When the centripetal force on an orbiting body disappears (e.g. if it the body is a ball and the force was exerted by a string and the string rips, or, more unrealistically, if the body is the earth and the sun suddenly disappeared), the body continues in linear motion. How is angular momentum conserved here?
I shall assume that the sun is at rest, so before the half mass disappear, the orbit of earth is an ellipse.
Here's my problem  I'm trying to do a cleanup of my hard drive, and I'm constantly seeing this scenario:  (1) Aggregate folder size: I check the folder size viewing properties right clicking the folder  (2) Individual folder/file sizes: I check the individual file/folder sizes belonging to that folder viewing properties on the individual file/folder sizes, and then adding them up (with hidden and system files shown).   These two figures are always different, sometimes by a large margin.   For instance, my C drive now, by method (1), shows a consumed space of 42.4 GB.   However, by method (2), by inspecting the respective files/folders, the total consumed space adds up only to 33.1 GB. This is a 28% discrepancy (over the smaller file size) and I have no idea where the rest of the space is going.   My hard drive is constantly in danger of being full (it's bootcamp on a mac with 50 GB of partitioned space), and there's not much I've got installed in terms of software, nor do I have large media files like movies, downloads, temp files, application temp files, torrents, etc.   Most of it's just consumed by the files in C:/Windows, but last I checked, I don't know enough to know which files to delete, and I've heard that you really can't do anything about the space boggling winsxs and system32.   Can you guys recommend a starting point to free up disk space?  Thanks!!  Baggio. Prove that $f'(a)=\lim_{n\to\infty}\frac{f(b_n)-f(a_n)}{b_n-a_n}$ under some conditions Prove that $f'(a)=\lim_{n\to\infty}\frac{f(b_n)-f(a_n)}{b_n-a_n}$, with $f: (k_1,k_2)\to\Bbb R$ differentiable at $a\in (k_1,k_2)$, $\lim b_n=\lim a_n=a$, and $a_n&lt;a&lt;b_n$.   My attempt: Im trying to write the above equality as a particular case of this other equality that hold by any interior point $a$  $$f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a-h)}{2h}$$  Because $a_n&lt;a&lt;b_n$ then I can write  $$f'(a)=\lim_{n\to\infty}\frac{f(b_n)-f(a_n)}{b_n-a_n}=\lim_{n\to\infty}\frac{f(h_n+a)-f(a-j_n)}{h_n+j_n}$$  Where $h_n=b_n-a&gt;0$ and $j_n=a-a_n&gt;0$, where these two new sequences tends monotonically to zero.  Now my problem: my thought is that $h_n$ and $j_n$ are particular sequences then I can build a sequence $k_n$ such that $j_n$ and $h_n$ are subsequences so I can write:  $$\{h_n:n\in\Bbb N\}\subset\{k_n:n\in\Bbb N\}\text{ and }\{j_n:n\in\Bbb N\}\subset\{k_n:n\in\Bbb N\}$$  (I can build this $k_n$ by example doing $k_{2n}=h_n$ and $k_{2n-1}=j_n$) and then I have the desired expression  $$f'(a)=\lim_{n\to\infty}\frac{f(b_n)-f(a_n)}{b_n-a_n}=\lim_{n\to\infty}\frac{f(k_n+a)-f(a-k_n)}{2k_n}$$  My question:   Is this proof right? In particular Im unsure of the implicit statement "if the limit exist for $k_n$ then it must exist too for the particular case with $h_n$ and $j_n$". The use of two sequences, $h_n$ and $j_n$ in the limit make me unsure about the validity of this manipulation and assumption. There is a simpler/different way to prove the same?   Thank you in advance. How to connect a GUI button to function? How to make a model cast shadows onto the world texture/background? In most of my models, I usually end up setting a world background using an HDRI image of some kind or another. This has always been really useful for lighting in general or as a background for my final render.  But what if I want my model to look like it is interacting or touching part of the background? For example, in the image below that I found, the modeled objects look like they're part of the scene, and even appear to cast shadows on the ground. How can I do this? Frying Oil Reuse What are some guidelines or rules with regard to filtering and reusing (vegetable/canola/sunflower) oil that has been used for deep frying?  Is it safe and acceptable under some circumstances to filter and keep oil? If so, does it depend on which foods were fried? How long can the filtered oil for be kept for?  Or, is it never appropriate? If not, are the reasons health or flavour related? or both? Do I need to take my weight off the bike to lock out the suspension? What can $E=mc^2$ do? In the famous equation , the variables stand for:     $E$ is energy, $m$ is mass, and $c$ is the speed of light (in vacuum).   And I understand the equation fairly but limited in knowing in its effects/outputs in reality, honestly.   Some example? Could anybody please give me some sort of examples in real world which are done by this equation? Show that $xy=0$ is a variety but not a manifold How do you set a default value for a MySQL Datetime column? How do you set a default value for a MySQL Datetime column?  In SQL Server it's getdate(). What is the equivalant for MySQL? I'm using MySQL 5.x if that is a factor. "I have never said" versus "I never said" I have never said this.   I never said this.    Is the usage of have in the first sentence justified or grammatically correct?   What is the difference in meaning?   When should I use one form over the other? Is the "omniscient-omnipotent-omnipresent" definition of God consistent? is commonly defined as an  (infinite knowledge),  (unlimited power),  (present everywhere) entity.  Is there any logical  in this definition?  I have seen several paradoxes like below     Does God know what he's going to do   tomorrow? If so, could he do something   else?" If God knows what will happen,   and does something else, he's not   omniscient. If he knows and can't   change it, he's not omnipotent.      "Can 'an omnipotent being' create a   stone so heavy that it cannot lift   it?"   Do they mean that the defintion of God ( as commonly held) is logically inconsistent? Showing that if a subset of a complete metric space is closed, it is also complete Let $(X, d(x,y))$ be a complete metric space. Prove that if $A\subseteq X$ is a closed set, then $A$ is also complete.  My attempt: I tried to prove that every Cauchy sequence $(b_n)$ of points of $A$ converges to a point $b\in A$. However could not figure out the exit way. Maybe I am on the wrong track. Could you please help me?  edit: More from my attempt:  Suppose $A$ is a closed set and let $(x_n)$ be a sequence of points $A$ such that $\lim_n x_n\to b$.  Suppose now that $A$ has the property that $b\in A$, whenever $x_n$ converges to $b$. We know that every element of $x_n$ which is convergent in $X$ also converges to a point in $A$. Since $x_n$ is a Cauchy sequence in $A$, it must converge to a point $y\in A$. But the limit of a convergent sequence is unique.   Take $x\in A$ and select an appropriate $n$, which enables $x_n$ to converge to a point $x$ in $X$. Since the limit is unique, it must follow that $x=y$. Thus $x\in A$ and $A$ is closed.   If $A$ is a closed subset of $X$, then any Cauchy sequence of a point in $A$ is convergent in $X$ and hence converges to a point in $A$. Thus $A$ is complete. Updating global js variable within function and sending updated to HTML document Is there a GUI IDE for programming python just like java has netbeans?  I wanted one where we could drag &amp; drop test boxes &amp; labels just as we do in swing components using netbeans or eclipse.  Or is there an add-on present in either netbeans or eclipse itself? What if the Earth stops in its orbit and instead falls straight towards the Sun? The Earth orbits around the Sun because it has angular momentum. If we stopped the Earth in orbit and then let it fall straight towards the Sun, how long would it take to reach the Sun in seconds?  Details and assumptions  The mass of the sun is 2*10^30 Kg. The mass of the earth is 6*10^24 Kg. The earth is 149,600,000 Km from the Sun. You may treat the Earth and Sun as point masses. Hole-and-nail paradox in special relativity How to share files through the local network? Efficiently storing query-affecting settings in a database I want to know how to efficiently store query-affecting settings in a large analytical database (MS SQL Server). By efficiency I mean no hardcode (1) and fast/optimal query processing (2).  My example  I've got a massive table of sales orders. Let's simplify it to two columns:  SalesOrder  OrderID      int      Primary Key  OrderDate    date   I need to retrieve the list of orders for the last X days and this X should be configurable. That's why I introduce a table of settings where I define it:  Admin_Configuration  SettingName   (PK)|      SettingValue  ------------------|------------------  DaysBack          |          30      It allows me not to hardcode this "30" in query definitions. Then I would run queries like:  SELECT OrderId   FROM SalesOrder as S   JOIN Admin_Configuration as A     ON A.SettingName = 'DaysBack'   WHERE [OrderDate] &gt; DATEADD(DAY, -A.SettingValue, GETDATE());   Performance  It avoids hardcoding, but leads to a performance problem. This query runs much slower than it would've, had I simply hard-coded 30 in the WHERE clause.  My understanding is that optimizer doesn't know how large my SettingValue is and assumes the worst - that all rows would be retrieved from the SalesOrders table, thus leading to a suboptimal query plan.  Trying a scalar function yielded even worse results. I'm desperate enough to have a separate view which stores these hardcoded settings but I've been refraining from it with hopes for a better solution.  Here is the original query plan (with a join): 4 seconds elapsed   Here is the plan with subquery, as :  SELECT orderid FROM [SalesOrder] as s WHERE s.OrderDate &gt; dateadd(DAY,    (SELECT -NumValue    FROM Admin_Configuration where SettingName = 'DaysBack'   )     , getdate())   7 seconds elapsed, slightly slower   As for the ideal performance: I just hardcoded 20, the query took 2 seconds. It needs mentioning that the Admin_Configuration table contains only 18 rows, hence it shouldn't affect performance.  SELECT OrderId   FROM SalesOrder as S   JOIN Admin_Configuration as A     ON A.SettingName = 'DaysBack'   WHERE [OrderDate] &gt; DATEADD(DAY, -20, GETDATE());   Using parameters would mean that I'd have to replace views (that I use now) with stored procedures/table-valued functions, which is unacceptable in my case I have three displays, a 4K display flanked by two 1080p displays. Physically, these monitors are roughly the same size. Furthermore, Windows 10 handles the DPI scaling of the 4K monitor separately from the 1080p monitors just fine. So while it appears as such to Windows    the real layout is three monitors of equal size side by side.  The problem is that the mouse keeps getting caught on the edges of display 2 and three when dragging from display 1. I would like the mouse to continue onto display 2 or 3 regardless of which part of the left or right edge of display 1 the mouse touches.  This question has been asked before (e.g. ) and a common answer is to suggest that this behaviour is intentional on the basis that mouse movement should reflect different size displays. Such an answer is not appropriate as it doesn't consider DPI and scaling. : So how many named special cases of each modal verb are there, in toto? There's   1. subjunctive should, and   2. mandative should, and   3. , at least;   and I s[up]pose 4. Should we? is suggestive should, and   5. Shouldn't we? is tag **should**, and I can think of a lot more.    I ask about only operative uses of this modal verb including literary and formal usage,  but please omit anything obsolete. I'm trying to classify them and learn the formal terms describing each. Are there any apt others in addition to the ones above? (I added the enumeration below)  1. What's 'subjunctive should'?  Whenever should is conjugated in the subjunctive mood? 5. What's a tag **should** ? 6. Where does  should/would  feature? What's it called? This should is detailed in   or  or .