How can I get an equation of trajectory of any particle moving in a circular motion? I mean I want to get an equation of a circle by using the conditions given for a circular motion. We know that if a particle enters in a force field $\vec F$ with velocity $\vec v$, where force $\vec F$ is perpendicular to $\vec v$, and if the force remains perpendicular to $\vec v$ always, then the trajectory of that particle becomes a circle. We can understand it physically. $\vec F$ is always perpendicular on $\vec v$ and hence on the displacement at every position of the particle. We know the force is always directed to a fixed point from which the force is created. At any point on the trajectory, the direction of displacement is given by the tangent on that point. so we can assume the force as the radius of a circle, as it is always perpendicular to the tangent on the trajectory, which is a property of a circle. So the trajectory will be a circle. But how can I get it mathematically? Can I get an equation of a circle for the trajectory of the particle by using the conditions(like $\vec F$ is always perpendicular to $\vec v$ and hence on displacement or something like that)?
If it is given that component of acceleration perpendicular to the velocity of a body has a constant, non-zero magnitude, how can we mathematically prove that the trajectory of the body will be circular?   I have basic knowledge of calculus, vectors and co-ordinate geometry. I can prove that the trajectory of a body in projectile motion is parabolic. I tried to follow a similar method to get an equation of circle but couldn't succeed. I tried forming parametric equations of x and y in terms of time t (the way we proceed when proving the parabolic path of a projectile) but could not reach the conclusion.  Edit: I have edited the question in order to improve it. However I still can not share my work on this question as I had asked the question almost 6 months ago and now I can not find where I tried solving it. Sincerely sorry for that.
What is a good asymptotic for $f_n = f_{n-1}+\ln(f_{n-1})$? I don't want to use the Super for that, I want to use Alt+Space instead. Integrating Knowledge into an existing website Many of the people have asked almost the same question.  I'm trying to render a big mesh + fur/grass file with a few characters on the screen.  when I hit render, after a number of processing, copying data to gpu etc... i get following error.  Mem: 2960.07M, Peak:2960.07M | CUDA error: Out of memory in cuMemAlloc(&amp;device_pointer, size).  I'm using Geforce GTX 690 which contains 4GB memory.   Can somebody tell me how can i get GPU rendering, while 2960M is far less then 4196M of memory.   I know somebody would suggest me to use CPU.. but i'm curios why not GPU?   I've already compressed texture, etc   Edited  I don't think any other application is using any of my 1GB remaining memory. everything is closed but blender is there, also the GPU utilities are showing 2000/2000 MB are free on one card and 1900/2000MB is free on 2nd card.    i'm using GTX 690. with two GPUs of 2GB each. If somebody thinks that one GPU should have memory matching to the cuda memory/or Peak memory then I'm doubtful. Because in preferences Blender does know that I have 2GPUs where I can select either first GPU or second GPU or even both.   When I render some of smaller objects/scenes both of my GPU renders simultaneously with two blocks.  I'm running following specs  Xeon E5-1660 0 24GB - 1600mhz GTX690 What's the difference between a method and a function? Feed lists answered questions as "unanswered" So I did a fresh install of Ubuntu 20.04 on a computer with the new RTL8125B Realtek network card and I cannot get the network to work. I was very suprised since this is supposed to be the new feature of the 5.4 kernel (I have 5.4.0-26-generic installed). Does anyone have any idea on how to get it working? Thank you. $ lspci -knn | grep Eth -A3 06:00.0 Ethernet controller [0200]: Realtek Semiconductor Co., Ltd. RTL8125 2.5GbE Controller [10ec:8125] (rev 04) Subsystem: ASUSTeK Computer Inc. RTL8125 2.5GbE Controller [1043:87d7] Kernel modules: r8169 07:00.0 VGA compatible controller [0300]: Advanced Micro Devices, Inc. [AMD/ATI] Ellesmere [Radeon RX 470/480/570/570X/580/580X/590] [1002:67df] (rev e7)  $ dmesg | grep -i r8169 [    0.911236] r8169 0000:06:00.0: unknown chip XID 641 Is it mandatory to have a link to the theme designer? I just wanted to know if it's ok to modify/remove links pointing to the designer of the theme.  In my case I have "theme by" and the site's logotype, would it be ok to remove the logotype and write the site's name? Or remove it altoghether?  Does it have to do with the particular designer of the theme?  Thanks How do I change player name in Minecraft smp in offline mode on a Mac? If $f: \mathbb Q\to \mathbb Q$ is a homomorphism, prove that $f(x)=0$ for all $x\in\mathbb Q$ or $f(x)=x$ for all $x$ in $\mathbb Q$.  I'm wondering if you can help me with this one? I want to prove this:  Prove that if $a \in \mathbb{Z}$ is divisible by $4$, then there exist $b$ and $c$ where $b,c \in \mathbb{Z}$ such that $a = b^2 - c^2$  I want to prove this directily:  $4\mid a $,  $a\in \mathbb{Z} \implies \exists b,c \in \mathbb{Z}, \quad$ s.t. $a=b^2-c^2$  Let $k\in \mathbb{Z}$. Then $4=a\cdot k$ and $k$ can divide $4$. Hence,  $\frac{4}{k} =b^2 - c^2 = (b-c)(b+c) \implies \dfrac{4}{k\cdot(b-c)} = b+c$  Is this proof correct? A list of inconveniences between quantum mechanics and (general) relativity? It is well known that quantum mechanics and (general) relativity do not fit well. I am wondering whether it is possible to make a list of contradictions or problems between them?  E.g. relativity theory uses a space-time continuum, while quantum theory uses discrete states.  I am not merely looking for a solution or rebuttal of such opposites, more for a survey of the field out of interest. So this is the situation:   My brother received a PS4 on his birthday. He doesn't have a PS+ account, and I don't have a PS4, so I activated it as my primary PS4. Based on what I've read on the FAQ, any user can play my digital games and do online gaming since I activated the system as my primary PS4. A friend offered his account so we can play his digital games. He has a primary PS4 activated.   The question is, can we really play his games without any issues? For example, can we download GTAV from his account and play it using any account on our PS4? how to avoid number repeation by using random class in c#? Trigger to check for duplicate contacts on lead conversion I'd like to write a Lead trigger, to check for duplicate Contacts before inserting the lead, or converting it.   Contacts fields I'd like to check:   Firstname - should not be the same as the leads name Lastname - should not be the same as the leads name Email - should not be the same as the leads phone   After checking these fields &amp; failing to find a duplicate, only then should the lead be converted. Yes, I'm a moron, and I tried to manually install a higher point release of python 2.7.16 to /usr/bin/  (i did ./configure --prefix=/usr) on my Ubuntu 16.04 machine. It seems to work fine EXCEPT for packages that depended on python, like apt-listchanges, which fails with messages like ImportError: No module named apt_pkg  there's no make uninstall available, I've tried manually removing files, and still when I try apt --reinstall install python, Nothing gets overwritten and the python executable still gives the new 2.7.16 version...   Any idea how to get the official repository version back?? Landscape topography from grayscale image Ridge & LASSO norms This post follows this one:   Here is my question:  As far as I know, ridge regularization uses a $\ell_2$-norm (euclidean distance). But why do we use the square of this norm ? (a direct application of $\ell_2$ would result with the square root of the sum of beta squared).  As a comparison, we don't do this for the LASSO, which uses a $\ell_1$-norm to regularize. But here it's the "real" $\ell_1$ norm (just sum of the square of the beta absolute values, and not square of this sum).  Can someone help me to clarify? "cron" processes spawning and piling up, but no crontab EDIT: I solved this, so see below for my own answer. While the bounty asks for a fix to the problem, I'd like to award it to someone who can explain why a cron process that ran for 6 years over multiple MacOS versions suddenly became a problem in Catalina.  First note, this is not a duplicate of  because I don't have active crontabs on my computer. Cron processes spawn, but they never finish, so they just keep piling up. I'm running MacOS Catalina 10.15.1  Cron jobs are spawning every couple of minutes and never dying. The oldest cron process always has a parent process "launchd", and I've searched for populated crontabs and can't find any. If I let this go on long enough, eventually I run into a scenario where I have hundreds of running cron processes and everything on my computer grinds to a halt, and I can't even open new apps (I get cryptic messages like ".app cannot be opened" when I try to open an app). My only solution at this point has been to keep activity monitor open and periodically killing off all active cron processes.  I've been able to create a new user account that seems to be immune from this issue. However, as soon as I logged into my apple ID on that account to set up iCloud drive, the problem started with that account, too. I logged out of my Apple ID and attempted uninstall iCloud drive, but the problem persisted. Clean installs of Catalina don't clear up the problem, I've reset SMC and PRAM multiple times, and repairing permissions (via this ). I've also ensured that cron has full disk access (via this ). I'm at a dead end with trying to troubleshoot this problem.  I'm not sure it's related, but my "system.log" file seems to get spammed the following few lines:  &lt;redacted&gt; AMPDeviceDiscoveryAgent[2132]: Entered:_AMMuxedDeviceDisconnected, mux-device:63 &lt;redacted&gt; AMPDeviceDiscoveryAgent[2132]: Entered:__thr_AMMuxedDeviceDisconnected, mux-device:63 &lt;redacted&gt; AMPDeviceDiscoveryAgent[2132]: tid:9a27 - Mux ID not found in mapping dictionary   ANY help is appreciated. I've been doing ASP.NET development for a little while now, and I've used both the GridView and the DataGrid controls before for various things, but I never could find a really good reason to use one or the other. I'd like to know:  What is the difference between these 2 ASP.NET controls? What are the advantages or disadvantages of both? Is one any faster? Newer? Easier to maintain?  The intellisense summary for the controls doesn't seem to describe any difference between the two. They both can view, edit, and sort data and automatically generate columns at runtime.  Edit: Visual Studio 2008 no longer lists DataGrid as an available control in the toolbox. It is still available (for legacy support I assume) if you type it in by hand though.