The relation between the spring constant and its radius I've observed that the spring constant of a spring with a smaller radius is higher than that of a spring with a larger even though they are made of the same material , their thickness and all other factors are same but the radius. Does this mean that the spring constant of a spring made of a given substance is not its property and is there a relation that governs such behaviour of a spring?
What factors influence the spring constant? For certain springs we have been taught that Hookes law, $F=-kx$ works. For such springs, what factors determine k. For instance, k is proportional to length. Can k be expressed in terms of dimensions, mass and other fundemental quantities, or is k a fundemental property of the spring itself?
Use early 2009 20" iMac as external display for 2015 rMBP I have a Word file which has some Excel plots in it. Unfortunately I lost the original excel plots and the word file is now 'de-linked'.   Is there a way to retrieve the lost data for the plots? Just copying the plot back to Excel does not seem to work, but I can see that the data is still there somewhere, because hovering over a dot on the plot shows the values. How can I integrate $$\ln \left( x+\sqrt{1+x^2} \right) $$   The problem is from IA Maron Calculus, Problem 4.3.17. The problem is under the heading 'Integration by parts'. I have tried it, but it just leads to 2 unsolvable integrals What is the difference between IQueryable and IEnumerable? What is the difference between IQueryable&lt;T&gt; and IEnumerable&lt;T&gt;?    See also  that overlaps with this question. What is the best backup strategy for macbook pro laptop? I have a lot of data on my macbook pro retina laptop and i am looking for the best backup strategy.  I have previously had a windows desktop with an external harddrive connected all of the time and a backup program that copied to the external harddrive incrementally.  I am trying to figure out the best backup strategy given its a laptop that i move around with and travel with alot (so can't really have a external hard drive always connected)  What is the recommended backup solution for map laptops? How to only append to clipboard (but not overwrite current data)? Let $R$ be a commutative Noetherian ring (with unity), and let $I$ be an ideal of $R$ such that $R/I \cong R$. Then is it true that $I=(0)$ ?  I know that a surjective ring endomorphism of a Noetherian ring is also injective, and since there is a natural surjection from $R$ onto $R/I$ we get a surjection from $R$ onto $R$, but the problem is I can not determine the map explicitly and I am not sure about the statement. Please help. Thanks in advance. Where to spend a night in the Mumbai airport My girlfriend and I have a layover at the Mumbai airport from 19:35 to 05:55 and we would like to sleep in a cheap place near to the airport (maximum 30 minutes away). Is there any cheap hotel near there? Or any cheap lounge where to sleep well? (flight at 05:55 is an intercontinental flight and we prefer to stay "fresh"). If not possible, is the airport a safe place to sleep? $a$ and $b$ are natural numbers and $p$ is a prime number. Prove that if $a^{p}-b^{p}$ is divisible by $p$, then it is also divisible by $p^{2}$.  My attempt: Based on Fermat's theorem $(a^{p}-a)$ and $(b^{p}-b)$ are divisible by p, therefore their difference is divisible by $p$. I.e. $(a^{p}-a)-(b^{p}-b)=(a^{p}-b^{p})-(a-b)$ is divisible by $p$. Since the left hand side of this equation is divisible by $p$, the right hand side also should be divisible by $p$. Based on the problem's assumption $(a^{p}-b^{p})$ is divisible by $p$ therefore we conclude $(a-b)$ must be divisible by $p$. Now We can factorize $(a^{p}-b^{p})$ $$(a^{p}-b^{p})=(a-b)(a^{p-1}+a^{p-2}b+....+b^{p-1})$$ we have to prove $(a^{p}-b^{p})$ is multiple of $p^2$ which means we have to show the right hand side should be multiple of $p^{2}$. But $(a-b)$ is multiple of p therefore we have to show  $(a^{p-1}+a^{p-2}b+\cdots+b^{p-1})$ is multiple of $p$ and I am stuck here. Any help would be appreciated. so I am creating a few posts about bearings and wheels, (story so far):      I have now got the wheels, bearings and freehub working.  Started riding to work this morning and I almost couldn't stop at a junction.  Thankfully wasn't going too fast.  I physically can not get the wheel to lock.  I have to use both brakes at full force to get the bike to roll to a stop.  If I only use one brake I would not stop at any junction in time.  Could the cause be any of the following:   Over greased bearings? Over greased axle? not tightened cone hubs enough? not tighted axle enough?   I do not have any grease on the disk brakes them selves.  Rather nervous getting on the bike now.  Don't seem to be doing too well with this bike home fixing thing.  over greased bearings?  EDIT  Forgot to mention that it is hydraulic disk brakes.  EDIT 2  So in the end I took it to the LBS as I didn't have much confidence in my self at this point.  They had to replace the front pads but where able to clean the rear and cleaned the rotors.  (I had tried to clean the rotors but obviously not vigorously enough).  He said it was a bit of a pain to do.  All is working again.  They double checked my wheels and said I had put them back together correctly which is reassuring.  Thanks for all the help Significance of the second focus in elliptical orbits Moderncv has a line of empty space between address and phone/email, like this:   MWE:  \documentclass[12pt]{moderncv} \moderncvstyle{banking} \name{Sherlock}{Holmes} \address{221B Baker Street}{London}{U.K.} % Your current address \phone[mobile] {+44~(0)71234 00000}    % Your mobile phone number \email{superdetective@mail.com}  \begin{document} \makecvtitle \end{document}   How do I remove/decrease that space by shifting the phone and email line up? What is the association bonus, and how does it work? Low-RAM Linux Distro Specifically for serving static sites If every pair of congruence equations admits solutions, then the entire system admits solutions Let a system of three linear congruence equations in integers be given;      \begin{cases}x\equiv b_1\mod c_1\\ x\equiv b_2\mod c_2\\ x\equiv b_3\mod c_3\\ \end{cases}   with $c_1,c_2,c_3\in\mathbb Z_{+}$. Show that, if every pair of congruence equations admits solutions, then the entire system admits solutions.    Now, if every pairs admits solutions, then for example $b_1\equiv b_2\equiv a\mod \gcd(c_1,c_2)$ for some $a$ then;   $x-a\equiv b_1-a\mod c_1$  $x-a\equiv b_2-a\mod c_2$  replacing $x$ by $x':=\frac{x-y}{\gcd(c_1,c_2)}$, we obtain:  $x'\equiv b_1'\mod c_1'$  $x'\equiv b_2'\mod c_2'$  and since $\gcd(c_1,c_2)=1$ by Chinese Remainder Theorem we have a unique solution $\mod{c_1'c_2'}$, so $x\equiv (\gcd(c_1,c_2)x'+a)\mod{\operatorname{lcm(c_1',c_2')}}\quad(1)$  We can do it with another pair (take $2$nd and $3$rd, since we know that such a pair admits solutions;  $x\equiv(\gcd(c_2,c_3)x''+a')\mod{\operatorname{lcm(c_2',c_3')}}\quad(2)$  Now how can I show that for the pair $(1)$ and $(2)$ also solutions exist ? I am trying to make folder that everyone can view and can't copy,modify.    For all answers.I thinking about specially ms office files. I tried following methods  Software  - copy protect    - anti copy    This method can't protect save as and upload  MS office macro    Users can enable,disable macros.  MS office restriction    This is best choice but can't protect from file uploading.   Almost I found the solution. Way is combination of MS office restriction and restrict access from browsers using firewall or antivirus. But I don't exact how to do this. How to prove this is a metric? I have 2 distance functions $d(x,y)=|x^2-y^2|$ and $d(x,y)=|x^3-y^3|$ and I am trying to prove that they are metrics on $\mathbb R$, or give a counterexample that they are not metrics on $\mathbb R$.  I managed to prove the first 3 properties for the 2nd function namely:   $d(x,y)\ge0$ $d(x,y)=0$ iff $x=y$ $d(x,y)=d(y,x)$   and I now have shown that the 1st function is not a metric.  But for the triangle inequality property, $d(x,y)\le d(x,z) + d(z,y)$, Im stuck on it. How would go about proving it for $d(x,y)=|x^3-y^3|$? Thanks. How can I remove duplicate values from an array in PHP? Google Sheets has several import functions   IMPORTDATA IMPORTFEED IMPORTHTML IMPORTXML   Sometimes the above functions returns errors like #N/A Imported content is empty and would like to be sure that there isn't a problem with the resource content to be imported.  How do I know if these functions are able to get the content that I looking to import?  I know that there also exist IMPORTRANGE but that function only is able to import content from Google Sheets spreadsheets What is moiré? How can we avoid it? What is the image defect called "moiré"? What causes it, and how can we avoid or reduce it? Is it related to "false color"?