Tension in string and Spring force Why is that a massless string has same tension throughout its length provided the string is inextensible. Similarly in spring block problems we take spring as massless as well and here too we consider spring force to be uniform throughout it's natural length. I actually want the derivation of how do we get this. Thank you.
Rigorously proving equal tension on both ends There is always one point in introductory mechanics that has been continuously bothering me ever since I had taken my freshman physics course - Why does a massless rope have its tension equal on both ends?  I have been searching for a good explanation on various textbooks at different levels, but none of them provided me a satisfactory explanation. They either   State directly "we assume the string is massless and the pulley has no friction, therefore the tension must be the same throughout the string" without further any elaboration and continue the calculation, Claim without any reasons that "whenever the rope and the pulley are massless and frictionless, the tension will be the same on both sides", or Explain using a lot of hand-waving arguments regarding infinite acceleration, non-zero moment of inertia, etc. , in one line or two.   Although those hand-waving arguments of type $(3)$ do sound quite reasonable to me and will convince me for a while every time I review them. But after a while, I will start feeling not well about this whole tension concept again.  Therefore, I am looking for a rigorous (in the physicist's sense) proof of exactly why and when do the tension in rope will be uniform, and hopefully from that I can clean up all of my conceptual difficulties.
Benefits of using QQ-plots over histograms In , Nick Cox wrote:     Binning into classes is an ancient method. While histograms can be useful, modern statistical software makes it easy as well as advisable to fit distributions to the raw data. Binning just throws away detail that is crucial in determining which distributions are plausible.   The context of this comment suggests using QQ-plots as an alternative means to evaluate the fit. The statement sounds very plausible, but I'd like to know about a reliable reference supporting this statement. Is there some paper which does a more thorough investigation of this fact, beyond a simple “well, this sounds obvious”? Any actual systematic comparisons of results or the likes?  I'd also like to see how far this benefit of QQ-plots over histograms can be stretched, to applications other than model fitting. Answers on  agree that “a QQ-plot […] just tells you that "something is wrong"”. I am thinking about using them as a tool to identify structure in observed data as compared to a null model and wonder whether there exist any established procedures to use QQ-plots (or their underlying data) to not only detect but also describe non-random structure in the observed data. References which include this direction would therefore be particularly useful. I absolutely hate it when I watch a video on YouTube and it automatically loads another video page after it's done playing.  This is especially annoying when I'm in the middle of typing a comment as it will cause me to lose what I have typed.  I'd like the video to simply stop and for no further action to take place once it is finished playing.  Is there any way to accomplish this? Does D&D 5e have a rule for character knowledge about monsters? In the 3rd and 4th editions of D&amp;D, there were explicit rules for determining if a character knew anything about a monster before them.  In 3rd edition, for example, use of the  skill with a general DC of 10 + the monster's HD allowed for determining one fact, plus one fact per 5 points over the check.  Looking at the Intelligence section of the 5e PHB, I don't see any similar notation.  Does 5e provide any guidance as to when a player could use their out-of-game knowledge about a monster, or when the player might be told things their character would probably know in-game? Add markdown toolbar to mobile web On the full site, there's that helpful Markdown toolbar above the post box:   This is conspicuously missing in the mobile site:   It's difficult just to get a bracket on my iPad. Backticks aren't even available. If there's a view that really needs it, i vote mobile. Vanishing of Taylor series coefficient How can I disable creepers' block damage on my Minecraft server? Please note that these are NOT acceptable solutions:   Completely disabling monsters (peaceful mode) Completely removing creepers Completely disabling explosions (TNT + Creepers) Disabling creeper explosions   More exactly, I want the creepers to be able to spawn and explode, but without doing their terrain damage (they should still be doing health damage to the players).  I'm fine with the use of mods but I'd prefer a built-in solution if possible. When installing or booting from USB flash drive screen goes blank When I was installing Ubuntu 11.10 64-bit on my new PC from a flash drive since the only computers I have do not have disk drives, I ran into a problem.  After the ASUS logo when switching on the PC, and pressing F8 to select the USB flash drive, there was the options thing as normal that allows me to select to either boot form the flash drive or install on the HDD, when I do either of these, there is a bunch of text on a black background, as I would expect, but after a few seconds the screen goes blank and the monitor says No signal.  I have previously installed Ubuntu with a flash drive to fix issues caused my Windows and ran into no problems.  Is it a problem with the pen drive I am using? Or maybe a problem with the system? I tried it in my USB 3.0 ports at the back and the USB 2.0 ones as well.  My Specs are:   &nbsp;&nbsp;AMD Phenom II x4 965BE &nbsp;&nbsp;ASUS AMD Radeon HD 6670 &nbsp;&nbsp;ASUS M5-L75 USB3 (typing this from memory, it's an AM3+ ASUS board with USB3.0 in the AMD700 chipet or something) &nbsp;&nbsp;Corsair XMS3 DDR3 4GB(2x2GB) 1333MHz RAM    I ran the memtest program on the flash drive as well, and it came out clean. How do I integrate the following? $\int{\frac{(1+x^{2})\mathrm dx}{(1-x^{2})\sqrt{1+x^{4}}}}$ Effect on loading a DC motor - load torque Could someone give a (detailed) explanation (perhaps with formulas) of how the load (torque) affects the armature current of a DC motor (separately excited or shunt or series)?  As the load torque increases the speed of the motor decreases, and assuming the terminal voltage stays constant, the EMF will become lower and thus the armature current will increase.  But could someone give me the formula that shows the relation between load torque and armature current?  Is it something like:   \$T_\text{developed} = T_\text{shaft} + T_\text{friction windage} + T_\text{load}\$ ,   where \$T_\text{developed} = K \, \text{flux}_\text{pole} \,I_\text{armature}\$ ?  But then you have  \$P_d = P_s + P_{f_w} + P_l   \$  where \$P_d = K \,\text{flux}_\text{pole}\, I_\text{armature}  \,\omega_m\$  In the last formula you can't see that as the increase due to \$P_\text{load }\$ the armature current increases because maybe someone can say the speed \$\omega_m\$ increases.  And how come at no-load the armature current and thus the torque developed is zero? Is it really zero, or do I have to assume there is some current flowing?  Could also explain me the principle of torque developed. Because at no load, the shaft torque is equal to torque developed, right? How to reset postgres' primary key sequence when it falls out of sync? I ran into the problem that my primary key sequence is not in sync with my table rows.   That is, when I insert a new row I get a duplicate key error because the sequence implied in the serial datatype returns a number that already exists.  It seems to be caused by import/restores not maintaining the sequence properly. How to force the browser to reload cached CSS and JavaScript files I have noticed that some browsers (in particular, Firefox and ) are very zealous in using cached copies of .css and .js files, even between browser sessions. This leads to a problem when you update one of these files, but the user's browser keeps on using the cached copy. What is the most elegant way of forcing the user's browser to reload the file when it has changed? Ideally, the solution would not force the browser to reload the file on every visit to the page.  I have found  and  suggestion to be useful. It turns out there is a term for this: auto-versioning. I have posted a new answer below which is a combination of my original solution and John's suggestion. Another idea that was suggested by SCdF would be to append a bogus query string to the file. (Some Python code, to automatically use the timestamp as a bogus query string, was .) However, there is some discussion as to whether or not the browser would cache a file with a query string. (Remember, we want the browser to cache the file and use it on future visits. We only want it to fetch the file again when it has changed.) Have you used any of the C++ interpreters (not compilers)? I am curious if anyone have used UnderC, Cint, Cling, Ch, or any other C++ interpreter and could share their experience. I noticed that there is a definition of the pvalue in my textbook is defined says the following:    I have no idea why it is written with a supremum. I've spent hours pondering this, does anyone have enough of a background to help me with this? Thank you! Favorite questions in the Android app I haven't been able to find my favorite questions in the official Stack Exchange Android app.  Looking at my accounts page gives nothing on them. How to type special/accented letters in LaTeX? How to type these special letters from European languages in latex? ä, é, and L'? What does the dot mean How to add chips from Material Components library to input field in android? Are there number systems corresponding to higher cardinalities than the real numbers? sample make file for creating more than one exe files with different directory Visual Studio Editor does not underline errors anymore My Visual Studio (2008) Editor has stopped to underline Errors (this nifty wavy red lines). I can't really tell when, but it can be related to the installation of .Net Framework 3.5 SP 1 or the MVC Beta (which I guess is unlikely). Furthermore have I installed and uninstalled both CodeRush and Resharper for evaluation purposes (decided not to keep either one of them).  Does anyone know the problem and how to restore this functionality again?