if force acts on a wheel away from the centre does it have both translatory and rotatory motion?( the wheel is not fixed.)  And  as per my knowledge,if force acts away from the line of axis it produces same acceleration as it produces when it is acting through the centre,so if in the 1st case if it produces rotation (due to torque) from where does the extra energy is getting to it??
Assume there is a rigid body in deep space with mass $m$ and moment of inertia $I$. A force that varies with time, $F(t)$, is applied to the body off-center at a distance $r$ from its center of mass. How do I calculate the instantaneous acceleration, rotational acceleration, and trajectory of this object, assuming it starts from rest?
Show that   $$\displaystyle1 - \frac{x^2}2\leq\cos x\quad\forall x \in \mathbb{R}$$   Let $f(x) = \cos x - 1 + \frac{x^2}2$; then we need to show that $f(x) \geq 0\quad\forall x \in \mathbb{R}$.  Since $$\lim_{x \to \pm\infty} f(x) = +\infty$$ and $$\operatorname{im}f'' = [0, 2]$$ (as $f''(x) = 1 -\cos x$), it remains to show that $x = 0$ is a minimum for $f(x)$, but I cannot do that because I would need to solve $f'(x) = x -\sin x = 0$, which I cannot. I think it would suffice to show that plugging in $x = 0$ we have that $f'(x) = 0$, and combining it with the fact that the function $f$ is convex, we would be done.    Is it enough? What's a better proof?  EDIT: It just occurred to me that there is a way to solve $f'(x) = 0$. Just observe that $f'(x)$ is an odd function and that $f''(x) \geq 0$ implies that $f'(x)$ is monotonically increasing. Since an odd function, if defined at $x = 0$, must be $0$ there, $x = 0$ is a solution to the equation and it's the only one. Prove that if $\mathcal P(A) \cup \mathcal P(B)= \mathcal P(A\cup B)$ then either $A \subseteq B$ or $B \subseteq A$. Chi-squared test when two vectors have different lengths I want to compare two groups (vectors) having different lengths to see if their distributions differ considerably. However, when I use chisq.test, I get an error due to inequalities of lengths:  &gt; chisq.test(factor(Group1), factor(Group2))   results in:  Error in chisq.test(factor(Group1[, col]), factor(Group2[, col])) :  'x' and 'y' must have the same length   Could someone help me with this? Thanks. root owned program with setuid bit on Ping is a a program owned by root with the user id bit set.  $ ls -l `which ping` -rwsr-xr-x 1 root root 35752 Nov  4  2011 /bin/ping   As I understand it, if a user runs the ping process, then the effective user id will change from the real user id (i.e. the user id of the person who launched the process) to the user id root. However when I try this and look at the output of ps to see if the ping process is running as the root user, I still get the real user id showing.  ps -e -o user,ruser,euser,cmd,args | grep ping sashan   sashan   sashan   ping -i 10 -c 1000 www.goog ping -i 10 -c 1000 www.google.com Apple Loops Copyright Selecting ARIMA orders based on ACF-PACF vs. auto.arima Why normal acceleration doesn't bring a change in speed? Suppose There's a particle of mass m moving with speed $v_{0}$, at certain moment a force starts acting on it (centripetal force) and it start uniform circular motion  Now every other reference states that centripetal force only brings about change only in direction  But plugging-in things in the equations paint a different picture  suppose  normal/centripetal acceleration = $A$  then $|v| = \sqrt{v_{0}^2+\frac{(A t)^2}{1}}$  we can certainly see that factor of $t$ will bring change in magnitude of velocity   Secondly, why the velocity is always tangential to the circle, but how can we mathematically show such? Intuitively for me if there's constant acceleration toward a certain then at some point of time, the body should 'slack' and fall toward the direction of acceleration   Here' a picture for a reference: how to hide mesh but keep hair particles How can I trigger a command when a player is hit by a Snowball? I am trying to make a snowball that triggers a command when it hits a player in 1.15, but I don't know how to do that. When I do execute at @e[type=snowball] run gamemode spectator @a[distance=..2] it only runs the command if it goes right next to them. I want to install Open JDK or Oracle JDK to my machine but I got an error during installation. The error is:  Selecting previously unselected package oracle-java8-installer. dpkg: warning: files list file for package 'unity-scope-gourmet' missing; assuming package has no files currently installed dpkg: warning: files list file for package 'unity-scope-guayadeque' missing; assuming package has no files currently installed dpkg: warning: files list file for package 'unity-scope-gmusicbrowser' missing; assuming package has no files currently installed dpkg: warning: files list file for package 'unity-scope-manpages' missing; assuming package has no files currently installed dpkg: warning: files list file for package 'unity-scope-firefoxbookmarks' missing; assuming package has no files currently installed dpkg: warning: files list file for package 'unity-scope-gdrive' missing; assuming package has no files currently installed dpkg: warning: files list file for package 'unity-scope-home' missing; assuming package has no files currently installed E: Sub-process /usr/bin/dpkg returned an error code (2)   I searched on Google and try to something for install/reinstall this packages. Finally i try to remove Unity. I faced this error for my every tries. I am using Xmonad. How can I fix this issue? Is it possible to set a DNS record to redirect all example.com/* traffic to www.example.com/*?  I'm running a  app on Heroku's .  I set up , and the documentation discusses . I also read about how to do this with . However, I'm wondering if there's a way to do this via DNS, i.e., without hitting my app. In wikipedia FKT algorithm is given for planar graphs. Not anything for complete graphs. I need to find the number of perfect matchings in complete graph of six vertices. How to make Arduino (Snap) access serial port /dev/ttyUSB0 (or /dev/ttyACM0)? I have Ubuntu 18.04 LTS installed.  I installed Arduino IDE v1.8.5 as a snap with  snap install arduino-mhall119   I have added my user to dial-out group and rebooted:  sudo usermod -a -G dialout $USER  $ groups  username adm dialout cdrom sudo dip plugdev lpadmin sambashare   I connected my Arduino Nano to the laptop, it is detected on port /dev/ttyUSB0 (ch341-uart).  $ ls /dev/ttyUSB0 -al crw-rw---- 1 root dialout 188, 0 авг 13 11:59 /dev/ttyUSB0   I tried to upload the simpliest Blink sketch and get the following errors in the terminal:  $ arduino-mhall119.arduino    avrdude: ser_open(): can't open device "/dev/ttyUSB0": Permission denied   Problem uploading to board.  See http://www.arduino.cc/en/Guide/Troubleshooting#upload for suggestions.**   Also I have installed arduino 1.8.5 with umake - it can access serial port in usual way.  Note: we already have  here, but the solution was to use , not fix for the Snap.  So my question is: How to make Arduino (Snap) access serial port /dev/ttyUSB0 (or /dev/ttyACM0)?  Update: connecting serial-port to the snap fails with the following message  $ snap connect arduino-mhall119:serial-port  error: snap "core" has no "serial-port" interface slots   Current interfaces are:  $ snap interfaces | grep arduino :home                            arduino-mhall119,gedit,organize-my-files,vlc :network                         arduino-mhall119,gedit,organize-my-files,pulsemixer,vlc :unity7                          arduino-mhall119,gedit,organize-my-files,vlc :x11                             arduino-mhall119,organize-my-files,pulsemixer,vlc -                                arduino-mhall119:serial-port   Changing permissions of serial port to 777 with sudo chmod 777 /dev/ttyUSB0 does not help.  The following do not work too (because of syntax errors):  $ sudo snap connect arduino-mhall119:arduino error: snap "arduino-mhall119" has no plug named "arduino"  $ snap connect arduino-mhall119:/dev/ttyUSB0 error: snap "arduino-mhall119" has no plug named "/dev/ttyUSB0"  $ snap connect arduino-mhall119 error: cannot resolve connection, plug snap name is empty   Information about YAML-manifest:  $ ls -l /snap/arduino-mhall119/current/meta/snap.yaml -rw-r--r-- 1 root root 353 июл 28 00:20 /snap/arduino-mhall119/current/meta/snap.yaml  $ cat /snap/arduino-mhall119/current/meta/snap.yaml |grep -A 12 slots  $ cat /snap/arduino-mhall119/current/meta/snap.yaml |grep -A 5 plug     plugs:     - x11     - unity7     - home     - network     - serial-port Isn't the const modifier here unnecessary? Prove  that the function $\displaystyle f(x)=\frac{x}{x^2+1}$ is continuous at $x=-1$. You should give a proof that is directly based on the  definition of continuity.   I saw many similar questions on this website, but none of them gave me a satisfactory answer. I more or less understand how epsilon-delta proofs work but my problem lies more in the algebraic manipulations probably. So here's my "attempt".  Definition of continuity; for $f:A \to \mathbb{R}$     $$\forall \varepsilon &gt;0 \ \exists\delta&gt;0\ \forall x \forall x_0 \in A:|x-x_0|&lt;\delta \Rightarrow |f(x)-f(x_0)|&lt;\varepsilon$$   So from what I understand, the procedure for epsilon-delta proofs are as follows: 1. Plug the point of interest $x_0$ into $|f(x)-f(x_0)|&lt;\varepsilon$ 2. Write $x$ in terms of $\varepsilon$ 3. Then plug $x$ into $|x-x_0|&lt;\delta$ 4. Find $\delta$ in terms of $\varepsilon$. Done    At step 1, we have $\left|\dfrac{x}{x^2+1}+\dfrac{1}{2}\right|&lt;\varepsilon$. Step 2 is where I'm stuck because of the 2nd degree polynomial in the denominator. First, I did this: $\left|\dfrac{x}{x^2+1}\right|&lt;\left|\dfrac{x}{x^2+1}+\dfrac{1}{2}\right|&lt;\varepsilon$. Then, I wasn't too sure about what to do, I thought about getting rid of the "$1$", but that would screw up my inequality. I thought about factoring $x^2+1$, but it doesn't have any real roots. I tried inverting the fraction, but that led me to nowhere. And I've already exhausted the tools I have. How do I proceed?  Apparently, epsilon-delta proofs are the most basic building blocks of analysis. And I'm already failing, this is both embarrassing and depressing... How to evaluate  $$\det\begin{bmatrix}1+x_1y_1&amp;\cdots &amp; 1+x_1y_n\\ \vdots &amp; \ddots &amp; \vdots \\ 1+x_ny_1 &amp; \cdots &amp; 1+x_ny_n\end{bmatrix}$$?  I tried factorisation but I have no idea. I could not factor anything out... I need a hint to kick off.   Edit: A conjecture due to multilinearity: if we let $v_j=(x_1y_j,\cdots,x_ny_j)^T$, then my guess would be $$\det\begin{bmatrix}1+x_1y_1&amp;\cdots &amp; 1+x_1y_n\\ \vdots &amp; \ddots &amp; \vdots \\ 1+x_ny_1 &amp; \cdots &amp; 1+x_ny_n\end{bmatrix}=\det(1,v_2,\cdots,v_n)+\det(v_1,1,\cdots,v_n)+\cdots+\det(v_1,v_2,\cdots,1)=\color{red}{0}$$ where $1$ denote $(1,\cdots,1)^T\in \Bbb K^n$,$\Bbb K$ the underlying number field. What is the difference between Clamav and Clamtk? I know that both of them are antivirus. How To Make Clickable Text In Minecraft I am trying to make clickable text in Minecraft in the chat box. It was used in the adventure map, "UP".   The idea is that you are given an option in the text to click, "Yes", or "No". Each word will give an output of a command.  How is this achieved? I would like to travel in the USA for a period of 3 - 6 months. But I want to spend some time working for my employer (remote work) and getting paid by the employer. Is this possible under a tourist visa (B1/B2)?  I hope this is OK since I will not be getting paid by a US employer. I am in the IT trade so all that I will be needing is my own laptop and a connection to the internet. The money I earn would actually help me pay for the expenses I get during the holiday. I am reading an older physics book that my professor gave me. It is going over Coulomb's law and Gauss' theorem. However, the book gives both equations with an $r^3$, not $r^2$, in the denominator. Can somebody please explain why it is given as r^3? An image is attached for reference.  Also for equation 1-24, can somebody please explain how the middle side is equal to the right side with the del operator?