven damped, in addition to SHM. Each of these articles should be at least as long as SHM and can be expanded quite a bit.  If you merged SHM into HM then one should merge damping since it describe the damped case as well.  With all of that the article won't be as long as the longest articles, but the separating the articles allows each of them to have a separate see also section and to go in more depth with important examples, etc.  To me, the longer HM article would be too unwieldy.  TStein (talk) 05:26, 9 May 2009 (UTC)


 * I disagree, I think that the very crux of this issue is that they only need be long articles if they are kept seperate, the overlap between, for example, SHM and Damped HM is huge, the only difference is the introduction of damping terms. You could instead of providing each one from first principles, give a general equation for HM, and show how this simplifes to SHM without damping or driving or reduce to other varieties of HM when other terms are dropped. I think it is important that we respect that SHM is not a phenomena in it's own right, but just a special case of a larger family. Elocute (talk) 11:59, 9 May 2009 (UTC)


 * I feel that while this merger would serve technically educated readers, it would not serve the much larger number of nontechnical readers who are simply looking for a basic definition of SHM. SHM is referenced in many basic articles that don't have anything directly to do with harmonic oscillators, such as Phase (waves), Angular frequency, Wave, Sine wave, Curve,  Lissajous curve, Motion (physics), Vibration, Eccentric (mechanism), Time in physics, Trigonometric functions, and Exponential function.  The simplified nontechnical explanations of amplitude, frequency, and phase that SHM requires are out of place in this already long article.  SHM is introduced in algebra classes long before the mathematics of harmonic oscillators. -- Chetvorno TALK 16:00, 9 May 2009 (UTC)

I agree with [User:Chetvorno|Chetvorno]] it seems to me that enough people will be interested in both SHM and damped harmonic motion without wanting to see the full blown treatment to make it worthwhile to keep those articles. This is not my main concern though.

I came here from User:Skysmith/Missing_topics_about_Physics trying to find a suitable article to redirect Driven oscillation to, but it took me a while due to the maze of related articles that sometimes linked to each other and sometimes did not. My main concern is that the articles become easier to navigate. I was hoping to use harmonic oscillator as a main page to summarize and link all of the other articles. Now I wonder if Oscillator might be a better page to do that with. Then HO can be merged into Oscillator with more specific details being merged to SHO and damping. We would also need a more technical article (with appropriate thisfor) that covers the fullblown mathematical case. Is there such a page? TStein (talk) 02:34, 10 May 2009 (UTC)

Whilst SHM could be preserved as it appears to be regarded as of specific interest is certainly believe a great number of the other articles (especially those regarding damping, driving, damping ratio, oscillators or oscillatory motion) could be merged into a single harmonic motion or oscillatory motion article. And on another related, but separate not; whilst I respect that Wikipedia is intended to be of use for everyone, for some reason their seems to exist a sect of users who interpret this to mean it must be of use to exclusively the technically uneducated, whereas I think it is also important that the encyclopaedia maintains some credibility and usefulness as a resource for those at who are well learned on their respective subjects. Elocute (talk) 15:34, 13 May 2009 (UTC)


 * We all agree that WP needs to serve both; what we disagree on is how that should be done. If we try to serve both with the same article, we end up serving neither well.  I am bring this up on the main physics page since the whole subject is a mess, IMO.


 * TStein (talk) 21:32, 13 May 2009 (UTC)

I suggest using the the article Vibration as the main article. I am not sure about the term Oscillator. Vibration is a more general term: it is the actual physical phenomenon. Oscillator seems like the object that exhibits vibration (oscillation). Having said that, I think it is necessary to merge Oscillation, Oscillator, Harmonic oscillator,Simple harmonic motion, Damping,Resonance,Resonator,Parametric oscillator,Q factor,Damping ratio,Vibration,Mechanical resonance,Electrical resonance,Acoustic resonance,Torsion spring,Oscillator phase noise, into one single article called Vibration. It may seem it will be a huge article, but I am expecting that not being the case. If it turns out to be too big, then the splitting can then take place. It is excellent that your are taking charge of organizing these articles in some way. sanpaz (talk) 02:50, 14 May 2009 (UTC)


 * Although I am opposed to this drastic merger, if it is done I suggest that the main article not be Oscillator. All of the subarticles to be merged are based on the idea of linear systems; the mathematical frequency domain techniques used in them are strictly valid only for linear systems.  However the topic Oscillator covers nonlinear oscillators, such as relaxation oscillators, that require different mathematical techniques.   A similar objection could be made to Vibration - it isn't limited to linear systems.   For a main article, I suggest one within the area of linear systems, such as this one, Harmonic oscillator.  -- Chetvorno TALK 16:47, 25 June 2009 (UTC)

What I am doing to reorganize (TStein)
Those of you have have been watching might have noticed me editing the harmonic oscillator related pages a fair amount. I thought it might be prudent to let you know how I am approaching this at this point since some of the edits might not make that much sense.

My goal is to unify most of the technical stuff to this article (Harmonic Oscillator) in a manner that similar to what Elocute suggested above: by starting with full equation then showing the others as special cases. Before I do that though I want to have one page (I don't know which one yet, but I am leaning toward Oscillator that will have a quick summary of all the major articles in harmonic oscillators. As an intermediate step I have been fixing the leads of these articles and copying them to Harmonic Oscillator as separate sections each with a main tag.  They may end up staying here, but at this point, my feeling is that they will eventually end up in Oscillator.

To respond to Sanpaz, I don't think that vibration would be a good home for the summary page because that page is mostly from the engineering perspective as it probably should be. I want a page dedicated to something closer to the physics perspective.

I have been merging a few pages as I go and hope to merge a few more. I would like to merge damping ratio and damping for instance. If I get ambitious I might rename damping to damped oscillator (leaving behind the musical instrument material). I don't want to make too many changes without giving others more chance for input, though. If I have done that already then please let me know. TStein (talk) 22:10, 15 May 2009 (UTC)
 * I see the Oscillator article as a repetition of Vibration, or vise versa. An Oscillator oscillates/vibrates, so it makes more sense to address the topic of oscillation/vibration in an article called Oscillation or Vibration and mention that an Oscillator shows that behavior. There is no point on repeating concepts and equations in two articles. Applications (engineering perspective) of vibration can be address at the end of the article and link to other articles if need be. sanpaz (talk) 22:45, 15 May 2009 (UTC)
 * There is a subtle and important difference between vibration and oscillation, IMO. Vibration is usually an unwanted side effect of very complicated systems of many coupled parts and is more akin to noise.    Oscillations are typically simple being one or two or at most a few coupled systems (oscillations in latices being an exception) and easier to describe mathematically.  There is of course considerable overlap between the two.  Nonetheless, there are a lot of vibration topics that physicists (and engineers) studying oscillations are not interested in like noise spectrums and vibration testing,..  There reverse is true as well.  I dont think the equations of motion and a lot of the math that only applies to simple system has any relevance to vibration.


 * This should not count as much, but oscillation is much closer to what I envision. Much of the work has been already done; for instance the see also and the reference section agree with what I want to put there.  Further, I don't need to move or rework any any material (such as vibration testing) which doesn't fit with oscillations.  TStein (talk) 04:08, 16 May 2009 (UTC)


 * I agree that vibration and oscillation should not be merged and the present content of each makes the distinction clear. Using Harmonic oscillator as the home for the detailed classical physics (as it currently stands) is a good idea. Oscillation should remain a top-level overview of everything, including non-physical systems. I did some cleanup on all the articles. Big tasks left that I probably won't have time for are to eliminate redundancy between the example sections of Harmonic oscillator and Simple harmonic motion, and cleanup/explain the math in both articles. --UncleDouggie (talk) 13:38, 29 August 2010 (UTC)

Technical Error
In the section titled "Damped, driven harmonic oscillator" there is an error in the equation introduced by: One might see that for a certain driving frequency, ω, the amplitude (relative to a given F_0) is maximal. This occurs for the frequency

The equation gives the frequency for the oscillitory part of the transient solution, but that is NOT the maximum of the driven resonance. The maximum of the driven resonance occurs when ωZ_m is a minimum, and that occurs for ω^2 = k/m - r^2/(2m^2).

199.106.52.17 00:43, 19 April 2006 (UTC) SRS

I think there is another technical error in subsection Step input of section Driven harmonic oscillators, where force is given as a step in frequency, which does not make any sense. I think it would be correct to write the following, but i'm not sure
 * $$ F(t) = \begin{cases} F\cos\omega _0t & t \geq 0 \\ 0 & t < 0 \end{cases}$$  — Preceding unsigned comment added by Snezak (talk • contribs) 23:33, 28 December 2010 (UTC)

Difference between time period ( T ) and frequency ( f ).
There is the foremost difference betweem time period and frequency, as below:- IN TIME PERIOD: In time period the no. of vibrations or oscilations are constant(i.e., one vibration or one oscillation) while the time is variable. IN FREQUENCY: In frequency the time taken or required is constant (that is one second) but the no. of vibrations or oscillations are variable.

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OR Please visit: "http://www.SMBaberKhanGuAS.com/information/physics/oscillation" —Preceding unsigned comment added by 116.71.208.57 (talk) 17:44, 18 April 2011 (UTC)

Proposed new page: Spring system?
Spring-mass system redirects to this page, but for the life of me I can't find a Wikipedia page on simple spring systems or spring networks (whatever you want to call them). I found Gaussian network model and Anisotropic Network Model and Stiffness matrix, all of which are related, but I feel there should be a page describing a simple network of linear springs in n dimensions, and the solution to such system, particularly that such a system becomes an overdetermined system of linear equations. Thoughts? —Ben FrantzDale (talk) 16:01, 3 May 2011 (UTC)

Rotational motion equivalence
I found the section relating to motion of a pendulum over a turntable to be confusing and not particularly relevent, so I deleted it. But there might be a need for a different section exploring the analogy with angular motion. David s graff (talk) 21:40, 30 August 2011 (UTC)


 * The odd "pulsation" term was added in 2004, here. I recommend we get rid of it, as I don't find it in sources; maybe some other bits there would be OK to keep though.  Dicklyon (talk) 07:47, 4 September 2011 (UTC)

Amplitude and phase
The article says:

"$$ x = A \cos {(\omega_0 t + \phi)} \, $$

where the amplitude $$A \,$$ and the phase $$\phi \,$$ are determined by the initial conditions."

Should the article say how are the amplitude and the phase determined by the initial conditions?

If the initial displacement is $$x(t_0)$$ and the initial velocity is $$v(t_0)$$ what are the amplitube and the phase?

Glome83 17:22, 4 June 2007 (UTC)

To solve that you'd substitute those $$t_0$$ and position/velocity values in giving you a system of two equations in two unknowns. I feel like for people who are familiar with transformations of sinusoids through constants such as amplitude and phase would find this process trivial. For example, say you had your initial position $$x(0)=2$$ and initial velocity (which I will be denoting with $$ \dot x$$) $$\dot x(0)=3$$ you could plug in position in the following manner:

$$x(0) = 2 = A\cos\left(\omega_0(0) + \phi\right)$$

$$ 2 = A\cos\phi$$

This is our first equation. By our initial velocity information, we have the following:

$$\dot x(t) = -A\omega_0 \sin\left(\omega_0t + \phi\right)$$ (by taking the first derivative)

$$\dot x(0) =3= -A\omega_0 \sin\left(0t+\phi\right)$$

$$3=-A\omega_0\sin\phi$$ This is our second equation. Dividing the second equation by the first allows us to solve for $$\phi$$.

$$\dfrac{3}{2}=-\omega_0\tan\phi$$

$$\phi = \arctan \left(\dfrac{3}{2}-\omega_0\right)$$

Substituting this into the first equation gives the following, which allows us to solve for $$A$$, if you recall that $$\cos(\arctan(\theta))=\dfrac{1}{\sqrt{\theta^2+1}}$$.

$$2 = A \cos \left(\arctan \left(\dfrac{3}{2}-\omega_0\right)\right)$$

$$2 = \dfrac{A}{\sqrt{\left(\frac{3}{2}-\omega_0\right)^2+1}}$$

$$A = 2\sqrt{\left(\frac{3}{2}-\omega_0\right)^2+1}$$ — Trevor K. — 15:05, 10 October 2011 (UTC)

Current reversal?
Hi! I was trying to find some information on Wikipedia about current and voltage reversal, which are common terms when dealing with RLC circuits, like those used in flashtubes. It's especially important when dealing with especially fast (underdamped) discharges, but also when dealing with AC as well. Capacitor manufacturers will almost always ask what the current reveral rating of the capacitor should be, as detailed here. I was actually trying to find a place to link the term "current reversal," as used in the flashtube article, but am not sure if it should direct here, the ringing (signal) article, or perhaps someplace else. Does anyone have any ideas? Zaereth (talk) 23:39, 10 November 2011 (UTC)
 * It would be a little extra work for you, but I think the ideal way to handle this is to create a sub section in the Capacitor called "current reversal". Write 2 to 3 sentences describing it. Then create a redirect page current reversal that redirects to the section you created. If you need help with this let me know. TStein (talk) 18:32, 11 November 2011 (UTC)
 * Done. Thanks very much, TStein, for your advice. Zaereth (talk) 02:47, 17 November 2011 (UTC)
 * I made some copyedits and stuff there. As your source is clear about, the main issue is voltage reversal, not current reversal.  Dicklyon (talk) 04:47, 17 November 2011 (UTC)
 * Cool. Looks good, thanks! (This is the wonderful thing about Wikipedia.) With flashtubes, it must be the reversal of current that causes the damage because it causes sputter to occur at the wrong electrode. At least, books on flashtubes discuss current reversal, but not voltage reversal. (Two sides of the same coin, perhaps?) Anyhow, it's nice to have someplace to link the terms to, so thank you both for your help. Zaereth (talk) 17:29, 17 November 2011 (UTC)

Auxiliary equation link under the steady state heading
Apply the "complex variables method" by solving the auxiliary equation below and then finding the real part of its solution: Is this line under the heading of steady state solution correct? Isn't the auxiliary equation the characteristic equation, where it is homogenous and RHS is zero? I.e. the auxiliary equation is solved to find the transient solution in the section above? — Preceding unsigned comment added by 131.111.185.81 (talk) 14:46, 22 November 2011 (UTC)

inspect the phase please
I do not get it. What is the phase at resonance frequency? The value calculated in the universal section is used to give the formula in the oscillator section but the universal section assumes cosine force whereas the oscillator section has sine force. The phase given is discontinuous at the resonance frequency; it can be either $&pi;/2$ or $&minus;&pi;/2$ but clearly not both at the same time. Also, the external resource gives the phase formula inverted, which gives phase $0$ in this case, which is what I would expect &mdash; but it assumes cosine force, apparently contradicting the universal section (the thing is, the universal section assumes a cosine solution, which is mathematically elegant but inconsistent with the usual exposition). --Yecril (talk) 16:29, 16 June 2012 (UTC)


 * The phase at resonance lags 90 degrees relative to the input. It doesn't flip from plus to minus as the naive interpretation of that arctan formula would suggest.  Your external source drives with a cosine and gives the response as a sine so that the arctan's singularity will be moved to where it doesn't confuse the matter; but that is confusing in itself.  Using an arc cotan would be a good alternative.  Or just interpret the arctan as having primary value in the range of 0 to 180 lag (actually, the argument is negative at low frequencies, so the phase is negative, consistent with lag).  Dicklyon (talk) 16:52, 16 June 2012 (UTC)

missing units
to add to above comment by Lincher, i arrived at incorrect result using the formula given for simple harmonic motion until further research on another page told me that the spring constant k must have unit N/m. my value is N/mm. with the units corrected, my calculation succeeded (result 3.47hz versus 0.11 hz btw)

the usefulness of this wiki page is limited by its completeness. (noob Carl) — Preceding unsigned comment added by 184.162.151.83 (talk) 21:45, 21 May 2013 (UTC)


 * In equations used in mathematical physics, units are generally not specified because any compatible set of units will work. Of course, if you use mm in one place and m in another, you'll get screwy answers.  The newton is an MKS unit (in F = ma, force F in newtons is mass in kg times acceleration in m/s/s, yes?, so mm would be a mixed up thing to use with N if you wanted to be sensible). Dicklyon (talk) 05:50, 22 May 2013 (UTC)

I too believe units should be specified. SI units for preference. The reason is that without them we are left with undefined symbols that confuse naive readers (like me). These symbols may be common knowledge to subject matter experts, but perhaps not to the average reader. — Preceding unsigned comment added by 118.208.102.157 (talk) 14:59, 25 December 2015 (UTC)


 * I agree. A common way to do it is to add a "definition of symbols" box at the bottom of the article giving for each variable its name and its SI units. -- Chetvorno TALK 02:56, 27 December 2015 (UTC)


 * In any physical equation, unit analysis is important, so yes, the units should be specified. However, specific units should be avoided in a general description in this article.  The individual should decide which specific units to use.  For a generic equation, like
 * $$ \vec F = -k \vec x \, $$
 * generic units should be used in this article, where $$ M $$ is mass, $$ D $$ is distance, and $$ T $$ is time. We then have $$ \vec F $$ in units of $$ (MD)/T^2 $$, $$ \vec x $$ in units of $$ D $$, and $$ k $$ in units of $$ M/T^2 $$. — Anita5192 (talk) 05:55, 27 December 2015 (UTC)


 * Added a "Definition of terms" box with both the dimensional units and SI units.   Thanks,   Lincher, 184.162.151.83, and 118.208.102.157, for bringing up a serious deficiency in this article. Hope that makes it more comprehensible. -- Chetvorno TALK 09:20, 27 December 2015 (UTC)


 * Thank you!  What you just added should make the article more clear to non-scientists.  I also added $$ a $$, the acceleration of the mass. — Anita5192 (talk) 21:24, 27 December 2015 (UTC)

Assessment comment
Substituted at 17:10, 29 April 2016 (UTC)