I've seen angular momentum defined as:   $$\ L=I \omega\ $$  In dynamics, the notation is different and states:   $$\ L_o = r  × (mv) \ $$  How are these definitions related, if they are describing the same thing?
Say we have a rod with a fixed axis of rotation. If we give it a push at some point other than the axis, it will start to rotate. Say our force is $F$, then we might write: $\Delta P=F\Delta t$. I’m wondering how to incorporate angular momentum here. We know that for a single particle it holds that $L=r\times p$. And for an extended object with a fixed axis of rotation, we generally might write $L=I\omega$. However, can I write $L=r\times p$ too for the extended object? I would think that I would need to integrate. So we have $p_\alpha=m_\alpha v_\alpha=m_\alpha r_\alpha\omega$, where we consider an infinitesimal mass $\alpha$. So should I write $L=\int r^2\omega\,dm$? I’m confused whether to work with this integral, or just use $L=r\Delta P$ at once.  The reason I'm asking, is because I'm trying to calculate the sweet spot of a baseball bat. So this is my attempt at solving the exercise.
Why is Excel truncating my 16-digit numbers? If I type in a 16 digit number (format: number, no decimals) it changes the number on me. Example: 1234567812345678 changes the view to 1234567812345670.    If I type it in as a general format it changes the numbers above so it displays 1.23457E+15 but if you click on the cell, the display shows the last digit as a 0 instead of an 8 once again.   I opened the file on a different computer and same issue now with it.  I have changed the auto correction and auto formatting all to no avail.  Help! help  displays information about builtin commands. What is the source of help? Does it maintain a database for builtin commands, or does it read some files of each builtin commands (similar to the manpage of each utilities)?  Sometimes I find its information seems to expand that by --help  $ cd --help bash: cd: --: invalid option cd: usage: cd [-L|[-P [-e]]] [dir]  $ help cd cd: cd [-L|[-P [-e]]] [dir]     Change the shell working directory.      Change the current directory to DIR.  The default DIR is the value of the     HOME shell variable.      The variable CDPATH defines the search path for the directory containing     DIR.  Alternative directory names in CDPATH are separated by a colon (:).     A null directory name is the same as the current directory.  If DIR begins     with a slash (/), then CDPATH is not used.      If the directory is not found, and the shell option `cdable_vars' is set,     the word is assumed to be  a variable name.  If that variable has a value,     its value is used for DIR.      Options:         -L  force symbolic links to be followed         -P  use the physical directory structure without following symbolic         links         -e  if the -P option is supplied, and the current working directory         cannot be determined successfully, exit with a non-zero status      The default is to follow symbolic links, as if `-L' were specified.      Exit Status:     Returns 0 if the directory is changed, and if $PWD is set successfully when     -P is used; non-zero otherwise.   I thought help extract the help part of the executable, but for a  python script , it isn't  $ help ./pdf-merge.py bash: help: no help topics match `./pdf-merge.py'.  Try `help help' or `man -k ./pdf-merge.py' or `info ./pdf-merge.py'.  $ ./pdf-merge.py --help usage: pdf-merge.py [-h] [-v] [--ask] [--output OUTPUT] [--title TITLE]                     [--author AUTHOR] [--keyword KEYWORD] [--pdftk PDFTK]                     [--gs GS] [--pdfmarks PDFMARKS] [--unicode]                     PDF [PDF ...]  Merge PDFs preserving bookmarks. Thanks to Larry Cai for suggesting that Unicode be supported and for discussion about the `--pdfmarks` option.  positional arguments:   PDF                  an input PDF to merge  optional arguments:   -h, --help           show this help message and exit   -v, --version        show program's version number and exit   --ask                pause for manual pdfmark tweaking   --output OUTPUT      name of the output PDF   --title TITLE        title of output PDF   --author AUTHOR      author of output PDF   --keyword KEYWORD    keywords for the output PDF   --pdftk PDFTK        path to the pdftk executable   --gs GS              path to the gs (Ghostscript) executable   --pdfmarks PDFMARKS  path to pdfmarks file. If not given, a temporary file                        is used. If given and the file is missing, execution                        will stop after the file is created (before the                        Ghostscript run). If given and the file exists, no                        attempt will be make to use pdftk to generate the mark                        file (I assume your input file is what you want).   --unicode            instead of merging PDFs, convert PDF-formatted unicode                        strings. For example `--unicode '&lt;FEFF03B103B203B3&gt;'                        \u03b1\u03b2\u03b3` Is there something like Linux console in Mac OS X? Like "terminal", but without running graphics environment. I have a complex command that I'd like to make a shell/bash script of.  I can write it in terms of $1 easily:  foo $1 args -o $1.ext   I want to be able to pass multiple input names to the script. What's the right way to do it?    And, of course, I want to handle filenames with spaces in them. Try to solve this puzzle:     The first expedition to Mars found only the ruins of a civilization.   From the artifacts and pictures, the explorers deduced that the   creatures who produced this civilization were four-legged beings with   a tentatcle that branched out at the end with a number of grasping   "fingers". After much study, the explorers were able to translate   Martian mathematics. They found the following equation:   $$5x^2 - 50x + 125 = 0$$   with the indicated solutions $x=5$ and $x=8$. The value $x=5$ seemed   legitimate enough, but $x=8$ required some explanation. Then the explorers    reflected on the way in which Earth's number system developed, and found   evidence that the Martian system had a similar history. How many fingers would    you say the Martians had?      $(a)\;10$      $(b)\;13$      $(c)\;40$      $(d)\;25$   P.S. This is not a home work. It's a question asked in an interview. Determining a tricky limit What is the limit of   $$\lim_{n \to \infty} \frac{\log n}{n^\delta}$$  where $\delta &gt; 0$? How do I install the latest version of EMACS and pick the appropriate UI? I'm running the latest Kubuntu (12.04 beta 2) and I would like to run the latest emacs (currently v24). The repos are one version behind. What's the best way to install v24 or later (and avoid future version conflicts)?  Also, is there any reason not to aways use the GUI version of emacs if X is running? For example, could I set the GUI emacs version as the default text editor and use it to edit cron jobs (crontab -e)? I'm assuming the answer is yes, but since I haven't done that yet (my default editor is nano), I want to check if there are reasons I should leave nano as the default editor.  Usually when I'm working on the command line I end up using nano. Now that I think about it, I have no idea why I keep doing that. Is there any downside to calling a GUI editor when working in an X terminal?  EDIT: I briefly tested these two versions   GNU Emacs 24.0.94.1 (x86_64-pc-linux-gnu, GTK+ Version 3.3.20) from  GNU Emacs 23.3.1 (x86_64-pc-linux-gnu) installed by default in Kubuntu.   .  Unfortunately (for me) the defaults installed version (23.3.1, 23.3+1-1ubuntu9) is the nox version.   Package: emacs23-nox Status: install ok installed Version: 23.3+1-1ubuntu9 Replaces: emacs23, emacs23-gtk, emacs23-lucid   The package with version 24 opens in GUI mode by default. That's what I prefer. Some of the version 24 changes that interest me are listed in the references below.   But there appear to be a multitude of different packages and versions I could install.  References:   What’s New In Emacs 24 (part 1) | Mastering Emacs  "    shell-mode uses pcomplete rules, with the standard completion UI.  Yowzah! There’s a lot of cool, new functionality hidden away in this gem of a change." EmacsWiki: Recent Changes Evaluating $\int_0^\infty \sin x^2\, dx$ with real methods? Grub not loading after Windows 8 Install My system was configured to dual boot Ubuntu 12.04.1 LTS and Windows 7. Today I got my hands on the MSDN release of Windows 8 and I installed it over my Windows 7. Now the computer just boots to Windows 8 directly without loading the GRUB screen.  So I followed the steps as suggested in: . Running this command:      ls -l /dev/disk/by-label/   gives the following output:      total 0     lrwxrwxrwx 1 root root 10 Sep 11 07:51 Entertainment -&gt; ../../sda2     lrwxrwxrwx 1 root root 10 Sep 11 02:45 PENDRIVE -&gt; ../../sdb1   Also fdisk -l command gives this as the output:      Disk /dev/sda: 320.1 GB, 320072933376 bytes     255 heads, 63 sectors/track, 38913 cylinders, total 625142448 sectors     Units = sectors of 1 * 512 = 512 bytes     Sector size (logical/physical): 512 bytes / 512 bytes     I/O size (minimum/optimal): 512 bytes / 512 bytes     Disk identifier: 0x1246aa23         Device Boot      Start         End      Blocks   Id  System     /dev/sda1   *        2048   319582199   159790076    7  HPFS/NTFS/exFAT     /dev/sda2       319582208   602906623   141662208    7  HPFS/NTFS/exFAT     /dev/sda3       602908672   625135615    11113472   83  Linux      Disk /dev/sdb: 1939 MB, 1939865600 bytes     64 heads, 63 sectors/track, 939 cylinders, total 3788800 sectors     Units = sectors of 1 * 512 = 512 bytes     Sector size (logical/physical): 512 bytes / 512 bytes     I/O size (minimum/optimal): 512 bytes / 512 bytes     Disk identifier: 0xc3072e18         Device Boot      Start         End      Blocks   Id  System     /dev/sdb1   *        2248     3788799     1893276    c  W95 FAT32 (LBA)   So I assume that I have to run this: sudo grub-install /dev/sda3 to get GRUB up and running. But I am getting this error:      /usr/sbin/grub-probe: error: cannot find a device for /boot/grub (is /dev mounted?).   Can anyone please guide me in the right direction? The current Ubuntu installation is far too customized to my needs to lose it to a boot manager issue! Any help is much appreciated! Let $x = (a, b), y = (c, d) \in \mathbb{Z}^2$. What is the condition on $a, b, c, d$ so that ${x, y}$ is a basis?   My answer: $ad\neq bc$ and $gcd(a, c) = gcd(b, d) = 1$.  The first condition ensures that they aren't the same vector; the second ensures that we can actually "get" all of the integer values/lattice points.  Is this correct?  Thanks. What happens if my shield breaks? Learn Bayesian inference applied to astronomy / astrophysics? Proof of Archimedes Principle Say I have an arbitrary body $\Omega$ and surface $\partial \Omega$ submerged in a hydrostatic fluid of density $\rho$ under the influence of gravity. How does one show ? i.e.  $$\int_{\partial \Omega}(P_0 + \rho g h)d\vec{S}=\text{vol}(\Omega)\rho g\vec{e_2}.$$ Entity Query field condition for Webform submissions Where is log file from rc.local? I have some commands to in my rc.local. I know that they are failing. How can I get log file with messages produced by executing rc.local? Where is it located?  I have checked the /var/log/boot.log. I know my messages are not there because I know already what is the reason of failure. But I still want to make sure from log file.  Note, I don't want to run script again, I could but I don't want. I would rather analyse wht happened during startup.  Thanks for any help.  Ubuntu 12.04 Desktop (if it matters) Installed Ubuntu 13.04 on a Dell laptop I had laying around.  Install goes well but when I'm done, I only get a command line, no GUI at all.  Tried several versions, tried 32 vs 64 bit all give me the same result.  Any ideas on what to do/try? I have noticed that the confidence interval for predicted values in an linear regression tends to be narrow around the mean of the predictor and fat around the minimum and maximum values of the predictor. This can be seen in plots of these 4 linear regressions:    I initially thought this was because most values of the predictors were concentrated around the mean of the predictor. However, I then noticed that the narrow middle of the confidence interval would occur even if many values of were concentrated around the extremes of the predictor, as in the bottom left linear regression, which lots of values of the predictor are concentrated around the minimum of the predictor.  is anyone able to explain why confidence intervals for the predicted values in an linear regression tend to be narrow in the middle and fat at the extremes? Are InnoDB Deadlocks exclusive to INSERT/UPDATE/DELETE? My fiancée and I are looking for a good Caribbean cruise in October and were wondering which islands are best to see and which Cruise line to take?  It seems like a lot of the cruises don't run in this month due to Hurricane season so I'm looking for other good options.  EDIT We'll be travelling in 2012. How to get UTF-8 working in Java webapps?