I've studied simple harmonic motion and the relation that force $\mathbf F$ is proportional to $-x$ and time period of motion. But if in some other motion, $\mathbf F$ is proportional to some other power of $x$ say $- \mathbf F$ proportional to $-x^3$, and we just doubled the amplitude of motion, then how would the time period change?
How would I find the period of an oscillator with the following force equation?  $$F(x)=-cx^3$$  I've already found the potential energy equation by integrating over distance:  $$U(x)={cx^4 \over 4}.$$  Now I have to find a function for the period (in terms of $A$, the amplitude, $m$, and $c$), but I'm stuck on how to approach the problem. I can set up a differential equation:  $$m{d^2x(t) \over dt^2}=-cx^3,$$  $$d^2x(t)=-{cx^3 \over m}dt^2.$$  But I am not sure how to solve this. Wolfram Alpha gives a particularly nasty solution involving the hypergeometric function, so I don't think the solution involves differential equations. But I don't have any other leads.  How would I find the period $T$ of this oscillator?
What is the reasoning behind restricting the number of migration paths? On the religion and culture stackexchange sites, many, many users will initially ask a question in a place that's kind of off-topic there, but are fully on-board elsewhere.    Christianity.SE, for example, gets a lot of general philopsophical questions that would be better on Philosophy, historical questions that might be better on History.SE, and hermenutical questions that are better on BH.SE.  It is, however, unreasonable in my mind for a user with a question to be required to have domain expertise on which sub-site each has. Indeed, even amongst users with a high amount of knowledge, there can be disagreement.  Likewise, , but as just about every one of those is in beta, the user would have to be very saavy (or would have to find that post) in order to nail it every time.  That said, recommending that a question be moved is much harder than it needs to be.  I have seen business rule logic that says, for example,    migration paths only exist to graduated sites (ergo politics, money, history, and quantitative finance wouldn't qualify) migration paths seem to be limited to 5.   Anything not on a migration path requires a moderator (and not just a high rep user) to get involved.  So, what is the reasoning behind this?  Note, I'm not asking for a solution (although the obvious solution would be to allow migration paths to any established site, beta or not, i.e. a VTC as off topic b/c it belongs on another site should allow you to pick any other site) - I just want to understand the business purpose in restricting migration paths to a small, predefined number for everyone other than a mod. What is the cause for the validity of Newton's third law? What, specifically, causes newton's third law?  For instance, if I push on a wall, why is it that I experience a force in the opposite direction?    I seem to vaguely understand that is has something to do with electronic repulsion or molecular compression (maybe that's completely wrong, I don't know).  As a related question, what would happen if two objects that were /infinitely/ immovable (at the molecular level, it cannot be broken or compressed) collided with each-other? When using sudo powers and calling gedit the top level menu with File Edit View Search Tools Documents Help is missing.  If I create user ID root and sign in once and call gedit will it create the necessary user configuration files that might fix this?  Will this also make the annoying error messages go away whenever I use sudo, gksu or pkexec as a regular user with elevated privileges togedit ?  Will there be other benefits with Nautilus and other Gnome derivative Ubuntu applications?  NOTE: by sudo powers you can google pkexec gedit 3.5k hits, gksu gedit 40k hits or sudo gedit 500k hits. I'm in the minority using the first method but I believe it will become the standard in Ubuntu 17.04.    June 12, 2017 Update Here are a list of errors you get when using pkexec gedit:  (gedit:13003): Gtk-WARNING **: Calling Inhibit failed: GDBus.Error:org.freedesktop.DBus.Error.ServiceUnknown: The name org.gnome.SessionManager was not provided by any .service files  ** (gedit:13003): WARNING **: Set document metadata failed: Setting attribute metadata::gedit-spell-enabled not supported  ** (gedit:13003): WARNING **: Set document metadata failed: Setting attribute metadata::gedit-encoding not supported  ** (gedit:13003): WARNING **: Set document metadata failed: Setting attribute metadata::gedit-position not supported   On this  there is a commitment to fix these distracting messages as of June 5, 2017. No indication is given for when the fix will be upstream nor if the functionality will be implemented or the error messages will simply disappear.  Note the bug report was filed over a year ago and over that time it only effects 18 people. I have a machine running a couple of vagrant VM. The problem I have is that sometimes I forget to shutdown those VM before I shutdown or reboot my machine. Because of that my machine get stuck with this message: waiting for vboxnet0 to become free  I searched about solutions and I found this page :    I tried what they for shutdown, but it doesn't work.  I wrote an sh file for that command:  #!/bin/bash  cd ~/workspace/git/mediaservice vagrant halt   any suggestions? How to prove $A+ A^T$ symmetric, $A-A^T$ skew-symmetric. How precise is the frequency of the AC electricity network? If I make a digital clock which first detects the AC frequency roughly (whether it is 50, 60, 100Hz, etc), then uses it as its clock pulse source, how precise would it be? Would it work world wide precisely?  For instance, in USA, what is the mean and standard deviation of the 60Hz AC frequency? A doubt regarding change of variables in Double Integrals. So I have recently learnt to calculate &quot;Double Integrals&quot; and I have a doubt on change of variables. Lets say we are evaluating a double integral of the form I = $\int\int _R F(x,y) dx dy $ , where $F(x,y)$ is an integrable smooth function and $R$ is the region of integration in the $xy$ plane. Now I want to convert to polar co-ordinates $x(r,\theta)=r \cos(\theta)$ and $y(r,\theta) = r \sin(\theta)$. Now the region $R$ in the $xy$ palne can be suitably converted to some region $R'$ in the $r \theta$ plane. The function $F(x,y)$ may take the form of some smooth and integrable $G(r,\theta)$. My main doubt is with the change of the area element. According to my textbook, $dx dy = \frac{\partial (x,y)}{\partial(r,\theta)} = rdrd\theta$. So the area element in the $r\theta$ plane is $rdr d\theta$. Now I want to experiment a little bit with my understanding : I write $dx = d(r\cos\theta) = -r\sin\theta d\theta + \cos \theta dr$  and $dy = d(r\sin\theta)=r\cos\theta d\theta + \sin\theta dr $. And now I multiply the two expression for $dx$ and $dy$ : $dxdy= -r^2\sin\theta\cos\theta d\theta d\theta - r \sin^2 \theta drd\theta + r \cos^2 \theta dr d\theta + \sin \theta \cos\theta dr dr$. Why isn't this expression equal to $rdrd\theta$ ? The $d\theta d\theta$ and $dr dr$ terms look very odd to me. Am I commiting some abuse of notation or something ? Also there are no higher order terms like $dr dr d\theta$ which I can ignore. I am totally confused with all this. Can someone explain this to me a bit clearly? Thanks in Advance :). EDIT 1 : I know differentials are not the &quot;usual&quot; kind of algebraic quantities and we can't just add and multiply them casually. I am just multiplying $dx$ with $dy$ with the spirit of multiplying some $\delta x$ and $\delta y$ and letting both $\delta x$ and $\delta y$ tend to zero. EDIT 2 : Is there a branch of Mathematics where addition and multiplication of differentials is very common? Is there something like differential of a differential ? Building a version of ubuntu without any USB functionality Proof of Convergence: Babylonian Method $x_{n+1}=\frac{1}{2}(x_n + \frac{a}{x_n})$ Google page title not matching html  tag Where does TREE(n) sit on the Fast Growing Hierarchy? I had a look at Knuth's The Art of Computer Programming, book 1. In chapter 1, section 1.2.2, exercise 25, he presents the following algorithm for calculating logarithm:  given $x\in[1,2)$, do the following:  L1: [Initialize] Set $y\leftarrow0$, $z\leftarrow x/2$, $k\leftarrow1$.  L2: [Test for end] If $x=1$, stop.  L3: [Compare] If $x-z&lt;1$, go to L5.  L4: [Reduce values] Set $x\leftarrow x-z$, $z\leftarrow x/2^k$, $y\leftarrow y+\log_b(2^k/(2^k-1))$, and go to L2.  L5: [Shift] Set $z\leftarrow z/2$, $k\leftarrow k+1$, and go to L2.  (The algorithm needs an auxiliary table which stores $\log_b2$, $\log_b(4/3)$, and so on, to as many values as the precision of the computer.  Then Knuth concludes: this exercise is to explain why the above algorithm will terminate and why it computes an approximation of $y=\log_b x$.  Well, I see that it works, but I am not able to explain why... How do I type Arabic text in Photoshop? I want to type Arabic text in Photoshop, but the text appears disjointed and the letters do not connect at all. For example محمد appears as م ح م د.   I'm using Photoshop CS6 on OSX Mountain Lion. System file windows open behind Unity launcher I'm on Ubuntu 16.04.1 at the moment, and system windows like the Files window, Terminal or System Software open behind the Unity launcher, which means they are partially hidden behind it. I can move them with Alt+F7, but I'd like a more permanent solution to this problem. Good practices for organizing notes Why "hide" Luke from his father somewhere Anakin knows? Overriding public virtual functions with private functions in C++ Obviously, most beers are carbonated with carbon dioxide, while some stouts/porters are "carbonated" with nitrogen.  I was curious if there has been any notable experimentation with other gasses to "carbonate" beer. If nitrogen can introduce a "creamy" texture to a beer, I would think that other gasses would also be able to introduce their own characteristics on beer. If any of these experiments have done, what characteristics do these other gases impart?  I'm sure cost would be an issue for certain gasses that would prohibit its use in beer, but for purposes of this question, cost is irrelevant.  As an aside, what is the correct term to use, as opposed to "carbonate", for a question of this nature? I was querying data from  using a LINQ query in LINQPAD and found that the data I got back is quite obsolete. I searched for Shekhar-Pro (me) by using a where clause on ID 399722 and the data returned was for 8 Jan 2011. Why was the data so old? Isn't the data updated in real time, or at least frequently? If not, where can I get realtime data? Proving by induction that $ \sum_{k=0}^n{n \choose k} = 2^n$ Prove by induction that for all $n \ge 0$:  $${n \choose 0} + {n \choose 1} + ... + {n \choose n} = 2^n.$$  In the inductive step, use Pascal’s identity, which is:  $${n+1 \choose k} = {n \choose k-1} + {n \choose k}.$$  I can only prove it using the binomial theorem, not induction.