Radial/centripetal vs. tangential/linear vs. angular acceleration I am thoroughly confused as to how exactly radial, tangential, and linear accelerations of points on a rotating wheel relate to each other. Every time I think I know what they mean individually, I find a source that tells me that no, that is wrong. I was actually writing down in this post what I think is correct, only to realize that I was writing the same thing for two obviously different concepts.  First, is radial acceleration the same as centripetal and tangential the same as linear?  Second, how do these different accelerations relate to each other on a spinning disk? I was reading through my textbook, and I think I understand angular velocity perfectly in this context. However, I do not understand, for example:     If a disk rotates at a constant angular velocity, why does a point on the rim have radial but not tangential acceleration? Mathematically, this makes sense (since $a=r\alpha$, so if $\alpha=0$, then so must $a$), but not conceptually. Does it have radial/centripetal acceleration simply because it is still exhibiting circular motion? Would the linear acceleration change––is there any linear acceleration?   Basically, I do not really understand how radial, tangential, and linear accelerations relate to each other? I have just a big salad of greek letters in my head.
Types of circular acceleration? To my knowledge there are three types of acceleration when a body (e.g. a rod) is moving in a circle about an axis. These are:   Angular acceleration : this is the rate of change of angular velocity. Tangential acceleration : this is the linear acceleration of the system in a tangential direction to the circle and equals the radius times the angular acceleration. Radial/centripetal acceleration : this is the linear acceleration of the system that is directed inwards towards the center of the circle.   I also think there are two types of velocity:   Linear velocity: this is its velocity in the tangential direction and is constantly changing Angular velocity or angular frequency: this is the rate of change of angle.   Is the above correct? And have I missed anything?
While editing some ad copy, and I was given the sentence, "Defending a lawsuit can be a big cost for your business." (My italics.)  I keep thinking the proper word to use is "expense" rather than "cost" as in: "Defending a lawsuit can be a big expense for your business." But I'm just not sure I'm correct.   Can anyone help with more definitive guidance? Can "cost" and "expense" be used interchangeably in a sentence like this?   Thank you! The Next tag badge block overlaps the badge score block. I'm running an Ubuntu 14.04 and want to install the Eclipse IDE for both Java and C/C++ programming. I installed the separate installation files from their official website. I followed this thread . (The top answer)   The steps helped me install the Java IDE but I don't know what changes I should make to the procedure to install the C/C++ IDE.   Another thread , recommended to install eclipse-cdt but the Eclipse version installed is outdated. (Doesn't allow me to install Pydev)   I have TWO questions:   Can the C/C++ libraries be downloaded from within eclipse, like Pydev? If not, how do I install both IDEs without interfering with each other?   P.S. I commented this query in the first thread but got no reply, that is why I'm asking a fresh question! $3^{n+1}$ divides $2^{3^n}+1$ Describe all positive integers,n such that $3^{n+1}$divides $2^{3^n}+1$. I am little confused about what the question asks-if it asks me to find all such positive integers, or if it asks me to prove that for every positive integer n,$3^{n+1}$ divides $2^{3^n}+1$. Kindly clarify this doubt and if it's the former part, please verify my solution-n=1. According to my logic,  $$\large\sqrt{x^2} = x^{2\times \frac{1}{2}} = x = x^{\frac{1}{2}\times 2}={(\sqrt{x})}^2$$  But when I look at the graphs of these guys, they're totally different.  Edit: Complex answers are okay, if you know what I mean. What is $\int_0^1\frac{x^7-1}{\log(x)}\mathrm dx$? free groups: $F_X\cong F_Y\Rightarrow|X|=|Y|$ I'm reading Grillet's Abstract Algebra. Let $F_X$ denote the free group on the set $X$. I noticed on wiki the claim $$F_X\cong\!\!F_Y\Leftrightarrow|X|=|Y|.$$ How can I prove the right implication (find a bijection $f:X\rightarrow Y$), i.e. that the rank is an invariant of free groups?  I am hoping for a simple and short proof, having all the tools of Grillet at hand. Rotman (Advanced Modern Algebra, p.305) proves it only for $|X|&lt;\infty$, Bogopolski's (Introduction to Group Theory, p.55) proof seems (unnecessarily?) complicated, and Lyndon &amp; Schupp's (Combinatorial Group Theory, p.1) proof I don't yet understand. It's the very first proposition in the book; in the proof, they say:     The subgroup $N$ of $F$ generated by   all squares of elements in $F$ is   normal, and $F/N$ is an elementary   abelian $2$-group of rank $|X|$. (If   $X$ is finite, $|F/N|=2^{|X|}$ finite;   if $|X|$ is infinite, $|F/N|=|X|$). $\square$   Is $N:=\langle w^2;w\in F\rangle$? What is an abelian $2$-group? Elementary? What and how does the above quote really prove? I'm guessing a free abelian group on $X$ is $\langle X|[X,X]\rangle\cong\bigoplus\limits_{x\in X} \mathbb{Z}$?  Can an isomorphism $\varphi:F_X\rightarrow F_Y$ not preserve the length of words? At least one letter words? Is there any UNIX variant on which a child process dies with its parent? Consider the sequence defined as  $x_1 = 1$  $x_{n+1} = \sin x_n$  I think I was able to show that the sequence $\sqrt{n} x_{n}$ converges to $\sqrt{3}$ by a tedious elementary method which I wasn't too happy about.  (I think I did this by showing that $\sqrt{\frac{3}{n+1}} &lt; x_{n} &lt; \sqrt{\frac{3}{n}}$, don't remember exactly)  This looks like it should be a standard problem.   Does anyone know a simple (and preferably elementary) proof for the fact that the sequence $\sqrt{n}x_{n}$ converges to $\sqrt{3}$? Why don't they use the transporter in the final fight in Star Trek Into Darkness? When Spock was chasing, and beating, Khan over a transport in San Francisco, Uhura orders them beamed both back  to the Enterprise, while Chekov says something like:     Its impossible, they are moving!                                   But in the first movie, it was Chekov that transported back to Kirk and Sulu, avoiding hitting the Vulcan planet when trying to disable the Romulan drilling platform. Insert into ... values ( SELECT ... FROM ... ) I am trying to INSERT INTO a table using the input from another table. Although this is entirely feasible for many database engines, I always seem to struggle to remember the correct syntax for the SQL engine of the day (, , , , and ).  Is there a silver-bullet syntax coming from an SQL standard (for example, ) that would allow me to insert the values without worrying about the underlying database? \end{abstract} conflicts with arabtext package I am using elsarticle  Everything was find before I used \usepackage{arabtext}.   When I compiled to PDF, I got the following error message.    And if I type "s", everything became fine again but I don't want to type an "s" every time I compile PDF preview.   Please anyone advise.  Thank you very much in advance for your help.  Here is the code  \documentclass[12pt, authoryear]{elsarticle}    % use "amsart" instead of "article" for AMSLaTeX format \usepackage[colorlinks=true,linkcolor=black, citecolor=blue, urlcolor=blue]{hyperref} \usepackage{geometry}                       % See geometry.pdf to learn the layout options. There are lots. \geometry{a4paper}                          % ... or a4paper or a5paper or ...  %\geometry{landscape}                       % Activate for rotated page geometry %\usepackage[parfill]{parskip}          % Activate to begin paragraphs with an empty line rather than an indent \usepackage{graphicx}               % Use pdf, png, jpg, or epsÂ§ with pdflatex; use eps in DVI mode                             % TeX will automatically convert eps --&gt; pdf in pdflatex         \usepackage{amssymb}  \usepackage{amsmath} \usepackage{tipa} \usepackage{arabtex} %SetFonts  %SetFonts   \title{Translating from English to Bahasa Indonesia: Which procedures should be taught to EFL students preparing to be a translator?} \author[mkl]{Mohammad Kholid}  \author[fai]{Faisal Mustafa\corref{cor1}} \ead{faisal.mustafa@unsyiah.ac.id}  \cortext[cor1]{Corresponding author}  \address[mkl]{Syiah Kuala University, Banda Aceh, Indonesia} \address[fai]{Syiah Kuala University, Banda Aceh, Indonesia} \date{}                         % Activate to display a given date or no date  \begin{document} %\section{} %\subsection{}  \begin{frontmatter}  \begin{abstract}   Abstract content is here.  \end{abstract}  \begin{keyword}  translation procedures \sep English to Bahasa Indonesia \sep teaching translation \end{keyword} \end{frontmatter} Rotating clears selected tags Changing the orientation of your device within the tag selection view while asking a question removes all selected tags and clears the textbox.     Steps to reproduce:     Ask a question (does not happen when editing) Edit tags Enter some text in the box  Select a tag from the suggestion list Repeat the previous two steps as many times as you care to Rotate the device (source / target orientation doesn't matter, just need to change it) All selected tags and text typed in the textbox are gone and must be re-entered   I've reproduced this using version 1.0.63 of the SE app on two devices:     HTC One E8 m2ss (the Hong Kong version) rooted and running stock 5.0.2 ATT Galaxy S3 i747, rooted and sim unlocked, running Slimkat 4.4.2.build.4.1.official ROM on Android 4.4.2 What would be the associated wavelength of the particle if its velocity is zero? How do I get a really wide tilde?  I need to cover at least  abcdefghijklmnopqrstuvwxyz  The \widetilde command does not work. Nor using \stackrel with \sim. Add layer to a QGIS group using Python Rest mass of the photon Why is the rest mass of the photon is taken to be zero?  If rest mass of the photon is non zero, then it leads to infinite energy of the photon (by $E=mc^2$ equation) and there is a violation of conservation of energy. Is this a reason?  Also, is there any thing called the relativistic mass of the photon? How to redirect STDIN of background process? In my Ubuntu, i run java application in background. I use bash script to run it and now it looks like:  nohup java -jar app.jar &amp; exit 0   The problem that i want to be able to write an input string to my application, without making it foreground, from different terminals/sessions. Something like  echo "mytext" &gt; /appdir/in   How should i change my script? Disable Tabbing in Firefox 3, Chrome 4 and IE8? Tabbed Browsing may be nice for reading news sites in one sitting, but I still think they are a usability nightmare if you need to keep the page open for a long time, e.g., for testing it while developing or as a reference for something.  The nice thing about the Windows Taskbar (before Windows 7 ) is that it's easy and natural to remember where things are. I can have 30 tabs in it, but because I know what each one is, I know where I have to click to open something - it's the most efficient way of working I've encountered so far.  Sadly, Tabbed Browsing completely breaks this efficiency because it is not possible to specifically click on a tab in the taskbar, I have to click on the browser, search the tab and then click it, which means at least 2 clicks.  Long story, short question: Is there an extension or setting to completely disable tabbing on Firefox 3, Chrome 4 and Internet Explorer 8 and have one Window/Taskbar entry for each open page? I know that in Chrome I can just drag a tab out of a Window to create a new one, but if there is a general way, that would be easier.  Edit: IE8 conveniently offers an option for that: Tools - Options - Tabs - Enable Tabbed Browsing. I'm wondering if it makes a difference in interpretation whether only the dependent, both the dependent and independent, or only the independent variables are log transformed.  Consider the case of  log(DV) = Intercept + B1*IV + Error    I can interpret the IV as the percent increase but how does this change when I have   log(DV) = Intercept + B1*log(IV) + Error   or when I have   DV = Intercept + B1*log(IV) + Error   ?