I'm very new to physic and I'm watching this physic lecture:   at the 22:05 he talks about dimensional analysis.  there is an apple and we drop it from height h and we measure how long it takes to hit the ground which we call it t  then he says that $$ t \propto h^\alpha m^\beta g^ \gamma $$ then he talks about dimension $$[T]^1 = [L]^\alpha [M]^\beta \frac{[L]^ \gamma}{[T]^ {2\gamma}}  $$ which if i understand correctly is the unit of the above statement. what i do not understand is that if my height is 3 meters then i square my height  $3^2$ = 6 meters for example the unit is the same so why the alpha in the first expression goes into the unit expression(the second expression)?  the first expression is kind of an equation of the time it takes for apple drop from height h to hit the ground at time t. for example if I square my height the function just return the $f(x^2)$ to me with the same unit(the unit will never change no matter what input I give to a function) the value that the function return will change not the unit.  I found a question about this story too  but this question do not ask the same question as I'm asking. Im asking about why you treat the unit statement as a function statement?  I mean in mathematic if you have a function f(x) and you square your x and put it into f(x) you get the same unit. The unit is not a function and the function is not a unit its a different story.  I feels like he transform a function into unit then calculate, then plug those units back into a function which seems invalid in mathematic.  Im very new to physic and I found physic world is so weird. can someone explain to me in very step by step and easy to understand with an example. Thank you.
Dimensional analysis, and the notion that quantities with different units cannot be equal, is often used to justify very specific arguments, for example, you might use it to argue that a particular formula cannot possibly be the correct expression for a particular quantity. The usual approach to teaching this is to go "well kids, you can't add apples and oranges!" and then assume that the student will just find it obvious that you can't add meters and seconds.  I'm sorry, but... I don't. I'm not convinced. $5$ meters plus $10$ seconds is $15$! Screw your rules! What are the units? I don't know, I actually don't understand what that question means.  I'm specifically not convinced when this sort of thing is used to prove that certain formulae can't possibly be right. Maybe the speed of a comet is given by its period multiplied by its mass. Why not? It's a perfectly meaningful operation - just measure the quantities, multiply them, and I claim that the number you get will always equal the current speed of the comet. I don't see how "but it doesn't make sense to say mass times time is equal to distance divided by time" can be a valid counterargument, particularly because I don't really know what "mass times time" is, but that's a different issue.  If it's relevant, I'm a math student and know extremely little about physics.
I'm playing around, trying to write some code to use the  APIs to shorten a URL.  After reading , I tried:     TRIM_API_URL = 'http://api.tr.im/api'    auth_handler = urllib2.HTTPBasicAuthHandler()    auth_handler.add_password(realm='tr.im',                              uri=TRIM_API_URL,                              user=USERNAME,                              passwd=PASSWORD)    opener = urllib2.build_opener(auth_handler)    urllib2.install_opener(opener)    response = urllib2.urlopen('%s/trim_simple?url=%s'                               % (TRIM_API_URL, url_to_trim))    url = response.read().strip()   response.code is 200 (I think it should be 202).  url is valid, but the basic HTTP authentication doesn't seem to have worked, because the shortened URL isn't in my list of URLs (at ).  After reading  I also tried:     TRIM_API_URL = 'api.tr.im/api'    password_mgr = urllib2.HTTPPasswordMgrWithDefaultRealm()    password_mgr.add_password(None, TRIM_API_URL, USERNAME, PASSWORD)    auth_handler = urllib2.HTTPBasicAuthHandler(password_mgr)    opener = urllib2.build_opener(auth_handler)    urllib2.install_opener(opener)    response = urllib2.urlopen('http://%s/trim_simple?url=%s'                               % (TRIM_API_URL, url_to_trim))    url = response.read().strip()   But I get the same results.  (response.code is 200 and url is valid, but not recorded in my account at .)  If I use query string parameters instead of basic HTTP authentication, like this:     TRIM_API_URL = 'http://api.tr.im/api'    response = urllib2.urlopen('%s/trim_simple?url=%s&amp;username=%s&amp;password=%s'                               % (TRIM_API_URL,                                  url_to_trim,                                  USERNAME,                                  PASSWORD))    url = response.read().strip()   ...then not only is url valid but it's recorded in my tr.im account. (Though response.code is still 200.)  There must be something wrong with my code though (and not tr.im's API), because  $ curl -u yacitus:xxxx http://api.tr.im/api/trim_url.json?url=http://www.google.co.uk   ...returns:  {"trimpath":"hfhb","reference":"nH45bftZDWOX0QpVojeDbOvPDnaRaJ","trimmed":"11\/03\/2009","destination":"http:\/\/www.google.co.uk\/","trim_path":"hfhb","domain":"google.co.uk","url":"http:\/\/tr.im\/hfhb","visits":0,"status":{"result":"OK","code":"200","message":"tr.im URL Added."},"date_time":"2009-03-11T10:15:35-04:00"}   ...and the URL does appear in my list of URLs on .  And if I run:  $ curl -u yacitus:xxxx http://api.tr.im/api/trim_url.json?url=http://www.google.co.uk   ...again, I get:  {"trimpath":"hfhb","reference":"nH45bftZDWOX0QpVojeDbOvPDnaRaJ","trimmed":"11\/03\/2009","destination":"http:\/\/www.google.co.uk\/","trim_path":"hfhb","domain":"google.co.uk","url":"http:\/\/tr.im\/hfhb","visits":0,"status":{"result":"OK","code":"201","message":"tr.im URL Already Created [yacitus]."},"date_time":"2009-03-11T10:15:35-04:00"}   Note code is 201, and message is "tr.im URL Already Created [yacitus]."  I must not be doing the basic HTTP authentication correctly (in either attempt).  Can you spot my problem?  Perhaps I should look and see what's being sent over the wire?  I've never done that before.  Are there Python APIs I can use (perhaps in pdb)?  Or is there another tool (preferably for Mac OS X) I can use? How to create a custom Linux distro that runs just one program and nothing else? How do I return the response from an asynchronous call? 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Is acceleration due to gravity constant? why each left ideal of this is minimal? For any prime $p > 3$, why is $p^2-1$ always divisible by 24? Let $p&gt;3$ be a prime. Prove that $24 \mid p^2-1$.   I know this is very basic and old hat to many, but I love this question and I am interested in seeing whether there are any proofs beyond the two I already know. QGIS 2.0 - can't select items in Print Composer the cop (who's) escorting him One of the cops escorts Tim out of the house and toward a police car. Curious neighbors have gathered outside. Tim spits on the cop (who's) escorting him.   Is &quot;escorts&quot; the more natural choice of verb to use in this context?  Is the emboldened sentence phrased naturally without adding &quot;who's&quot;? This question follow's JeffE's comment :     In most industries, a PhD has negative value.   I am curious about what potential negatives there may be with having a PhD, particularly in industries.  For example, my PhD has applications for image sensors, and the potential of working in that field is appealing—would having a PhD be a potential barrier? Would it be a case of how explicit the practicalities of the PhD can be made?  Would this make any difference? Find all integer solutions to $x^2+y^2+z^2=2xyz$ Does a photon instantaneously gain $c$ speed when emitted from an electron? Animated movie that had black robots that turned people into stone late 1960s to 1970s It was a weird sci-fi animated movie I saw only a little bit of it. This is what I recall:   It was old but had color, so I assume it was late 1960s to early 1970s. The animation and style reminded me of  but it wasn't as psychedelic. The setting was an alien planet with all sorts of bizarre creatures. The tech was advanced but not a dominant aspect of the planet's culture. The people lived simple and peaceful. The protagonist is a man with bluish skin like the others. The antagonists were humanoid shaped black robots that shot a beam that would turn people into stone. Recurrence relation for number of ternary strings that contain two consecutive zeros The question is: Find a recurrence relation for number of ternary strings of length n that contain two consecutive zeros.    I know for ternary strings with length one, there are 0. For a length of 2, there is just 1 (00), and for a length of 3, there are 5 (000,001,002,100,200).    I did a similar problem, finding a relation for the number of bit strings of length n with two consecutive zeros: $$a_n = a_{n-1} + a_{n-2} + 2^{n-2}$$    Since you can add "1" to the end of all the $a_{n-1}$ strings, "10" to all the $a_{n-2}$ strings, and "00" any string of size $n-2$.    For the ternary string problem, I'm pretty sure you would replace the $2^{n-2}$ with $3^{n-2}$, but confused about the other terms of the relation. 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