Let's keep things simple and assume that the wheel does not slide. Assume you are in a car that moves with constant speed. Obviously, the wheels exert a tangential force to the surface on the road in the points of contact, P. According to Newton's third law action equals minus reaction so the road exerts a force to the wheels in the direction of motion (please, see fig. 1). So the friction acts in the direction of motion of the car. It seems reasonable since if there were no friction then the wheels would be turning but the car would not move. However, there is one thing I can not understand. Let's now imagine a wheel rolling (without slipping) on a smooth surface with initial velocity V0 (here I mean that t). The same argument should hold (the wheel exert a tangential force to the surface on the road in the point of contact, P. According to Newton's third law action equals minus reaction so the road exerts a force to the wheel in the direction of motion). So the frictional force should again act in the direction of motion. Nevertheless, the wheel slows down with time which means that frictional force should point in the opposite direction to the direction of motion. Could someone give a QUALITATIVE explanation of the seeming paradox? I would also ask to use JUST the &quot;language&quot; of forces (not to apply the concept of the torque since it makes the explanation less vivid).
Consider this situation:  A ball is moving forward and undergoing rotation. Assume that it is not slipping. Eventually, the velocity and rate of rotation of the ball decrease, and it comes to a halt. But if you observe the direction of friction (when the ball is rotating clockwise), you will see that the friction should have provided a clockwise torque to the ball and the angular velocity of the body should have increased. But this doesn't happen. Why?
Is colour, as represented using primary colours, accurate only to humans? Slightly biological, hopefully physical enough to be answered.  Suppose a magenta hue is represented by a mix of red and blue pigment. This is all very well for a creature with red and blue photoreceptors, but suppose it was seen by a creature which had a magenta-sensitive receptor, but no red or blue. Would the colour appear the same (insomuch as a qualititave concept can appear the same, I suppose I mean 'Would they interpret it as colour of the same wavelength?')?  The crux of my question is, do the particular bands in which photoreceptors are activated affect vision of additive, as opposed to pure, hues?  Finally, completely off-topic, but if there happens to be a biologist around, do animals on the whole have similar photoreceptors, or are they placed largely randomly?  Thanks, Wyatt Where can I get the original Unix? I'm new at Linux and I just download Ubuntu 12.04 LTS.  I found a spot that showed me how join my Windows Active Directory.  It has me add couple lines into the 'hosts' but I'm not able to edit the file because I do not have the permission.  I'm not the owner.  I use chmod with it rwx and still it will not let edit the file.  How do you change the permission?  Thanks Bats and rats are mammals. 1 and 2 are positive integers. Oaks and maples are trees. Carrots and cake are food.   I seek a word to describe this, the process of finding a concept that encompasses several given concepts. Is there such a word? Is there a way to transfer money and items from iOS to Android? I've been playing Jetpack Joyride on iOS, and I just got an Android device. Can I move my things, such as money and unlocks, from my iOS device to Android? I have a blender file where a sphere is orbiting a force field. I want it to be more red when it's closer to the center, and more blue when it's further. I already have the math figured out, I just need to know how to get x/y coordinates in nodes.  Here's the file: I am debating whether or not to get the Potato Sack (yes, I realize I've been debating for several weeks now) bundle through steam.  I already own: Defense Grid: The Awakening, Super Meat Boy, and Audiosurf, all of which are part of the sack pack. What happens if I buy the Potato Sack?  Do I get 2nd copies of the three above mentioned games to gift elsewhere?  Is the price of the bundle reduced by the games I already have?  Do I get nothing special for already owning those 3 games? Is smell a bad way of determining whether meat is still good? Compute the limit of $\sqrt{1-a}\sum\limits_{n=0}^{+\infty} a^{n^2}$ when $a\to1^-$ I need some suggestions, hints for the limit when $a \to 1^{-}$ of $$\sqrt{\,1 - a\,}\,\sum_{n = 0}^{\infty}a^{n^{2}}.$$ Let $f:R\rightarrow R$ be a function such that all its successive derivatives exist in all $R$ and also $f(x)f''(x)\leq 0$ everywhere.  If $\alpha$ and $\beta$ be two successive roots of $f(x)=0$.   Then prove that $f'''(x)=0$ for atleast one $\gamma \in(\alpha,\beta)$.  My Attempt:  I began by taking an example $f(x)=(x-1)(2-x)$ and let $x=1$ and $x=2$ be its two successive roots.The statement is trivially true.  Then I took the function $f(x)=e^{-x}(x-1)(2-x)$.The statement is true here also so on and so forth.  But what would be the exact proof I wonder. It appears to be a question of Rolle's Theorem. While studying I read about the principle of underdetermination of scientific theories. I made some researches online but I am more confused than before. I read about the Quine-Duhem holistic thesis, but I do not know if is the same as underdetermination. Finite Summation of Fractional Factorial Series Is this true?  If $A$ and $B$ are sets, then $A=\emptyset $ if and only if $B = A \bigtriangleup B$.  If $A=\emptyset$ then $B=\emptyset$ too?  Could someone help me please? Methods of speeding up a huge DELETE FROM  with no clauses Using SQL Server 2005.  I am performing a huge DELETE FROM  with no where clauses.  It's basically equivalent to a TRUNCATE TABLE statement - except I'm not allowed to use TRUNCATE.  The problem is the table is huge - 10 million rows, and it takes over an hour to complete.  Is there any way of making it faster without:   Using Truncate Disabling or dropping indexes?   The t-log is already on a separate disk.  Any suggestions welcome! Why isn't this sudo mv operation with wildcard working? How to unplug a network cable that's stuck? Gauss's law in a uniform charge distribution extending infinitely in all directions Let us assume the universe filled with positive charge. About a particular point, all the positive charged particles will be symmetrical. Now consider a sphere of radius $r &lt; \infty$ and apply Gauss law. According to Gauss law the electric flux through the sphere will be some finite value as it contains some positive charge. But by symmetry the electric field through the sphere and hence electric flux will be zero. This is a contradiction. How to log commands within a "sudo su -"? If I:   [user@notebook ~] sudo echo 123456uu 123456uu [user@notebook ~]    Then I can see that in the logs:  [root@notebook /var/log] grep 123456uu * auth.log:Jan  9 17:01:51 notebook sudo: user : TTY=pts/3 ; PWD=/home/user ; USER=root ; COMMAND=/bin/echo 123456uu [root@notebook /var/log]    but if I:   [user@notebook ~] sudo su - [root@notebook ~] echo 1234567zz 1234567zz [root@notebook ~]    I cannot see it in the logs:   [root@notebook /var/log] grep 1234567zz * [root@notebook /var/log] echo $? 1 [root@notebook /var/log]    My question: How can I turn the logging on for the commands within the "sudo su -"?  OS is an Ubuntu 12.04 but the question is in general.   UPDATE#1:   [user@notebook ~] sudo su - [sudo] password for user:  [root@notebook ~] echo zizizi zizizi [root@notebook ~] cd /var/log [root@notebook /var/log] grep -iIR 'zizizi' * [root@notebook /var/log] grep 'COMMAND=/bin/su -' * auth.log:Jan 10 15:42:42 notebook sudo: user : TTY=pts/1 ; PWD=/home/user ; USER=root ; COMMAND=/bin/su - [root@notebook /var/log] Function to determine if two numbers are nearly equal when rounded to n significant decimal digits I have been asked to test a library provided by a 3rd party. The library is known to be accurate to n significant figures. Any less-significant errors can safely be ignored. I want to write a function to help me compare the results:  def nearlyequal( a, b, sigfig=5 ):   The purpose of this function is to determine if two floating-point numbers (a and b) are approximately equal. The function will return True if a==b (exact match) or if a and b have the same value when rounded to sigfig significant-figures when written in decimal.   Can anybody suggest a good implementation? I've written a mini unit-test. Unless you can see a bug in my tests then a good implementation should pass the following:  assert nearlyequal(1, 1, 5)  assert nearlyequal(1.0, 1.0, 5)  assert nearlyequal(1.0, 1.0, 5)  assert nearlyequal(-1e-9, 1e-9, 5)  assert nearlyequal(1e9, 1e9 + 1 , 5)  assert not nearlyequal( 1e4, 1e4 + 1, 5)  assert nearlyequal( 0.0, 1e-15, 5 )  assert not nearlyequal( 0.0, 1e-4, 6 )    Additional notes:   Values a and b might be of type int, float or numpy.float64. Values a and b will always be of the same type. It's vital that conversion does not introduce additional error into the function. Lets keep this numerical, so functions that convert to strings or use non-mathematical tricks are not ideal. This program will be audited by somebody who is a mathematician who will want to be able to prove that the function does what it is supposed to do. Speed... I've got to compare a lot of numbers so the faster the better. I've got numpy, scipy and the standard-library. Anything else will be hard for me to get, especially for such a small part of the project. R: loading a table with number as the column heading all were wrongly added a X, how to solve it?