I'm learning about astrodynamics on my own and I was wondering why the $r$ is cubed in the vector notation for of Newton's Law of Universal Gravitation:  $$\vec{F}_g=\frac{Gm_1m_2}{|\vec{r}|^3}\vec{r}$$  I am familiar with Newton's Law of Universal Gravitation of the form:  $$F_g=\frac{Gm_1m_2}{r^2}$$  Is there something obvious I'm missing?
I am reading an older physics book that my professor gave me. It is going over Coulomb's law and Gauss' theorem. However, the book gives both equations with an $r^3$, not $r^2$, in the denominator. Can somebody please explain why it is given as r^3? An image is attached for reference.  Also for equation 1-24, can somebody please explain how the middle side is equal to the right side with the del operator?
In the Harry Potter world, there's the term "muggle" for people with no magical abilities, "muggle-born" for those with magical abilities who's parents are muggles and "squib" for those with no magical abilities who's parents are magical.  But is there any collective term for all magical people (as opposed to wizards for males and witches for females)?  I don't recall seeing any word that refers to all men and women and children with magical abilities, and it's the only group left without noun or collective noun to describe them. Can Mystique create objects? .NET String.Format() to add commas in thousands place for a number Timeline vote summary shows votes for the 24 hours following the timestamp, which is confusing The new timeline is pretty snazzy. But there's one thing I've noticed that's a bit jarring; the timestamp for vote summaries is always 0:00:00 on a particular day, and the summary covers the rest of that day. So you end up with a summary that always shows up earlier than the events that presumably triggered the votes that show up in it.  A simple solution is to have it display 23:59:59 instead, and sort accordingly. This way the summary comes in after the events it summarizes. I had always kind of wondered why we didn't see interference in things like windows -- we were taught that the condition is that the thickness of the film/slab/medium just has to be an integral number of wavelengths, $d=m\lambda$. My logic was, if visible light has wavelengths ~400-780nm, any window pane has got to be an integral number of wavelengths (or close to integral) for some of the wavelengths in the visible.  For example, if a window is, $6mm = 6\times10^6nm$ thick, then you should see interference for wavelengths (and some range around them) 200nm, 250nm, 300nm, 400nm, 500nm, 600nm... Because 6mm is an integral number of each of those.  Obviously that would be really annoying and isn't the case. I found  (warning: PDF) source that seems to explain it and I was wondering if what they're saying is sound. On the 10th page (p171 for them), it says that the reason we don't see this interference is that no window is that smooth, it has a variance in the thickness of at least several wavelengths, so the interference effect is effectively average out among all those thicknesses, which usually gets rid of it. They do the math and show that.  Is that the reason?  If so, that leads me to two questions: Could we, if we were really careful and used something like MBE, make a glass window with that sort of accuracy, and then see interference effects in the optical range? I know that quartz crystal monitors can easily give you nanometer accuracy for evaporators and sputterers, but they also have pretty limited growth rates.  My second question is, then, do we see interference effects in glass windows with electromagnetic radiation of a comparable wavelength, like microwaves?  Thanks! I've removed the poll style answers in favour of open discussion.  I realise now this isn't the best way to have a proper discussion about this issue - sorry to anyone who commented/voted.    This is mostly a followup on this question:    As I've answered, the reason this is happening is that the root database (in the production Stack Exchange database server) is still called StackExchange.Programmers - a database name that propagates to SEDE and through it to the data dump.  The question here is what to do about it - the current situation is somewhat broken for users of the data dump.  These are the options we have:   Do nothing Update the root database name (requires downtime - so very undesirable) Update the update scripts so the SEDE database name is the new name, change will flow to the data dump export (breaks any queries using the old name) Update the export to the data dump so only it gets the new name Something else we haven't thought about   I have added all of these options as answers below - please vote on the one you most prefer, we will try to accommodate the top voted answer. I'm just trying out Arduino Uno for the first time with 2 blinking LEDs on a breadboard. All the tutorials on the Internet seem to use a resistor. I do know the function of resistors, but does it really matter here? These LEDs are working just fine without a resistor. Since the version of the word "that" might matter to the correct answer, I'll provide the actual title in question:  Title:  Existing Solutions that Didn't Work  If someone could explain the why of the answer, I would be appreciative.  EDIT:  It has been suggested that this is a duplicate, and that may be, in a sense.  However, the after reading the suggested prior question, I'm no closer to my answer.  I think it may have to do with my use of the word "that."  Perhaps it is incorrect?  Using these definitions of the word "that:" I'm having trouble figuring out which definition of that matches my title.  Am I even using it correctly?  Perhaps I should use which instead:  Possible Title:  Existing Solutions which Didn't Work  If my original usage is incorrect, and I should use which, then the answer is simple since which is always a pronoun or adjective, and those are both capitalized in a title.  However, I'm having difficulty understanding which part of speech that is in my original title. Convert char to int in C# How do I learn to become a good GM? I have played and tried to GM games since I was in elementary school, beginning with games like Shadowrun, Earthdawn, and Champions. Recently, I've gotten dack into gaming with a variety of other games (newer Shadowrun and Earthdawn, plus TOON! and Outbreak:Undead).  I've found examples of actual play and advice on how to actually GM a game in games to be lacking. I can only help but feel I am not alone as someone who isn't already a LITERAL dungeon MASTER.  I fully realize that there are an unlimited number of GM styles, personalities, etc. I do not think I am a very good GM though. No one else wants to be the GM, and few care enough to learn the complex rules that I often have to assist with and explain AS WE PLAY. Being left with the responsibility is quite overwhelming unless I create dynamic encounters or make it all up on the spot. I'm often left having no idea what to do.  Examples of Why I Think I Need Help  This question isn't about these examples so please don't dig into them in the answers - they're here to explain the basic things that leave me feeling like I don't know how to proceed as a GM.  I've read preparation is important, but last time I tried to make a story, the players immediately killed the primary character (a friendly person.) within about 2 minutes of starting the game. While I realize now that I let the dice take control over the story, it was rather difficult to continue a story that relied on this friendly character, with the players making enemies with the character's faction. It is still discouraging when my only other attempt at a prepared story ended with the players wanting to go in a complete opposite direction of where I planned. Both times, the hours of work I put into it were completely wasted, leaving me as ignorant as I always was before.  My group often wants to play Shadowrun, Earthdawn, or L5R- but I feel as though I am a horrible GM. The last time we played Shadowrun, I was bored out of my mind as we rolled hundreds of dice to do an obvious straight-forward "kick down the door" mission.   If I make security as I believe it should or as the forum threads describe their security or a few example missions I can scrounge up (examples of stories, not actual gameplay), my players literally feel completely hopeless as to how to achieve their mission without total failure.   One time when I was a teenager, I had players blow up at me saying "We have no idea what would work. You're telling us that our character's experience or the ally operative says that would get us killed, but what else can we do?" Since there are no examples of what players can do - I am left with the same feeling as the players. As an adult, I've had a similar occurrence where a player said to me, "I don't know what to do. I don't think we can even begin this mission." I had to make something up and just go with it, creating an NPC group (that fit in with the story) that goes in before the players to do something stupid so they don't have to. Mainly because I ALSO didn't know how to accomplish the mission, so I acted as the players would using my NPC group. Extremely boring and IMO a failed dungeon run.  I've had successful stories, but I've had more boring encounters than successful ones. Seeing as how roleplaying takes endless hours to complete a story, I do not see benefit in an activity that wastes more time than is enjoyable.  How Do You Learn To GM?  Where are the resources? Books, blogs, gameplay examples? Am I missing them? How did YOU learn to be a good GM? Or do you feel as bad at this as I do, even when it's a successful, fun encounter? What if the Earth stops in its orbit and instead falls straight towards the Sun? The Earth orbits around the Sun because it has angular momentum. If we stopped the Earth in orbit and then let it fall straight towards the Sun, how long would it take to reach the Sun in seconds?  Details and assumptions  The mass of the sun is 2*10^30 Kg. The mass of the earth is 6*10^24 Kg. The earth is 149,600,000 Km from the Sun. You may treat the Earth and Sun as point masses. If column matches another file, print every line with match (awk/grep) How do I solve this limit: $\lim _{x \to 0} \left(\frac{ \sin x}{x}\right)^{1/x}$? Java and manually executing finalize I want to make Morrocan-stye pickled lemons, but I was only able to find , and it says:  In On Food and Cooking, Harold McGee suggests that a solution of five to 10% salt is needed to achieve a good North African-style preserved lemon  I make sauerkraut with 20 grams of salt per kilogram of cabbage, so 2%. So lemons taking much more salt than cabbage is surprising and confusing; since they start out acidic, I would expect them if anything to require less salt. I'm not worried about harmful microbes growing; I'm worried about the fermentation microbes not being able to grow. It's my understanding that higher salt concentrations with cabbage not only slow down fermentation, but actually .  had the same issues:  I've never quite understood the need for such excessive salting when making preserved lemons. They still ferment great at normal levels of salt, and you then need to worry less about them being overly salty to use in future recipes. As far as I can tell it's solely from tradition. Also, from checking out On Food and Cooking, it seems that McGee only points out that originally they are done with 5 to 10% salt content. I'm curious where you read/saw that he recommended such?  Does anyone have any text- and/or experience-based opinion agreeing or disagreeing with the &quot;5 to 10%&quot; figure? How to make a circular array using offset object as center? induction proof: $\sum_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$ I encountered the following induction proof on a practice exam for calculus:  $$\sum_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$$  I have to prove this statement with induction.  Can anyone please help me with this proof? Maps from subsets of $\mathbb{R}^2$ that are either open/closed/continuous I'm self studying J. Lee's Introduction to Topological Manifolds, and after doing all other exercises on chapter 2, I can't seem to come with the proper counterexamples for this one.      For each of the following properties, give an example consisting of two subsets $X, Y \subset \mathbb{R}^2 $, both considered as topological spaces with their Euclidean topologies, together with a map $f: X \rightarrow Y$ that has the indicated property.         Open but not closed/continuous   Closed but not open/continuous    Continuous but not open/closed         etc... (all cases)   I think the reason for my difficulty is a lack of geometric intuition of what open, closed and continuous maps look like as deformations of shapes in $\mathbb{R}^2$ (I can construct counterexamples from artificial made up spaces).   I tried squashing balls and 'pumping space' into lines, as well as joining strips in annulus shapes, but all the examples I seem to come up with are either too well behaved (i.e. all 3) or too misbehaved.   Instead of a case by case answered I would prefer a couple worked out cases and some intuition on how to visualise open / closed maps in this context. Why does signing on to one site not sign me on to all of them? I've seen this question asked 3 times, but in none of those threads did I find a satisfactory answer.  I'm not asking about reputation: it makes sense that different sites should have different reputations.  However, consider the StackExchange drop-down menu at the top left of the screen.  When looking through the "Hot questions", the links are often to sister websites of StackExchange (programmers.stackexchange i.e.).  It doesn't make sense to me that if I go to one of those sister sites, I'm no longer logged in.  I feel like it's counterintuitive to click a link to go answer a question, only to find that you can't answer that question until you log in.  What is the reason for this?  Is there an option to do as I'm suggesting? How do you correctly say large numbers