Consider the following figure. A man sits in a pan and tries to raise himself by pullingthe rope downwards
Why are there two tensions on my free body diagram?
Determine Which of the following Sets is a Vector Space Determine which of the following sets is a vector space.  $V$ is the line $y=x$, in the $xy$-plane: $V =\{[x,y]: \; y=x\}$  $W$ is the union of the first and second quadrants in the $xy$-plane: $W=\{[x,y]: \; y\geq0\}$  $U$ is the line $y = x+1$ in the $xy$-plane: $U=\{[x,y]: \; y=x+1\}$    I know that the answer is that "$V$" is the only vector space. However how do I come up with a quick proof to show that $V $is a vector space and that the other $2$ aren't? Standard example where Jacobson radical not equal nilradical I don't know what exactly it's called, by cache refresh I mean, refresh the page after clearing its cache. I don't want to clear the entire browser cache.  I can't seem to cache refresh my pages. In Firefox, I know it to be Shift+Refresh.  In Chrome, I've tried Ctrl+R, Ctrl+Refresh, Alt+Refresh, Shift+Refresh but none of them work. Understanding etoolbox \ifdefempty and \ifdefvoid I am trying to understand the  \ifdefempty and \ifdefvoid macros. The documentation says that \ifdefempty     Expands to true if the control sequence is defined and is a   parameterless macro whose replacement text is empty, and to false   otherwise.   Similarly \ifdefvoid     Expands to true if the control sequence is undefined, if is a macro   whose meaning is \relax, or a parameterless macro whose replacement   text is empty, and to false otherwise.   From these descriptions I thought that \def\myempty{}, \def\myrelax{\relax}, and \def\myemptyempty{\myempty} would all be both "empty" and "void", but \def\myempty{} is the only one that is either "empty" or "void". What am I missing?  \documentclass{minimal} \usepackage{etoolbox}  \begin{document}  \def\myempty{} \def\myrelax{\relax} \def\myemptyempty{\myempty}   \ifdefempty{\myempty}{Empty}{Not empty}  \ifdefempty{\myrelax}{Empty}{Not empty}  \ifdefempty{\myemptyempty}{Empty}{Not empty}  \ifdefvoid{\myempty}{Void}{Not void}  \ifdefvoid{\myrelax}{Void}{Not void}  \ifdefvoid{\myemptyempty}{Void}{Not void}  \end{document} Flying Turkish Airlines, what are credit card rules? I have return tickets flying out of SFO on Turkish airlines for the whole family. The credit card I used to book the tickets has since been replaced by the bank with a new card.   In order to board, can I present the new credit card (same name) at the checkin counter, or must I present the original credit card (which is no longer valid and has been discarded)?  By 'discarded' - I mean cut up and thrown away in trash. It was no longer valid, since it was replaced with a new card. So can I use the new card to travel on a booking that was made using the original card?    The new card has a new number; and the ticket was purchased through an agent. In ideal gas model,  is the measure of average kinetic energy of the gas molecules. If by some means the gas particles are accelerated to a very high speed in one direction, KE certainly increased, can we say the gas becomes hotter? Do we need to distinguish the random vibration KE and KE in one direction?  Furthermore, if we accelerate a block of metal with ultrasonic vibrator so that the metal is vibrating in very high speed with cyclic motion, can we say the metal is hot when it is moving but suddenly become much cooler when the vibration stop? How do I read in the contents of a directory in Perl? An elementary question regarding a multiplicative character over finite fields Typing the ZIP code doesn't yield the company page Can we re-request features that were declined a sufficient while ago? On the nilpotence of the matrix $AB-BA$ Starting scheduled task by detecting connection of USB device I know it has been discussed that it is not possible to start an application from an usb drive upon connection due to limitations of autorun (or autoplay??) in Win 7. But it is possible to create a scheduled task having an event type trigger. Surely there must be an event occurring when the drive -- or any USB device, for that matter -- is connected.    Does anybody have the slightest idea which Event ID should I use? Or at least what type of event? Where can I find the event in event viewer? I need to format a drive containing an old install of Ubuntu 10.04, to install Vista (not my idea!) Vista setup apparently refuses to format non-FAT32 or NTFS partitions. However, I can't seem to format it from my live CD (Ubuntu 11.04) at all; I keep getting "device is busy" errors. What is the simplest way to format this drive? Down vote changed to up vote caused reputation meter to pass daily maximum I just experienced a minor visual bug.  I had an answer that earned 2 down votes and 22+ upvotes, earning me the daily maximum of +200. Upon making some edits to the answer at the behest of the comments, one of the users who down voted changed their vote. Now my reputation meter shows that I gained +202 for the day.    This correctly did NOT affect my reputation, as I still only gained 200, so it is merely a minor display bug.  This appears in the achievements drop down and my reputation tab on my profile. How do Minecraft biomes work? I've never seen any information about the distribution of biomes in a generated world.   First, are all types of biomes, guaranteed to exist in a world? Will I always find snow somewhere, and desert somewhere? I've got one world that doesn't seem to have cactus (so I guess the sand is beaches, no actual desert).  Second, is there any kind of pattern or relation to biomes? Is snow to the North, and desert South, for example?  Third, how about density of biomes? If all types do exist, would I find them all within distance X of my spawnpoint, or have to travel for weeks to find a given type? When to use an object pronoun or a possessive adjective before a gerund Android Debugging, how? How do I find the index of an element in a list in Racket? Let $A\in M_{n\times n}(\Bbb R)$ so that $I\notin span(A,A^2,...,A^n)$. Prove that $\det(A)=0$. Best way to put commas into large numbers I've started a new job and it involves looking at a bunch of big numbers. Is there an easy way to add commas to an int or decimal field to make it readable?  For example, SQL Server outputs the column on the left, but for my own sanity, I need it to look like the one on the right:  2036150 -&gt; 2,036,150     ...or would I have to write some heinous   left(right(vandalized_data),6),3) + ',' + right(left(vandalized_data),6),3)   function?  The perfect thing would be commas in the display grid, then plain integers in the output.