Suppose I put a pendulum of metal ball and very thin rope in highest achievable vacuum. What would keep it from not swinging forever?
I am attempting to settle a friendly bet.  Would a pendulum swing indefinitely in a hypothetical vacuum (i.e. no air resistance) having a hypothetical frictionless bearing (i.e. no energy lost due to friction) assuming the following   The frictionless vacuum is on Earth (9.8 m/s^2). The pendulum is already in motion and no other external forces other than gravity act on the pendulum.
Forcing decimal numbers to use comma in ArcGIS Field Calculator? How to remove a repository? Prove that $\Vert f(b)-f(a)-f'(a)(b-a) \Vert\leq \sup_{x\in [a,b]} \Vert f''(x)\Vert\Vert b-a\Vert^2$ Let $\Omega$ be an open set in $\Bbb{R}^n$ and $f:\Omega\to \Bbb{R}^m $ be of class $\Bbb{C}^2.$ Let $a,b$ be in $\Omega$ such that $[a,b]\subset \Omega.$ Prove that  \begin{align}\Vert f(b)-f(a)-f'(a)(b-a) \Vert\leq \sup_{x\in [a,b]} \Vert f''(x)\Vert\Vert b-a\Vert^2.\end{align} where $$f'(a)(b-a)=\big&lt;\nabla f(x),(b-a)\big&gt;=\sum^{n}_{k=1}\frac{\partial f_i}{\partial x_k}(a)(b_k-a_k).$$  I  think the idea of Mean Value Theorem for $C'$ functions in $\Bbb{R}^n$ () will help me in showing it but I don't know how to! Please, can anyone show me a proof or reference? A well-supported wireless PCI adapter for Ubuntu 12.04? I have 3 tables and I have to inner join Table A with Table B but a left outer between Table A and Table C.   Can I combine outer and inner join in the same query? I can nest queries and attain the desired result but I am unable to do both the joins in the same query. It appears that in other SQL languages, the order of joining is important. Is this the case in SQL Server as well?    Ok, here's the scenario.  Consider 3 tables. Table A, Table F, Table D.  I will need the recordset to contain all rows in D irrespective of whether it exists in F (after it's inner joined with A). So, a outer join comes to mind. What I would need is:   First do a inner join between A and F to get a set (this may be a null set) Then do a outer join with the recordset in (1) with D How to select multiple files with keyboard only in windows explorer Questions like  dealing with Ancalagon in particular, and Middle-earth's dragons in general, clearly show that Ancalagon was ridiculously, absurdly, unimaginably enormous.      The non-canonical image above shows one person's impression of how the different dragons of Middle-earth stacked up, with Smaug looking absolutely tiny at the far left, and Ancalagon on the right, too big to even fit in the picture.    Below, a presumably canonical picture, painted by Tolkien himself, shows Smaug in relation to Bilbo.  The rough measurements I took from this image suggest that Smaug is about 11 times longer than Bilbo is tall.  Depending on how tall Bilbo is - almost certainly somewhere between 3'6" and 4' tall - Smaug is approximately 38-44 feet long.      If the first picture is even vaguely accurate, Ancalagon would appear to be a minimum of something like 720 feet long from the tip of his nose to the end of his tail:  the visible portions of his head and body are at least 6 times longer than Smaug is from end to end, and we have to assume that the portion of his body that is out of frame is at least 3 times longer still, perhaps even 4 times or more (so the upper range of his length might exceed 1,000 feet).  Does Tolkien give us any way to estimate how big Ancalagon was?  It isn't his normal practice to describe things in great detail, I realize, but he often gives a few hints that allow for some rough approximation to be made.  Does anything he says suggest that the first image is anything like what he had in mind?  Note:  I haven't read anything about Ancalagon from Tolkien's own pen, except a few quotes on this site;  the most important bit of information I've gleaned is that he smashed 3 volcanoes to pieces when he died (or something like that).  How accurate is this, and what else did Tolkien say?    This question is not a duplicate of "", because none of the answers to that question address Ancalagon's size in any detail - except my own, which is based on what little information I was able to glean from the two pictures I included in this question.  All the other answers address how Ancalagon was killed, not how big he was.  I have no interest in how he died, only how big he was, or rather, what hints we can find that might give some indication of how big he was.  I wouldn't ask this question, linking that question in my own, if it contained the information I am interested in.    Note 2:  Regarding the relative sizes and impressiveness of Smaug and Ancalagon:  Imagine a meeting between Smaug and Ancalagon in Dragon Heaven: Smaug: "How'd you die?" Ancalagon: "A flying boat full of fire made by the Angels and driven by a Magic Elf smashed into me, so I fell onto a mountain and broke it, and I died. The Magic Elf lives in outer space now, and the boat turned into a star. How'd you die?" Smaug: "Um... Some dude shot me with an arrow." A case of optional “that”: “check the” vs. “check that the” Consider the following use case:   Please check the username and password are correct. Please check that the username and password are correct.   In this case, I would say that that is required because it feels more natural to me and so the flow is better. However, is the lack of that wrong in this case?  The question boils down to: when is that optional? See . A quick summary:   That is optional if the pronoun is the object. That is required if the pronoun is the subject.   I think that we fall into the second case: that is required because the relative pronoun is the subject. But I'm not sure if I'm interpreting these rules correctly. Can someone please advise? I understand using phone while it is being charged by a normal charger is not advised. What about using phone while it is being charged by a portable charger (powerbank)?  I own a HTC one and i was told by a mobile technician HTC’s own portable charger only should be used for HTC one. How much truth is there to it as HTC powerbank is not available in my city and i am thinking of using Sony portable charger for charging my HTC One? I have an Android device which is running Android 4.0.3 (or something like that).  My little brother decided to try and guess my lock screen pattern the other day, and ended up locking me out.   I have since been unable to log back in to the tablet using my Google Account details. I provided the correct login details, however it says they are incorrect. Could this be connected to the fact I have no internet connection?  I have no internet connection as the device is WiFi only, and the WiFi is currently turned off. I cannot turn it on, as I am locked out.  How can I log back into my tablet without wiping the user data? appears to have been question banned because of two unpopular Meta suggestions. (He says so in )  Look at the two contributions that appear to have gotten him banned. Like those of many others, they may not be of perfect quality, but they are not actively harmful to the site, either. As it stands, you can get question blocked on Meta simply for asking questions that aren't popular with the Meta crowd.  Why does the team insist on keeping the question ban active on Meta?  Why can't real problem users be dealt with through manual moderator action? Don't tell me it's a problem of scale; as far as I can see, actually toxic users are very rare on Meta.  People will frequently complain that Meta actively discourages the expression of unpopular viewpoints and suggestions. With the threat of question bans in place, I find those complaints difficult to refute. I hate that. Uniform Circular Motion Why does the object not go inward, into the circle if the acceleration is inward? I think its because the velocity to outward? So they sort of cancel each other out? But if the speed is kept constant and the acceleration is inward, won't the object eventially move inward?   Also, what keeps the direction of acceleration always point inward? What is the meaning of modular fractions? What is the meaning of $$ \\\frac{m}{n}\equiv k\pmod l $$ and $$ \\\frac{m}{n} \mod l $$ $m, n, k, l \in\mathbb N$ and $\gcd(m, n)=1$. I am using App Backup &amp; Restore for creating backup of my apps but with KitKat its using the internal SdCard as Storage.  When I explicitly set the external SdCard path it gives the error that it is unable to create folder,  I have also tried with creating folder under android/data/[packagename]/files/ but it is still not able to do the same.  Is there any solution so that apps can use external storage for writing files.  Note : I don't want to ROOT my Phone. How do I get the Ender Dragon egg and what does it do? So I finally killed the Ender Dragon! As I was climbing the portal back to the world, I noticed that on top of the column there was an egg. So I attacked, mined, clicked and nothing happened, it just teleported away. What does the egg do and how do I get it? Given a countable collection of metric spaces $\{(X_n,\rho_n)\}_{n=1}^{\infty}$.  Form the Cartesian Product of these sets $X=\displaystyle\prod_{n=1}^{\infty}X_n$, and define $\rho:X\times X\rightarrow\mathbb{R}$ by  $$\rho(x,y)=\displaystyle\sum_{n=1}^{\infty}\frac{\rho_n(x_n,y_n)}{2^n[1+\rho_n(x_n,y_n)]}.$$  Show that $\rho$ is a metric on $X$ whose induced topology is equivalent to the product topology on $X$.  So basically what this problem is saying is that there's a canonical way to define a metric on the countable product of metric spaces.  I showed in a previous problem that the topology induced by $\rho_n$ is equivalent to that induced by $\frac{\rho_n(x_n,y_n)}{1+\rho_n(x_n,y_n)}$.  And thus we can go ahead and just assume that $\rho_n&lt; 1$ for all $n$ and replace our infinite series by  $$\rho(x,y)=\displaystyle\sum_{n=1}^{\infty}\frac{\rho_n(x_n,y_n)}{2^n}.$$  Now comes the interesting part: how should I go about showing that the product topology on $X$ and the topology induced by $\rho$ are equivalent?    The basis for the product topology given to me in my book's definition is that of cartesian products made up the $X_n$ except for finitely many which are $O_n$ for some open subset of $X_n$.  However I believe I was able to improve upon this and show that I could decompose these into a basis where the $O_n$ were all open balls induced by their respect $\rho_n$ metric.  For $\rho$ I'm using the basis of open balls that it induces, as I can see no other reasonable choice.    However I can't seem to match these two bases up.  There are many different points $\{x_n\}\in X$ which make my infinite series less than a certain value and there is so much freedom in which terms in the series I choose to reduce in size that it seems hopeless to try and fit an open ball induced by $\rho$ into any one of my basis elements for the product topology.    Is there a more appropriate strategy for proving these two topologies are equivalent? How can I report bugs with or request features for the Stack Exchange mobile apps? I'm using the official Stack Exchange mobile app for iOS or Android, and  My app crashed! It doesn't look good on my new smartphone. It won't run after I updated my device. I noticed another bug. I have an idea for a new feature which is currently not in the app.  How do I report bugs with or request features for the iOS and Android apps? Also:  I've used the built-in functionality of the app to report a bug or submit a feature request. Did I do the right thing? The app prompted me to post on Meta after an error occurred. Should I post about it here? I'm wondering why the latest version is from 2017. I've heard rumors that the apps are no longer supported; is that true? I can't find the app on my device's app store anymore! Where did it go?     We get a lot of bug reports from users about the mobile apps, which are unsupported, and at the moment, there is no centralized Q&amp;A pair for those to be closed as duplicates of (the relevant answers are spread across multiple unrelated questions). This question is an FAQ proposal, to serve as a directly-related duplicate target for those questions. onKeyUp not firing for ESC in IE Derive equation for a cantilever in SHM What's the best way to communicate between view controllers?