Will the total work done by friction on a system be always zero if friction is static and internal to the system? Explain if possible by man boat problem as system and no friction is being provided by water to boat.  Friction only exists between man and boat and man is walking without skidding to the other end.
As I am seeing this,  the man and boat will have different displacement from ground frame,  so there should be a net work done by friction .  The books says,  total work done by friction is zero when friction is static and internal to system. Please explain..
A vessel is filled with liquid,3 parts of which are water and 5 parts syrup.How much of the mixture must be drawn off and replaced with water so that the mixture may be half water and half syrup?  What is the best possible way to solve this question? Can someone explain how combinatorial proofs work?  I've included an example questions that's been giving me a hard time.  Any insight on the topic would be great.  $$\sum_{k=1}^{n}k{n \choose k} = n2^{n-1}$$  Another one:  $${2n \choose 2} = 2{n \choose 2} + n^2$$  The solution says:  LHS: Pick two people from a pool of n men and n women. RHS: Pick two women, pick two men, and pick one of each.  But I don't see how this applies.  How does $n^2$ show that we're picking "one of each?"  Thanks for the help :) I'm using a flow to set a field to a certain value (01, 02, 03) depending on the value of another field on the same record. I initially wanted to do it with process builder but because of the mapping it seems I would need three 'criteria nodes' or a formula field. Furthermore, it seems Salesforce is recommending using flows instead of process builder for before update actions as it is faster.  The problem with Flows is that I found a way to trigger it when the record is new or updated but not when the specific field is changed so it triggers every time the whole record is updated even when the field stay the same.  On Process Builder, you can use isChanged and then call the flow with specific record but that kinds of defeat the point of using flows for before Update.  So is there a possibility to do the same thing directly in flow ?  Thank you for the help ! Apex REST API - Unable to set StatusCode to 429 I am trying to set the status code to 429 for Too many request using the code below:  if(Limits.getQueries() &lt; Limits.getLimitQueries()){               upsert UpdateData;               }               else if((Limits.getQueries() &gt;= Limits.getLimitQueries()))               {                   RestContext.response.statusCode = 429;                   UpdateError[] err = new UpdateError[]{                    new UpdateReturnError(UpdateData.Id, 'Too many Requests', 'Error updating MessageID ' + msg.MessageId)                 };                 RestContext.response.responseBody = Blob.valueOf(JSON.serialize(err));                 RestContext.response.addHeader('Content-Type', 'application/json');                 break;               }             }   I am getting the following error in the raw response:    Can someone help me out with this? Why do all SSH-RSA Keys begin with "AAAAB3NzaC1yc"? How to find the sum $$\sum^n_{k=1} (k^2+k+1)k!$$  What I tried as follows :   $$\sum^n_{k=1} (k^2+k+1)k!$$  =$$\sum^n_{k=1} (k^2)k!+ \sum^n_{k=1} (k)k! + \sum^n_{k=1} (1)k!$$  Now we can write $\sum^n_{k=1} (k^2) $ as $$\frac{n(n+1)(2n+1)}{6}$$ but what to do with $k!$ please guide thanks....... Add leading zeroes to number in Java? Is there a better way of getting this result? This function fails if num has more digits than digits, and I feel like it should be in the library somewhere (like Integer.toString(x,"%3d") or something)  static String intToString(int num, int digits) {     StringBuffer s = new StringBuffer(digits);     int zeroes = digits - (int) (Math.log(num) / Math.log(10)) - 1;      for (int i = 0; i &lt; zeroes; i++) {         s.append(0);     }     return s.append(num).toString(); } Let $f$ be an entire function and $L$ a line in $\mathbb{C}$ such that $f(\mathbb{C})\cap L=\emptyset$. Show that $f$ is constant function.  If $f$ is not constant  then $f(\mathbb{C})$ is dense set in $\mathbb{C}$, but how can I use that line does not intersect the image set? What happens when ability scores reach 0? D&amp;D 5E does not seem to have rules for ability score damage (that I could find), although it does mention that if CON is decreased, your maximum HP is reduced. In addition, the Shadow monster in the Monster Manual has a Strength Drain attack that deals 2 Strength damage. The attack says that if a character's Strength reaches 0, the character dies. Are there any general rules for ability damage in 5th Edition? How to prove $\vdash\neg P\to (P\to Q)$? I am trying to prove,  $$\vdash\neg P\to (P\to Q)$$  where $P$ and $Q$ are arbitrary but otherwise fixed formulas. The only axioms and rule of inference that I can use are,  $\color{crimson}{\text{Axiom 1.}}\  P\to (Q\to P)$ $\color{crimson}{\text{Axiom 2.}}\  (S\to (P\to Q))\to((S\to P)\to (S\to Q))$ $\color{crimson}{\text{Axiom 3.}}\  (\neg Q\to\neg P)\to(P\to Q)$ $\color{crimson}{\text{Rule of Inference.}}$ .  I know that in Angelo Margaris's book First Order Mathematical Logic there is a proof (page 53) but as I have mentioned , the dot notation seems very confusing and translating them to the nested parentheses notation seems very difficult to me. So, I am trying on my own to prove the result. Can anyone help? Stack Overflow in ES, PT, RU, JP have mixed-language descriptions A few months ago I set full disk encryption during installation of Ubuntu 17.10. Now, I've decided to upgrade. The upgrade ran to the end without any problems. However, after reboot I am not able to log into my encrypted disk.  Where can be the problem? I am 100% sure I press the "right keys" on the keyboard, but technically I don't know what I am writing because of "*" characters and maybe after upgrade my keyboard layout changed. I use some characters which might be somewhere else on the keyboard. What language is default after upgrade?   By the way, I've already tried Caps-lock, but still no luck.  Help will be really appreciated. I am not a fan of installing my system and all backups over and over again. How do I know when to use "the" versus "a" versus "∅" as an article on a noun? With proper nouns, we don't use the except for river names, newspaper names, etc.  I want to know why we use the with White House. I mean, under which rule can we categorize it? What could other similar examples for that rule be? If a derivative of a continuous function has a limit, must it agree with that limit? Is it correct to say e.g. "I like your way of wearing" or "the way you wear looks perfect"?  Is it politely? Is it in use or not? May I use "the way you wear" referring to whole wearings including dress, scarf, hat, bracelet. jewellery, etc?  Thank you for your suggestions in advance. Why does stdafx.h work the way it does? As usual, when my brain's messing with something I can't figure out myself, I come to you guys for help :)  This time I've been wondering why stdafx.h works the way it does? To my understanding it does 2 things:   Includes standard headers which we might (?) use and which are rarely changed Work as a compiler-bookmark for when code is no longer precompiled.   Now, these 2 things seems like two very different tasks to me, and I wonder why they didn't do two seperate steps to take care of them? To me it seems reasonable to have a #pragma-command do the bookmarking stuff and to optionally have a header-file a long the lines of windows.h to do the including of often-used headers... Which brings me to my next point: Why are we forced to include often-used headers through stdafx.h? Personally, I'm not aware of any often used headers I use that I'm not already doing my own includes for - but maybe these headers are necessary for .dll generation?  Thx in advance Are all algebraic integers with absolute value 1 roots of unity? I have just finished my homework.   I just finished my homework.      I think there must be a difference in meaning. Could anyone tell me the difference in meaning sentence 1 and sentence 2? How to reset the Xorg / xserver? Xserver keeps crashing on my general user account both in Unity 3D and 2D. It doesn't do it on the guest account.  That's why I want to reset / rollback everything xserver related, hoping this is going to fix it.  Most answers on this involve sudo dpkg-reconfigure xserver-xorg. However, this does nothing for me (generates no output) and I read in other forums, that it's deprecated.  So, what's the current way to reset all xserver / xorg options; what's the alternative to sudo dpkg-reconfigure xserver-xorg? X crashes during work, not before login or so. And the effect of a crash is, that I get thrown out of the session, not a freeze or machine crash. Deleting comment removes "Add comment" link on Mobile If I delete a comment on the mobile version of the site (at least if it's the only comment on an answer), the "Add comment" link will vanish along with the comment, so I have to reload to leave another comment.     but got declined as , which is why I'm reposting this (as it may well be a new bug with the same symptoms).  I'm using Firefox 32.0.3 on Android.