I recently asked my physics teacher this question, and he could not give me a clear answer: How gravity acts on a streched rope? Consider example like this one below: if we strech a rope of mass m between points distance d apart, how low will the rope hang? And what will be the tension on the rope? Is rope a perfect parabola in this scenario? I just find it extremely quaint and I couldn't find the answer anywhere.
The height of a string in a gravitational field in 2-dimensions is bounded by $h(x_0)=h(x_l)=0$ (nails in the wall)  and also $\int_0^l ds= l$. ($h(0)=h(l)=0$, if you take $h$ as a function of arc length) .  What shape does it take?  My try so far: minimise potential energy of the whole string,  $$J(x,h, \dot{h})=\int_0^l gh(x) \rho \frac{ds}{l}=\frac{g \rho }{l}\int_0^l h(x) \sqrt{1+\dot{h}^2} dx$$  With the constraint  $$\int_0^l \sqrt{1+\dot{h}^2} dx- l=0$$ If it helps, it's evident that $\dot{h}(\frac{l}{2})=0$.  Generally, this kind of equation is a case of a constrained variational problem, meaning that the integrand in  $$\int_0^l \frac{g \rho }{l}h(x) \sqrt{1+\dot{h}^2} +\lambda(\int_0^l \sqrt{1+\dot{h}^2} dx- l)dx$$  Must satisfy the Euler Lagrange equation. The constraint must also be satisfied.  But, in truth, by this point I am clueless. $\lambda$ is worked through $\nabla J  = \lambda \nabla(\int_0^l \sqrt{1+\dot{h}^2} dx- l)$. I have tried this , but get nonsensical answers.  Is this method the best? If so, in what ways am I going about it wrongly thusfar?
Groups of order $56$ Let $G$ be a group of order $56$. (We do NOT assume the Sylow-$7$ subgroup to be normal.) Then either the Sylow-$2$ subgroup is normal or the Sylow-$7$ subgroup is normal. How to prove?  My idea: consider the case $n_2=7,n_7=8$. Then the number of elements in all Sylow $2$ -subgroups and Sylow $7$-subgroups is at most exactly $(7-1)*8+(2-1)*7+1=56$. This cannot give a contradiction. We need to prove that $G$ is simple. Then $|G||(8!/2)$. To prove $G$ simple, we need to prove there does not exist normal subgroups of order $2$ and order $4$. But I do not know how to prove. \draw not working inside axis environment Judging from the pgfplots manual I expected the following code to work:  \documentclass{article} \usepackage{pgfplots} \begin{document} \begin{figure}   \begin{tikzpicture}     \begin{axis}[       xmin=0,       xmax=4,       ymin=0,       ymax=4,       ]       \draw (1,1) -- (3,2);     \end{axis}   \end{tikzpicture} \end{figure} \end{document}   However, it doesn't - meaning that it seems the \draw command does nothing at all. I just get an empty axes with no line.    Can anyone shed some light on this? I guess the solution will be really easy... I'm using a class I've derived from DocumentPaginator (see below) to print simple (text only) reports from a WPF application. I've got it so that everything prints correctly, But how do I get it to do a print preview before printing? I have a feeling I need to use a DocumentViewer but I can't figure out how.  Here's my Paginator Class:  public class RowPaginator : DocumentPaginator {     private int rows;     private Size pageSize;     private int rowsPerPage;      public RowPaginator(int rows)     {         this.rows = rows;     }      public override DocumentPage GetPage(int pageNumber)     {         int currentRow = rowsPerPage * pageNumber;         int rowsToPrint = Math.Min(rowsPerPage, rows - (rowsPerPage * pageNumber - 1));         var page = new PageElementRenderer(pageNumber + 1, PageCount, currentRow, rowsToPrint)                        {                            Width = PageSize.Width,                            Height = PageSize.Height                        };         page.Measure(PageSize);         page.Arrange(new Rect(new Point(0, 0), PageSize));         return new DocumentPage(page);     }      public override bool IsPageCountValid { get { return true; } }      public override int PageCount { get { return (int)Math.Ceiling(this.rows / (double)this.rowsPerPage); } }      public override Size PageSize     {         get { return this.pageSize; }         set         {             this.pageSize = value;             this.rowsPerPage = PageElementRenderer.RowsPerPage(this.pageSize.Height);             if (rowsPerPage &lt;= 0)                 throw new InvalidOperationException("Page can't fit any rows!");         }     }      public override IDocumentPaginatorSource Source { get { return null; } } }   The PageElementRenderer is just a simple UserControl that displays the data (at the moment just a list of rows).  Here's how I use my Row Paginator  PrintDialog dialog = new PrintDialog(); if (dialog.ShowDialog() == true) {     var paginator = new RowPaginator(rowsToPrint) { PageSize = new Size(dialog.PrintableAreaWidth, dialog.PrintableAreaHeight) };      dialog.PrintDocument(paginator, "Rows Document"); }   Sorry for the code dump, but I didn't want to miss something relevant. Precedence between implication and bi-implication I came across this question:  Let p, q, and r be the propositions  p : Grizzly bears have been seen in the area. q : Hiking is safe on the trail. r : Berries are ripe along the trail. Write these propositions using p, q, and r and logical connectives (including negations).     If berries are ripe along the trail, hiking is safe if and   only if grizzly bears have not been seen in the area.   I see an If r,q iff ~p where If r,q is the part "If berries are ripe along the trail, hiking is safe" and ~p is "grizzly bears have not been seen in the area"  and I came up with (r -&gt; q) &lt;-&gt; ~p  But the answer given is: r -&gt; (q &lt;-&gt; ~p)  Can someone please explain the difference in precedence? Was Louis stalked days in advance too? IIS7 - The request filtering module is configured to deny a request that exceeds the request content length How can I take a self-portrait without using a remote?  My camera is a Canon Powershot SX210 IS. I do not have a remote, but I do have a tripod. How do I install Ubuntu from an USB drive / create a live USB-stick in Windows? Is there a DSLR fixed lens camera? What happens to the holes in the skull where teeth were developing? In Skeptics.SE, a question regarding the skulls of children arose. Depicting a scary skull where the permenent teeth are "hidden" inside the jaws and the milk teeth are in the their place.  After all milk teeth fall out, and the permanent teeth "rise", do the holes left by them filled up, or do we go around with holes in our jaws?    The picture: Regular expression to match a line that doesn't contain a word Beamer: how to justify the whole text? Sci-Fi Movie/Show Where a Soul is Captured in a Glass Vessel Why is "No new release found" when upgrading from a LTS to the next? If I run:  sudo chown -R user:user /   Can I revert it to what it was before I ran it? Matryoshka words Since it's Friday and others are playing games, here's another one to take us into the weekend.  Everyone knows about those Russian nesting dolls (called matryoshka dolls, I believe). A small doll nests inside a larger, which itself is inside a larger doll, and so on.   This game does the same with words. You start with a three-letter word, add a letter to make another word, and so on. Longest word wins. The catch? Each additional letter has to form a legitimate word.  Here's one to start you off:     bur   burr   burro   burrow   burrows     Good luck. How to record both screen and sound with Quicktime on El Capitan? I'm trying to record both what's happening on screen and the audio being played by the applications. To my surprise, QuickTime did not record the sound. I believe it worked before in previous OS X releases, but that is not something that I use often so I have not bothered to try earlier.  What is the current situation on El Capitan ? Have I messed with something that prevents QuickTime from capturing the sound, or has it never worked ?  Here is my view of Quick Time  / New Screen Recording on a Mac Mini:   On my Macbook Pro, I have the option select None or Internal Microphone. Internal microphone actually records the speakers of the Mac Book as heard from its own microphone, which obviously sounds terrible.  This is a MacMini 7,1 (late 2014) and there is nothing in Input Sources. How come such different methods result in the same number, $e$? How do I connect the initial highway to my six-lane road? How to use stars and bars for solving number of integral solutions to an equality?