so let's suppose there is a wheel, which rotates thus moving on a flat plane, without any slopes, and the gravitational pull throughout the plane is the same as well. The height of the center of  mass of the wheel does not change as well. So my question is does the wheel do work against gravity?  Personally, I believe it does because when you take the contact point of the wheel to the plane, its height changes as the wheel rotates, thus changing the potential energy of the point while rotating. However, I couldn't be sure if my hypothesis was correct by researching in various websites and couldn't conduct any simulations or experiments. So I would love to hear your opinions on the topic.
Perhaps this is trivial: how does one prove that the work done by a homogeneous gravitational field on a system of particles is equal to the work done on a point mass with the total mass of the system, located at the center of mass of the system.
How can I serialize an object to JSON in JavaScript? I lost my .keystore file? how I can continue this limit resolution?  The limit is:   $$ \lim_{n \to \infty}{\sqrt[n]{\frac{n!}{n^n}}} $$  This is that I have done:  I apply this test: $ \lim_{n \to \infty}{\sqrt[n]{a_n}} = \frac{a_{n+1}}{a_n} $  Operating and simplifying I arrive to this point: $$ \lim_{n \to \infty}{\frac{n^n}{(n+1)^n}} $$  I've done something wrong? Thanks! Is it possible to redirect all of the output of a Bourne shell script to somewhere, but with shell commands inside the script itself? Redirecting the output of a single command is easy, but I want something more like this: #!/bin/sh if [ ! -t 0 ]; then     # redirect all of my output to a file here fi  # rest of script...  Meaning: if the script is run non-interactively (for example, cron), save off the output of everything to a file.  If run interactively from a shell, let the output go to stdout as usual. I want to do this for a script normally run by the FreeBSD periodic utility.  It's part of the daily run, which I don't normally care to see every day in email, so I don't have it sent.  However, if something inside this one particular script fails, that's important to me and I'd like to be able to capture and email the output of this one part of the daily jobs. Update: Joshua's answer is spot-on, but I also wanted to save and restore stdout and stderr around the entire script, which is done like this: # save stdout and stderr to file  # descriptors 3 and 4,  # then redirect them to &quot;foo&quot; exec 3&gt;&amp;1 4&gt;&amp;2 &gt;foo 2&gt;&amp;1  # ...  # restore stdout and stderr exec 1&gt;&amp;3 2&gt;&amp;4 "The product key entered does not match any of the Windows image" I have a HP with Windows 7 installed. I want to do a clean installation of Windows 8 from a bootable DVD.   I purchased my Windows 8 32-bit core from the Microsoft store and I received my product key. I then burned the .ISO file to a blank DVD using imgburn. When I restart and change my BIOS to boot from internal CD/DVD and then boot from my internal DVD, I start the installation process.  After I pass the screen where I select the language and location, and click Install it stays there for a minute and then it says:     The product key entered does not match any of the Windows image   I did my research and learned that I need to add a PID.txt file to my source file containing the Windows 8 .ISO but the thing is: I do not know how to add files to the .ISO. The guide I researched is for a bootable USB and I am using a DVD. Ubuntu 18.04 bond with netplan duplicate MAC address after cloning How can I get debug log in Mavensmate TeXnicCenter & Adobe Reader DC How to configure TeXnicCenter to work with Adobe Reader DC? When I building and view current file .tex the following error "Can not execute the command" shows up and the PDF is not generated.    If there are any related post, could help. Thanks a lot! What is a valid use case for an "execute only" file permission? I was reading up on chmod and its .  I saw that 1 is execute only.  What is a valid use case for an execute only permission?  To execute a file, one typically would want read and execute permission.  $ echo 'echo foo' &gt; say_foo $ chmod 100 ./say_foo $ ./say_foo bash: ./say_foo: Permission denied $ chmod 500 ./say_foo $ ./say_foo foo $x^3-9=y^2$ find integral solutions Find all integral solutions  $x^3-9=y^2$  I tried many times but still no idea how to solve it. I will be grateful for any help. Why does Microsoft Windows' performance appear to degrade over time? Windows XP/2003 and earlier (can't attest to Windows Vista, but I suspect it's the same) all appear to become more sluggish over time as applications are installed and uninstalled.  This is not a scientifically tested observation, but more of a learned-through-experience piece of wisdom.  (I've always suspected the registry as being behind the issue.)  Is there any concrete evidence of this degradation occurring, or it just an invalid perception of mine? The kernel and range of the powers of a self-adjoint operator Suppose $L: V \rightarrow V$ self-adjoint (on a finite-dimensional space $V$); $b$ is a real number. Show   The minimal polynomial has distinct roots; $\ker(L) = \ker(L^k)$ for $k\geq1$; $\operatorname{im}(L) = \operatorname{im}(L^k)$ for $k\ge 1$; $\ker(L-bI) = \ker((L-bI)^k)$ for $k\geq1$;     My thoughts:  1) I think I'm supposed to assume root is repeated and reach a contradiction but I'm not sure how to proceed mathematically with that.  2-4) pretty sure there's some huge theorem that I'm missing. can't create a jobs user profile I'm stuck on this page even after linking stackoverflow and github accounts. For which values $r, s \in \mathbb R$ is the set $\{u, v, w\}$ linearly independent? Reverse polarity protection Use induction to prove the following: $1! + 2! + .... + n! \le (n + 1)!$ What kind of material is the Robert Downey Jr. Iron Man's armor supposed to be made from? I know he crafted the prototype from the things he had on hand, but once he was able to build the next one(s) in his own lab, what hi-tech material did he use for the movie armor? I use  on Windows 7 and I want to know if there is a good substitution for it on Ubuntu.   I want something that is good for editing html files. Surjective morphism of affine varieties and dimension I'm trying to do the following exercise from the book An Invitation to Algebraic Geometry:     Show that if $X \to Y$ is a surjective morphism of affine algebraic varieties, then the dimension of $X$ is at least as large as the dimension of $Y$.   Could anyone give me a hint? I'm looking for a proof that avoids commutative algebra since it is not mentionned in the book at this point. Why is `[` a shell builtin and `[[` a shell keyword?