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    Falling Faster Than Freefall: A Lesson In Didactics And Critical Thinking
    By Sascha Vongehr | January 16th 2011 07:46 PM | 69 comments | Print | E-mail | Track Comments
    About Sascha

    Dr. Sascha Vongehr [风洒沙] studied phil/math/chem/phys in Germany, obtained a BSc in theoretical physics (electro-mag) & MSc (stringtheory)...

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    Remember the neat little experiment whose result may be counter intuitive to some of those who embrace the “all bodies fall the same way inside earth’s gravity” ‘doctrine’ without quite understanding it? Here it is again - if the video should not come up fast enough, here it is on the MIT website.

    It is a simple set-up and great idea for a project involving children. Older ones may be able to calculate the required lengths of the wooden angle's arms and the required position of the cup.


    I challenged readers to come up with a short and philosophical reason for why the end of the arm reaches the table faster than the free falling ball although much of the force onto the wooden plank just presses on the hinge without being able to accelerate it.


    ‘Philosophical’ means: Get to the vital core and convince with fundamental insight – no formulas.


    The being faster is necessary for the ball to separate from the golf tee and land in the cup. But the cup is not important – it is a gimmick much like the guy wearing a lab coat. Important is a deeper understanding of the physics involved:


    In old ages long ago, people thought that a feather does not fall as fast as a cannon ball. Nowadays, people know that the feather only falls slower due to air resistance. This in turn has become somewhat of a 'doctrine' in the sense that many who 'know' it do actually not really understand the underlying reasons and thus apply the doctrine where it does not apply. They are surprised by the experiment in the linked video, because the wooden plank’s tip experiences a downwards acceleration larger than if in free fall! (Look at the golf tee where it supports the ball: They separate!)


    I would like to thank all the readers for their willingness to participate and stick their neck out. I learned a lot from the comments. Especially: I saw how widespread this misconception is that mathematics is necessary to understand deeper connections. People seem to be ready to be duped into accepting something as correct because it is full of formulas. It goes so far that even those who are well on the way to correct answers do not trust their own abilities merely because they have not added equations.


    Most important lesson today: Mathematics is a language. You use it to communicate content. If the content is not correct, the fact that it is expressed in form of formulas only makes it worse!


    I often experienced students or colleagues telling me something impossible and when I immediately object they are annoyed if I don’t even look at their calculation. Why should I? It is wrong. Why spend an hour trying to figure out where the formulas are not applicable? Life is too short.


    Do never trust people who try to tell you that you just do not understand their argument because your mathematics isn’t good enough. It is actually only people who do not understand their own field properly who ever do that.


    Anyways – John Marretada came closest. He did not have the confidence to go all the way, but he had three important ideas.


    I do like his considering the mass to be close to the hinge (1). Whatever the length of the plank, whatever the starting angle (as long as it is below 90 degrees), if all the mass is close enough to the hinge, the massless tip can reach any velocity, even light velocity and beyond.


    Another fact holding true regardless of lengths, cups, masses, whatever, is that if all the mass is in the plank’s tip and the starting angle is 0 degree, the ball’s acceleration and the tip’s initial acceleration will be exactly identical (2) (because both are then free falling bodies, at least for an infinitesimal short time). This second one is all that John was missing.


    I like his total avoidance of equations. What I did not like about his argumentation is that it too soon and too often mentions aspects that are (a) not quite equal or should be different neglecting this or that and (b) dependent on going a certain fraction (of length or angle) here or there. Even if it is all true, such is never as convincing as stuff that is obviously (a) identical especially if it is so moreover in (b) extreme situations. That’s why (1) and (2) above are didactically superior:

    (1) In the extreme limit of all mass at the hinge, the tip's acceleration is identical to infinity (which is yet another extreme that overshadows all doubts about something maybe not being quite enough to do the trick).

    (2) In the extreme position of all mass being in the tip and the starting angle being 0 degree, the accelerations are identical.


    John’s further idea is also very good, namely considering to take some of the mass from the tip into the plank. In other words, ‘smear’ the mass along the plank. All mass away from the tip also ‘wants’ to fall as freely as the tip, too, but cannot because it is connected to the tip, and so it must drag the tip along. Thus, the tip obtains additional acceleration. Given the extreme position (2) implies that the accelerations are identical already. Hence, the additional acceleration makes the tip’s acceleration larger than that of the ball. Now we have already proven the main point: the tip can be faster than in free fall.


    Somebody else may prefer a more mathematical "the smearing of the mass leads the acceleration monotonically from extreme limit (2) into the direction of extreme limit (1)" without mentioning physical forces at all. I consider this a matter of taste and both versions should be pointed out.


    Again: We have proven that the homogeneously-smeared-out-mass plank is faster. Only problem left is that our starting angle is still zero degrees! Note: the plank’s acceleration is identical to zero at the extreme of 90 degrees and increasing the starting angle approaches that extreme monotonically: So if you want to prove that the tip is faster also at a finite angle above zero degrees, you need to make convincingly sure that the added acceleration from smearing the mass along the plank is not overcompensated by the slowing down due to the starting angle being larger than zero degrees.


    So again: Only problem left is that our starting angle is still zero degrees! But we can change the angle because the tip’s acceleration is larger than the ball's acceleration by a finite (i.e. not just infinitesimal) amount. Therefore, we must increase the starting angle by a finite amount if we wanted to compensate for the additional acceleration. So let us compensate in this way, increasing the starting angle until the acceleration of ball and tip are identical again (the ball is of course always experiencing the same acceleration regardless of where it starts).


    Now we are basically finished because after compensating, we have a finite angle at which the tip and ball have identical accelerations (always try to stick to identical if at all possible). Take half that compensating starting angle instead and the tip will be again accelerated more than the ball but at a finite angle (> zero degrees). That is it: QED.


    How large is that angle? Is it close to 35 degrees? Where does the cup go? I do not care. It is not important for answering the question posed. Those details probably need calculations with formulas, but that is exactly why they are not interesting.


    UPDATE: The short version was provided by Colin in his comment below.

    Comments

    colinkeenan
    But, hasn't the author already provided the explanation?  He has. 

    Ok, I will work on copying and pasting just the answer, removing any extraneous remarks, and post it, but it won't be my answer, it is Sascha's based on the other guy's answer that I haven't read.
    vongehr
    Straw-man: “But Sascha’s argument is wrong. At zero starting angle, the tip does not know that it is connected to the center of the plank. At t = 0, tip and center are equal; there is no difference between them being connected or not. There are no forces that could pull vertically along a strictly horizontal path (the plank at zero degrees). The instantaneous acceleration a is due to the actual force, F = m a, thus all parts also of the plank with smeared out mass fall exactly with a = F/m = g at an starting angle of zero degrees.”
    colinkeenan
    Good point, straw-man. 

    I guess I was always thinking about the plank being almost but not exactly at 0. 

    In my comments added to Sascha's argument, I said it would help to imagine the table is not there so that the plank could fall below 0 degrees.  Taking that argument further and starting at an angle of -90 degrees, the acceleration is once again 0.  That would suggest the acceleration at the tip becomes less than g somewhere between 0 and -90 and that the maximum acceleration is at 0 degrees.  It's clear that the acceleration must be greater than g at any angle almost 0, and so the maximum acceleration is greater than g at an angle of 0, but I can't immediately figure out what's wrong with straw-man's point.
    colinkeenan
    OK,

    After sleeping on it, I think I have an answer for straw-man.  The tip always knows it is connected to the middle regardless of angle and always feels force applied to the middle.  To be convincing, a thought experiment will be needed. 

    Imagine the plank is attached to the wall by the hinge and there's no table or that the table has a hole in it so that the plank can fall past 0 degrees as I said in my last post. 

    Now imagine the entire room containing the plank is moved into free-fall in space so that gravity no longer accelerates the plank.

    Now push the plank down at the center with a constant force.  Your hand is not even touching the end of the plank, so if it were true the tip did not know it was connected to the center at 0 degrees, the tip would not move.

    Straw-man, you know from experience, regardless of angle, except for the end attached to the hinge, the entire plank will move including the tip, so your assumption that the tip does not know it is attached to the center at 0 degrees was mistaken.  The tip will always feel the force applied to any part of the plank because the plank is solid.
    colinkeenan
    And now we know why the acceleration is maximum at 0 degrees.

    The tip always feels force applied to any part of the plank regardless of angle because the plank is solid at all angles.  The net force is greatest at all points of the plank when the angle is 0 because the hinge can't push back against the weight at that angle.
    colinkeenan
    Comments in brackets are my own.  I also paraphrased one part of the proof and left a couple of words out of the conclusion.  Here's Sascha's proof that there must be a starting angle greater than 0 where the end of the plank will accelerate more than the ball experiencing free fall:

    In the extreme position of all mass [of the plank] being in the tip and the starting angle being 0 degree, the accelerations [of the tip and the ball] are identical. [May help to imagine the table is not there so the plank and ball are free to fall below 0 degrees]

    ...‘smear’ the mass along the plank.  All mass away from the tip also ‘wants’ to fall as freely as the tip, too, but cannot because it is connected to the tip, and so it must drag the tip along. [If you imagine the plank starting to fall below 0 degrees it's easy to see the rest of the plank is confined to go slower than the tip even thought it 'wants' to fall just as fast due to gravity.]

    Hence, the additional acceleration [from the plank's mass closer to the hinge dragging the tip along as it falls] makes the tip’s acceleration larger than that of the ball. Now we have already proven the main point: the tip can be faster than in free fall.

    Only problem left is that our starting angle is still zero degrees!

    Note: the plank’s acceleration is identical to zero at the extreme of 90 degrees... [It's easy to imagine balancing the plank straight up so that it doesn't move at all - that's 90 degrees]

    I will paraphrase the next part of Sascha's proof:  Since the acceleration of the plank's tip is greater than the ball's acceleration at 0 degrees and the acceleration of the plank is nothing at 90 degrees, there must be some angle between 0 and 90 where the plank's acceleration is the same as the ball's acceleration.

    Back to Sascha's proof:  Now we are basically finished... we have a finite angle at which the tip and ball have identical accelerations.  Take half that ... starting angle instead and the tip will be again accelerated more than the ball but at a finite angle(> zero degrees). That is it: QED.
    colinkeenan
    When I first watched the video, I was also saying to myself, what's the big deal?  The ball started higher than the bottom of the plank and higher than the cup it lands in so of course it's able to fall into the cup.  Maybe the video isn't as convincing as it could be, but it is convincing nonetheless.  Watch it again and pay attention to the vertical distance between the ball and the tee at the end.  Also pay attention to the distance between the ball and the top of the cup at the start and end.  The ball is obviously experiencing less acceleration downward than the tee and cup.  And finally, as Sascha pointed out many times, if the tee would fall with the same acceleration as the ball, it would have pulled the ball to the side a little.  The ball falls straight down, taking the shortest possible path to the table.
    colinkeenan
    The 'surprising' thing about the experiment is that the tee/plank accelerate faster than the ball and we are trained to think everything falls with the same acceleration at the surface of the earth neglecting air resistance.

    The total distance traveled isn't relevant - the fact that the tee travels in an arc while the ball goes straight however makes it obvious the tee also travels further than the ball.

    Without air, a ball dropped straight down will hit the ground at the same time as a ball shot horizontally from the same height.  After shooting the ball horizontally, it experiences no other horizontal acceleration but keeps moving forward at a constant speed due to inertia.  Gravity accelerates it towards the ground with the same acceleration as the ball that was not shot - the acceleration due to gravity. 

    It is because of learning about things like this that some people can be surprised the tee at the end of the plank accelerates faster than the ball, which it clearly does, because when the tee stops moving, the ball is much higher than the top of the tee, but the top of the tee started at the same height as the ball.

    Sascha suggests you try some experiments yourself.  If you start at an angle of 85 degrees, as he suggests, you will clearly find the plank accelerating very slowly compared to the ball which will hit the ground long before the plank.  The point is that the plank is not in free-fall as the ball is and as already argued the tip can experience more or less acceleration than that due to gravity depending on the starting angle.

    If there is some part of the proof that you don't understand or disagree with, you should post your disagreement here.  If your problem is with the experiment, you should perform some of your own until you are convinced that the plank does not accelerate the same as a freely falling ball.
    Consider the inverse construct:
    "Gravity" pushing up on the plank and the hinge and the fook'n ball.

    Remove gravity and replace with induced force. Would this have any effect on the affected components?
    I don't have a lab, so I'm just askin~

    colinkeenan
    Not sure what you mean by induced force.  Nothing relevant in Wikipedia but from Google found this link to a book from the 70's:

    http://www.springerlink.com/content/k8438521t4408384/

    The book is titled "Induced forces in the gravitational field".  I didn't try reading it but at first glance it seems to be starting from a relativistic description of gravity to derive induced force on a particle.  None of that is necessary for our analysis since we already know results from Relativity agree with Newtonian gravity at the surface of the Earth.  If it didn't, Relativity as a description of gravity would've never been accepted.  All of the analysis done so far is using a Newtonian understanding of gravity because it's the easiest to apply and is correct for the situation being analyzed.
    colinkeenan
    I get what you're saying now.  Remove gravity and slam the table up into the ball and plank!  That's great because it greatly simplifies the situation while staying completely equivalent to the gravity situation.

    Now, without gravity, we can more easily see what's important, which is the hinge of course.  The ball and plank can be positioned any way we want without gravity, so put the ball floating off to the left of the tip of the plank.  When the table is accelerated up into the ball and plank, the ball experiences no force at all before hitting the table and never moves.  The only force on the plank (as a whole) is at the hinge.  That's so much easier to understand.

    Imagine that instead of a plank we had very soft and wet mud shaped like a plank.  As the table slammed up into the mud, the mud would flatten along the table leaving the tip unmoved next to the ball.  But the plank is rigid and can't flatten along the table, so the tip can't stay unmoved with the ball!  So, what happens?

    The upward force from the table being slammed up pushes on the plank where it meets the table at the hinge.  Part of that force is aligned with the plank, pushing it up with the table, and part of that force is at right angles to the plank, causing the plank to spin.  If the starting angle is too close to 90 degrees, most of the force from the hinge is aligned with the plank and less is at right angles causing the plank to spin very slowly at first as it is quickly lifted up past the ball; the ball hits the table first.  But, if the angle is low enough, most of the force from the hinge is at right angles to the plank, causing it to spin down very quickly, possibly beating the ball down to the table.

    Now the problem is reduced to showing that it is possible to have an angle greater than 0 that will cause the tip to spin down to the table before it is lifted above the ball.
    vongehr
    If I were the mean guy people make me out to be, I would scream "equivalence principle!" and tell you that it is only making everything more complicated (i.e. the table would have to be slammed in a specific way, namely with constant acceleration). However, the approach as such, i.e. no formulas, the originality (mud-plank?!?!), and especially the attempt to apply symmetry to the problem (here equivalence between gravity and acceleration) make this idea exactly the kind of thing that I was looking for. I wonder of how much schools can teach such methodology - I get more and more the feeling that it is few who are able to think this way. Reminds me of a school teacher trying to introduce sin and cos. He build up the tension like a good teacher should, then asked his key question about a triangle, to let us sweat for a while in the difficulty of it all, but I immediately answered "hb=cd so c=hb/d because the area A=hb/2 is the same" instead of even fucking around with angles (he should have used a less symmetric triangle). He looked at me stunned for like what felt forever, seeing his whole lecture, the teachable moment he was after and had prepared so well, falling apart in front of his inner eye. But then, after he explained it to the class, nobody else appreciated the difference.
    I can't believe how rude some people are here. Particularly you, Sasha. To be honest your 'identical' and 'extreme' arguments aren't even that good.

    Here's my short reasoning:

    The tip accelerates faster than free-fall because it is being pull down by gravity, and by part of the plank. The centre of mass wants to fall at g, but it can't. It's partially held up by the hinge. It's also partially held up by the inertia of the plank tip (for an example of this, find a video of a chimney collapsing - it breaks near the middle, or video a tipping stack of blocks). If the centre is pulled up by inertial forces from the tip, switch reference frames and then the tip is pulled down from the centre. They trade forces and therefore acceleration.

    Really what's going on is the centre of mass of the plank falls half the distance and a lot slower than the ball. Shear within the beam pulls and pushes each slice so that it remains a solid body.

    Hope that helps, Helen.

    vongehr
    Oh yea I am sooo rude. Where am I rude? Because I say you are wrong?
    "You are wrong" is not rude, it is science. If you cannot stand the heat, get out of the kitchen.
    The tip accelerates faster than free-fall because it is being pull down by gravity, and by part of the plank. The centre of mass wants to fall at g, but it can't. It's partially held up by the hinge.
    Wrong! You say yourself that the tip AND the hinge hold back the middle, so how can you prove that the middle still overcomes both and accelerates the tip faster than freefall? It does not at a larger angle. How you know there is a smaller angle? Exactly, use master Sascha's proof.
    Samshive
    Hi Everyone,
    I think perhaps things are getting a little too heated here. Everyone should just relax a little bit. First of all - Sascha's explanation does explain the effect adequately (certainly more adequately than my attempted explanations). That being said, I think a lot of the discontent is with the relatively harsh way Sascha (for lack of a better word) "debunks" the attempts.

    Helen, the tee is relevant for the ball to fall into the cup. However, the tee is not relevant to explain why the board falls faster. The height of the tee as probably adjusted using a trial and error in conjunction with the position of the cup. The reason why the tee is not relevant to the effect is because the ball and tee (connected to the board) separated immediately when the board was released. This directly implies that the tee (hence the board) experienced a larger vertical acceleration than the ball - and although Sascha's wording probably left a lot to the imagination, this was the effect that he was asking us to explain. 

    Steve, as to the idea that the tee and ball will separate immediately even if the vertical accelerations were equal is probably valid if you disregard any and all other physical effects that the ball would experience because of the tee (For such effects to be discarded, you would probably need a tee with a pin like point where the ball would rest on). In any case, this is again not directly relevant. Note that As the angle of the plank decreases, the component of the gravitational force acting to vertically accelerate the plank increases to a maximum at 0 degrees. In other words, the vertical acceleration of the plank increases from the moment it is released (regardless of angle). So even if you do the experiment in such a way that the instantaneous accelerations of the plank (tee) and ball are identical, the plank will still fall away faster because its acceleration increases while the ball's is constant. 

    Finally, Sascha, I think perhaps you should tone down the sarcasm a bit. And as to your strawman above, all I can say is that levers are wonderful things. 

    As an afterthought - I think this whole discussion (including the original post) demonstrates both the good and the bad of science quite well. All the way from phenomenon, to competing hypotheses, to ardent support of one's ideas, and of course the hostilities that come with the territory. 

    John Marretada

    I posted some attempt to explain and maybe it was missed among so many comments :)

    http://www.science20.com/comments/57072/I_will_Try_1_plank

    Will try a short(??) explanation here:

    1) The plank makes all its points arrive on the table at the same time (it is supposedly rigid)

    2) A given point is slowed down when compared to a point to its left (closer to the tip) because it is obviously traveling less distance in the same time (Doesn´t matter how long a plank takes to fall).

    3) A point in the plank almost on the hinge falling at free fall speed  would make the tip of the plank fall at an impossible speed and so there are points near hinge slowed down when compared to a free fall object.

    4) There must be a point that is falling at least the same average speed of a free falling object. It happens because is impossible that ALL POINTS are falling SLOWER than a free falling object because it would mean that potential energy is disappearing. (Or that the plank wouldn't fall at free falling average speed if detached from the hinge and lifted and then realead)

    (obviously I'm ignoring  dissipation (friction, air resistance, hinge friction, etc) and a plank too light with too much surface could fall slower than the ball but it is not the case here as seen in the experiment)

    5) The tip cant be the point that has the same average speed of a free falling object. It is because all points to its right (closer to the hinge) would be SLOWER than a free falling object (these points would be slowed down and no point speed up meaning energy disappeared!). So the tip is to the left of this point ( that falling at the same average speed of a free falling object) and therefore the tip is falling FASTER than a free falling object!

    6) The tip of the plank (and therefore the whole plank)arrives faster than a free falling object and therefore the cup (on the plank) will arrive sooner than the ball (doesn't matter the angle!). It is just a matter of choosing an angle that gives the desired effect (as the cup will be there wating for the ball :).

    If an angle too acute the ball would not enter the cup (because the cup's edge). The angle (and tee and cup position) may be chosen that the cup appears to fall at the same speed of the ball(maybe an interesting option :) .

    Very short(?) version:

    All points of the plank (and any free falling object) are subjected to the same gravity force (acceleration).

    Putting a hinge cant create or destroy these forces it only redistribute them.

     So (in a non dissipative system) there must be at least one point X  falling at a free falling (average) speed (if ALL points are slower than free falling then some these forces would be disappearing).

    Given that point X, the hinge makes all points to its right (closer to the hinge) fall slower than free falling speed (less distance in the same time of a free falling object) and therefore (this lack of acceleration)  must be compensated by points more accelerated to  the left of point X (the hinge makes these points travel a greater distance in the same time of free falling object).

    As the tip is to the left of point X and is faster then it is faster than a free falling object( again travel a greater distance in the same time of free falling object)

    Now is just a matter of  choose the angle, ball and cup position to give desired effect.

    I hope it was useful!

    colinkeenan
    "6) The tip of the plank (and
    therefore the whole plank)arrives faster than a free falling object and
    therefore the cup (on the plank) will arrive sooner than the ball (doesnt
    matter the angle!)."

    Oohps, you made a mistake somewhere.  If you try the experiment yourself and read through the blog again you'll realize the angle does matter.  Although the tip of the plank eventually experiences acceleration greater than gravity, it can start off at such a low acceleration that the ball hits before the plank ever reaches an acceleration as big as gravity.  Just try it - at angles close to 90, the stick falls much too slowly at first to reach the ground before something free-falling from the same height.
    Hang on. It's all very well avoiding maths and dismissing irrelevent perturbations but you do have to allow for quantum gravity effects. These become very significant at...

    ... wait for it ...

    ... the plank length!

    Its all about perspective people. You are the ball. You and your buddy, the plank, are free falling. There is no relative motion between you. At some point one end of the plank is restricted. What do you see? One end moves up, the other moves down.

    The key(devil) is in the wording. Any body can be be reduced to a point of the same mass at it's center of gravity to analyze its motion in a gravitational field. No one ever said that every point in a body will experience the same acceleration, especially under the influence of additional forces like tables.

    Stick with the math people :)

    blue-green
    Hoo Boy, I didn’t know this discussion was still alive. I have a mental picture of an apparatus that I think can help one get to a very compact explanation. I hope you like it Helen. The “plank” is viewed edgewise as being extremely thin, yet perfectly stiff. Reduce it to very thin and rigid pole or rod with all of its mass concentrated on a Bead that can be slid to different positions along the rod. If the bead is set at the bottom, very close to the hinge, then I think one can imagine how the pole will swing down very fast. It would be a bit like setting the weight of a Metronome all the way down at its fastest setting. Surely some of you tried this while enduring piano lessons. If the bead is instead shifted all the way up to the top end, then the end of pole (rod, plank) is going to mimic free-fall (with a minor correction for friction and hesitation in the hinge). This is the slowest setting … like putting a metronome’s weight at the very top … where it will lazily swing back and forth. The crux is the center of mass and where it is located. Starting angle would be the next most significant factor.. Steve focused on this right away. (In the ideal limit, the rod is mathematically thin and there is nothing material at the far end, so it can move faster than c without violating Special Relativity. Another example of this would be the moving point at the intersection of two long knives crossing each other like scissors. At a certain distance away from the pivot point, the mathematical cutting point of intersection is racing away from the pivot point faster than the speed of light. More examples like this are in the 1964 Spacetime Physics book by Taylor and Wheeler.)
    vongehr
    Hey - the blue-green kitty is back. I like your metronome picture, though it is a little misleading. What drives the metronome is a big weight below which does not change position while the bead is only slowing it down. What drives the plank is the bead you invoke itself, so the driving force also changes if you displace the bead.
    John Marretada

    The simplest explanation I could:


    It is an ideal model but should explain the effect



    1) I suppose the system is not dissipative ( no friction, air resitance, hinge friction, etc))


    2) Different points of the plank start at DIFFERENT HEIGHTS


    3) Plank makes ALL  its points arrive at the SAME TIME on the table


    4) It is impossible that ALL points of the plank are FASTER than a free falling object starting at the SAME height. (This would mean that forces are coming out of from nothing!)


    5) It is impossible that ALL points of the plank are SLOWER than a free falling object starting at the SAME height. (This would mean that the system is dissipating and contradicting with 1)!)


    6 ) From 5) we infer that some points of the plank are slower than free falling (from the same height) and from 4) we infer that some points are faster than free falling (from the same height). (They cant all be falling at the free falling speed because they cover different distances in same time)


    7) The fastest point of the plank is the tip that is traveling the greater distance in the same time and from 6) we infer that it is FASTER than free falling (from the same height)





    It is very hard to explain the effect of the video without using an ideal model. For instance if the plank is very light and have a  sufficient great surface it will be slower than a free falling object (think of a high surface cardboard plank)! Would this experiment prove that the tip cant fall faster than free fall?? NO !!! It would show that there are factors that can undermine the effect!


    If we stick to consider all factors we would have to consider air resistance, hinge friction and even the latitude the experiment was carried!



    So the idea here is not to take everything in account, just the underling principle. If you cant explain the effect (falling faster than free fall) AT LEAST ideally how could we find an explanation for a real case far more complex?


    So I believe we should forget about some real aspects behavior of the plank, the tee, the height of the cup, any kind of dissipation, etc and focus on a simple model (that was what many here have done). Having understanding this we can pass to a more complex case.


    Ladislav Kocbach
    One picture is worth thousand words, and I have about 20 animated. They show snapshots of a simulation of this fascinating little problem. It looks like a video, but there are about 16 different snapshots taken of a running PHUN toy simulator (the engine of it is no a  toy, though, it is a very efficient toy, based on a MSc (not PhD as I wrote before) in computer science / numerical mathematics).

    My little contribution (worth about 16 000 words, but much much shorter) is at our department local webserver (do not be afraid to click here, uib.no means University i Bergen, Norway) -
    http://web.ift.uib.no/AMOS/faster_fall/
    Picture no. 209 (you can stop and type it into the text field) shows the moment when the tip of the plank reaches the table. The end of the plank arrives before the ball, and is ahead a much longer distance than the original length of the golf tee. But the simulation also shows that it can often be a close race. The best advice, before starting perhaps long mathematical or logical analysis, play a little bit with the PHUN ( see PHUN (download), scroll a bit down) and have fun. Everybody can do the simulations, as you can see from the pictures, the sketches are all you need

    The above addres contains links to the original MIT video, PHUN, and ALGODOO. I strongly recommend all the participants, as well as Sascha, to get PHUN for fun and understanding.

    I have already posted this at the other discussion and I have got a very illustrative answer from Sascha there.  Illustrative about the nature of Sasha's approach to didactics. It appears that the best didactical approach is bullying your audience. ( I expect some escalation here. )
    vongehr
    This is bullying???:
    Fun, but I reject the idea that a simulation is better than the real thing. We can just pause or slow down the video and all that your simulation shows can be much better seen in the video without fearing somebody has introduced laws of nature of a parallel universe via 'too clever' coding.
    Especially, I reject your last statement about that these tools is what I need for understanding. I got the solution to the question by using my head, not video or simulation. If video/simulation do not confirm what logic tells me, they will be wrong. This is the attitude to take in the age of photoshop and video manipulation software.
    I severely doubt you know what bullying is. This is not it.
    I think the simple answer is this:

    The center of gravity of the board falls at the approximate speed of freefall.
    Since one end is attached at a fulcrum, and the CG experiences 1G of force, because of leverage the end of the board experiences greater than 1G of force, therefore the end accelerates faster than freefall, while the ball with it's compact CG will only experience 1G.

    vongehr
    The center of gravity of the board falls at the approximate speed of freefall.
    No, it is much slower. Also, at a larger starting angle even the tip is much slower.
    Okay, that does seem reasonable.

    So, I think at the CG of the arm it would be
    cos(angle) x 1G

    and the end would be something like
    cos(angle) x 1G x len/len cg from pivot

    scientists can be their own worst enemies. This experiment rally garnered my interest as a layman. I have a high school education. I am an electrician, and a pilot so i have a rudimentary understanding of physics. I can also observe nature. I thought I was going to get an answer that i could understand and not get put off by scientific pontification. wasnt that the whole point of the thread. make the layman understand it? get me to trust scientists again? I do not fall int the catagory of people put off by mathematical scientific explanations, but it is human nature to not trust what you dont understand, and i dont understand the math, or the terminology. years ago my dad told me a bullet dropped and a bullet fired from a gun would reach the ground at the same time, removing the effects of air resistance and currents...yada yada. everything falls at the same rate he said. I had a hard time understanding it or believing it, so he took me to the museum of science in boston, There was a display there with two clear tubes about 50 feet high. feathers in one tube and a bag of sand in the other. the tube with the feathers was evacuated. the feathers fell at the same rate as the sandbag. Viola! it all came together. simple and elegant. not so philosophical tho. Your target audience is me, the typical american, a little dumb around the edges from a public education in the 60s and 70s. make me understand it and i think youve reached your goal.

    vongehr
    wasnt that the whole point of the thread. make the layman understand it? get me to trust scientists again?
    No, exactly the opposite. The point was to get you to adopt a highly critical attitude, not just to trust anything that nowadays comes under the label "science" - there is too much shit under that label. Among many points that are also relevant, one is that we should not be satisfied with wrong explanations just because they happen to satisfy most laymen. Right explanations often need more effort than most laymen are willing to invest, but then not understanding laymen is still better than people who mistakenly think they understand.
    ok i missed the point of the question. but i disagree (sorry to hijack the thread here) that "not understanding laymen" is still better. It is not. not understanding laymen create a lot of problems. Its polarizing. check out the hundreds of threads on the airplane on a treadmill. Half of people cant see the answer, and they cant be convinced. The not understanding laymen (NULS for short...ha i like it, intonates a void of some type!) dont understand a hole in the ozone layer. the understanding laymen may understand it, or may just believe the math. either way someone is missing the point. critical thinking needs coaching, and clear explanations when the topic is over the receivers head.
    The explanations for the falling plank are over my head, so i am a NUL here. I can clearly see how it is true, and i can peice together bits from posts to get an idea of the science, but im not positive like i am about the feathers.

    vongehr
    I said not understanding laymen is better than mistakenly sure ones. As somebody famous once said, a little bit of knowledge is a dangerous thing, or some such.

    Did you read Colin's explanation of my explanation (see above)? If that does not do it, than maybe you just can't grasp it, which does not make you a bad person. On the contrary, being aware of one's own limitations is one of the most important things to know.
    I read the explanation, and I resent the insult, as badly couched as it was. if it was badly couched on purpose...shame on you. but i still dont have a satisfactory explanation. I do grasp it, but have to accept smearing, which is akin to accepting math i dont understand.
    so let me ask a few questions.
    the ball is irelevant, it is a point of reference.
    everything falls at 1g
    what part of the plank falls at one g? why?
    is the hinge necessary? will a loose plank fall at the same rate?
    can you equate this to a lever some how? weight x arm = moment is easily understandable without complex math

    vongehr
    Sorry, but if you cannot bear that there are things you may not be able to understand and feel insulted if somebody points this out, we cannot communicate. I am a scientist; it is my job to go to the edge of my knowledge and intellectual ability all day and deal with my own limitations all the time. You will have to be able to do so at least once in a while or you are wrong here. My blog is not about rubber balls or wooden planks! It is about critical thinking, criticism against overenthusiastic scientism and naive evaluations of emerging technologies etc.

    (BTW: Smearing is not maths. It is just imagining a handful of mud at the tip of the otherwise massless plank and smearing it along the plank.)
    well then.. shame on me for interpreting it as an insult then, my apologies.. I did aplolgize for hijacking the thread earlier, i know this is all tangetal to your intent. it is painfully clear that we cannot communicate.
    i live by the pilots maxim there are old pilots and bold pilots, but no old bold pilots...know your limitations and adhere to them. so i am well aware of the necessity of admitting them. im vexed by the answer, but its not outside my ability to grasp, so with that i will bang out, and get my answers from mythbusters ;) mike

    John Marretada
    Lets take as a challenge to explain  this subject in a simple way  that most  people understand!
    Lets start from very simple premisses that we agree  on and construct a system that adds complexity until it reaches a level that can explain what happened with the ball and plank!

    Lets start!!

    Do you agree that a rigid, dissipationless and extremely thin plank is a valid theoretical model
    to start to think about?


    Do you agree that the gravity acts equally on all points of the plank regardless its movement?
    colinkeenan
    Yes.  By the way, if you are going to follow your reasoning already posted further up in this blog, please also include a discussion of the starting angle.  Otherwise, you give the impression the plank will always land before the ball, and that is not true.  The plank only lands first for "low" angles.  Calculations to figure out how low the angle needs to be aren't necessary - just show that there must be a starting angle above 0 degrees where the plank lands first.  This is what Sascha did in the first place. 

    Any discussion about this event, no matter how idealized, must take into account the starting angle.  At high starting angles (closer to 90 degrees) the tip starts off with acceleration lower than free-fall.  That's due to the table supporting most of the weight of the plank so that the net force on the center-of-mass is far less than needed for free-fall.  As a result, at such a high angle, the ball lands long before the plank attains high acceleration.  Your result in the post above, that the tip must have higher acceleration than free-fall is still true, but only as the plank reaches lower angles, and by then, the ball can already have landed. 

    Your earlier post did not take into account that the starting angle is important.  It is so important that you can see a thread under the plank presumably put there specifically to make sure the starting angle was optimal.

    Sascha's proof is the only one to take the starting angle into consideration without introducing any math.  Do you have another proof, also taking into account the starting angle, that more people will agree with?  (Probably unrealistic to think everyone will ever agree.)
    blue-green
    Yes John, same gravity …. same great Earth below.
    Did you see my reduction to a bead on a rigid and mathematically “thin” rod? I said it was a bit like a metronome, just to create a memorable image (although I later realized that in this digital age, not so many people here have played with a purely mechanical metronome).

    The bead embodies all of the weight. By sliding it all the way down next to the hinge, one can make the far end of the rod come down faster than the speed of light, especially if the rod is really long.

    Do you see any violation of Special Relativity at the far end?

    This little experiment tests people’s assumptions about both free fall and light cones.

    As I wrote in a nearby article on entropy … .
    “The irony here is that Sascha, the drama queen of “no mathematics please”,
    is the only one [in the entropy article] whose comment is purely mathematical.
    Go figure. … As [the bead] is shifted … from the hinge to the far end,
    the full range of possibilities is covered.” (besides starting angle)
    blue-green
    Try relaxing the rigidity requirement for the bead at the bottom of a rod. Sex it up and imagine the bead is the head of a sperm-like organism. The rod is its long and quivering tail. How fast can the tip of the tail move? We now are close to modeling a bullwhip. It is grasped firmly in the pivoting hand and flicked. CRACK goes the sonic boom at the far end. Can this image work for an open string with a bob of mass at one end? How fast can the far end of the filament move? Can it break out of its own light-cone and make a cosmic crack?
    blue-green
    Many exercise books and sections begin with the recommendation that one should first guess at each answer (to a first order approximation) and only then start calculating. In this way, one develops a practiced way of seeing. Be that as it may, it is absolutely clear that Newton could never have made his impact if he had not given his intuition a foundation in mathematics. The same can be said for Einstein, Schödinger, Heisenberg and Dirac (gratuitous name-dropping …). Without their core mathematics, these individuals would never have gotten traction. Imagination leads, yet it is mathematics that does the underpinning. If you want to explain something difficult to me like quantum entanglement, you had best put it in the language of Dirac’s bra-kets.
    blue-green
    Whatever you are trying to say alpha Sascha, has been expressed before with more power and within that eloquence, more grace. Your fresh reminders, are welcomed. For a guy sitting in the cat bird’s seat of the most powerful country of the world, you ought to get out more. In 2009, my wife and daughter enjoyed the new train service from Beijing to Lhasa (capital of Xizang). We then journeyed a few days into eastern Tibet. I wish I had time to go further and deeper into the jungle. Don’t squander your chances Sascha. It is too big a world out there to remain bitter. Tibet for China is like the West was for the USA. Savor the century. I’m totally envious of your natural resources. Get your papers in order and live. Cat Bird Seat :: how to express that in Chinese Characters?
    vongehr
    Whatever you are trying to say alpha Sascha, has been expressed before with more power and within that eloquence, more grace.
    Given that it apparently did not have sufficient effect however, maybe the grace and eloquence were the wrong approach, just making the coffee house bourgeoisie stick their heads up their asses a little deeper, and a more rational (is that what you confuse with "bitter"???) approach is in order.
    logicman
    The short description of what we see in the video:

    An object is placed to one side of and higher than a target.
    The object falls vertically, the target in an arc, both accelerated by gravity.
    The target, falling through less vertical distance, lands before the object.
    The target, falling in an arc, moves so as to be under the object before the object lands.


    The long description:

    The plank is an inverted pendulum.
    The object is a bouncy ball, so a cup is needed to keep it in place.
    The target is the bottom of the cup.
    The golf tee is needed for two reasons:
    1 - to raise the ball so that it will clear the rim of the cup;
    2 - to keep the ball from rolling away.
    The illusion of a cup falling faster than a ball arises because we focus on the cup as a whole object, whereas in the experimental context it is only the base of the cup that counts.

    The experiment would work just as well with an inelastic object and a sticky target.
    It would then be obvious that the object starts higher up than the target,
    but the 'magic illusion' would not be so much fun.
    vongehr
    Wow, there is so much wrong here, I don't even want to get started. BTW, these articles have a comment section, you noticed? You can read them before commenting. It is a great feature really. Helps not ending up with about ten different mistakes made by others before.
    logicman
    , there is so much wrong here, I don't even want to get started.

    Hand waving is not science.
    Please state where I am wrong and why.
    If you don't want to respond to comments, don't write.

    BTW, these articles have a comment section, you noticed? You can read them before commenting. It is a great feature really. Helps not ending up with about ten different mistakes made by others before.

    Sarcasm is not science. 
    Having read the articles - both of them - and the comments, I decided to contribute some basic physics to the debate.  Do you have something against physics?
    vongehr
    Come on - your objections have been discussed in the comments - all of them. The golf tee, whether the cup is necessary, all of it.
    colinkeenan
    I didn't want to comment again repeating everything I've already pointed out a long time ago, but there is something else I can say about this experiment.

    I'm pretty sure this experiment is introduced to freshmen physics students after they've studied translational dynamics including circular motion and the pendulum, but before they've studied rotational dynamics of a rigid body.  When looking at a rigid body like the plank, it is clear force can be applied to move the center of mass, but also to rotate it moving the ends without moving the center of mass.  It is expected the freshman physics student, having worked so hard to understand circular motion will forget what he already knew about rotating objects even before taking physics, and will make the same mistake you did in thinking it's like a pendulum.  Once they see the experiment clearly demonstrate the end in fact experiences acceleration faster than free fall, they realize their mistake (hopefully) in not working with the center of mass (which is not at the end of the plank) and also realize it's not reasonable to have to take into account internal forces to figure out the motion at the end of the plank.  This hopefully motivates the freshman physics student to try and understand rotational dynamics of a rigid body (the next topic to be covered in the class regardless) so they can properly calculate the motion of the end of the plank without having to worry about internal forces. 
    logicman
    Sascha: you asked for a brief philosophical explanation of what was shown in the video.  That is what I gave you.

    I applied a heuristic: disregard all factors which do not influence the outcome of the experiment.

    Yes, we can look at the plank as a rotating object with a center of gravity at a height less than the height of the ball.  We could even assemble an experiment to demonstrate exactly this effect.  However, in the video the experiment is not well designed to show this effect.

    If you take two planks and join them with a hinge you have a slap stick.  This device converts motion into sound energy by air displacement.  In the experiment as shown in the video, any 'speed up' effect due to the rotation of the plank will be offset by the air displacement 'slow down' effect.

    I suggest that the rotation effect is negated by the air displacement, hence the fact that the ball starts off higher than the base of the tumbler is sufficient and necessary to explain what is seen in the video.

    I may, of course, be wrong.  I am open to discussion of this possibility.
    colinkeenan
    You have not observed what happens in the experiment very closely.  Here I'll copy and paste what I pointed out a couple of times in this discussion already:

    "At the start of the experiment, there is no difference in height [between the top of the tee and the bottom of the ball], but at the frozen scene at the end, when the tee stops moving, there is a huge difference in height between the bottom of the ball still in free-fall way up above the top of the cup, and the top of the tee way below the top of the cup.  In order for any separation to occur between tee and ball, the tee (and therefore the plank it is attached too) had to accelerate more than the ball."

    The experiment is reasonably well constructed to demonstrate the effect they wanted to demonstrate.
    logicman
    You have not observed what happens in the experiment very closely.
    But I have.  I observed that the fast falling of the tee has nothing to do with the outcome.
    Let me explain that by dealing with the matter in two parts:
    1 - why the ball lands in the cup.
    2 - why the tee, which starts under the ball, does not impede the vertical fall of the ball.

    1 - why the ball lands in the cup:

    As I stated before - the difference in height between the ball and the cup at the start of the experiment and when the plank comes to rest is virtually unchanged, as shown in this composite screen shot.



    2 - why the tee, which starts under the ball, does not impede the vertical fall of the ball.

    ... when the tee stops moving, there is a huge difference in height between the bottom of the ball still in free-fall way up above the top of the cup, and the top of the tee way below the top of the cup.
    Agreed, but irrelevant.  What you say is much the same as Sascha's explanation:
    (Look at the golf tee where it supports the ball: They separate!)
    Sascha also says:
    "The being faster is necessary for the ball to separate from the golf tee and land in the cup"
    I do not agree that the fast falling is either a necessary or sufficient condition for the separation of the ball and the tee.

    To determine why the ball and tee separate we must examine every condition that might cause or impede the separation: from the setting up of the experiment to the instant of horizontal separation of the tee and the ball.  What the tee does after that cannot affect the outcome of the experiment and is of entirely academic interest.

    We observe in the video that the ball bounces: it is elastic.  At the start of the experiment the ball is held on the unmoving tee by gravity and so must be deformed, even if only by a small amount.

    At the instant of release, nothing is moving.  However, the plank, cup, tee and ball are all in free-fall.  The force causing the elastic deformation is no longer applied, so at the instant that things start to fall the stored energy in the ball is restoring the roundness.  This means that a force - however small - is being exerted between the ball and the tee.  Since the tee and plank have more mass than the ball, the ball will be accelerated more by its stored energy. The force acting on the ball at the instant of release of the plank acts in opposition to gravity.

    In the first instant of motion of the tee it commences to move in an arc.  The motion in an arc has a vertical and a horizontal velocity.  Since the tee is in contact with the ball the horizontal motion must exert some force, however small, on the ball.  This force, acting at a tangent to the surface of the ball will cause it to spin.  During the elastic rebound of the ball it will continue to be in contact with the tee until the velocity of the tee tending to remove the tee from the ball exceeds the velocity of rebound which is maintaining contact.

    For as long as the ball and tee remain in contact the tangential motion of the tee with respect to the ball will impart a spin to the ball.  The ball will spin clockwise from the camera's viewpoint.  A spinning ball in a concavity tends to climb up the concavity, in exactly the same way as a shaft rotating in a bearing.  The spinning ball will climb to the right relative to the tee, which has a leftward velocity.

    As the tee moves left, the ball rises relative to the tee due to a combination of elastic rebound and spin.  The net effect from the observers viewpoint is that the ball is not displace laterally by the tee and doesn't begin falling until the tee has moved to the left and is no longer in contact with the ball.  At this point, discussion of the vertical velocity of the tee is otiose.
    colinkeenan
    The tee is roughly at the end of the plank.  The ball is roughly at the end of the plank.  The tee at the end of the plank reaches the table faster than the ball.  The cup has nothing to do with the problem posed.  The question to be answered is why does the tee reach the table long before the ball?  You postulate the reason has to do with elastic energy stored in the ball and some other minor forces.  I postulate those forces you are concerned with are negligible compared to the simple rotational dynamics of the plank.  I suggest you work out the rotational dynamics in detail, assuming every object concerned is a rigid body that has no elasticity.  If the result of that calculation contradicts in some way what is shown in the video, please let us all know.  I'm confident the experiment is satisfactorily explained by rotational dynamics of a rigid, inelastic body.  Someone early on had already done the work in Sasha's original post and I think he came up with the end of the plank hitting first as long as the starting angle was below 35 degrees.  I don't really care about the details of the calculation because Sascha's explanation in this post already proves that even if all those forces you talk about were not present, the end of the plank with tee attached would still hit the table first at low enough starting angle and experience an acceleration greater than free fall.

    By the way, at the start of the experiment, nothing is in free fall because the ball is supported by the tee which is supported by the plank which is partially supported by the table!  Part of the plank is on the table to begin with!  The table through the hinge exerts a force along the length of the plank partially supporting it against gravity.  The only place where the center of mass of the plank/cup/tee reach an acceleration equal to free fall is just as it all hits the table (i.e., if the plank could fall through the table while still attached to the hinge, the center-of-mass of the plank would momentarily experience free fall as it was falling through the table).  At every higher angle, the plank is partially supported by the table.  Of course, the tee instantly separates from the ball and at that point the ball is in free fall, but the plank still is not in free fall.  If you start from a high enough angle, the ball will hit long before the plank.  Just think about how a tree falls if you don't want to simply try some experiments yourself.
    colinkeenan
    I like your composite picture.  The fact the ball and cup end with the same vertical separation as they started with means the cup was carefully positioned to create that effect, causing the illusion that the plank was experiencing free fall the whole time.  It is ironic the illusion was the opposite of what you first said it was.
    Have enjoyed reading through this discussion- when I read the initial problem I admit I was as confused as many of the people who offered solutions that where way off. However I cannot grasp why after Sascha has proved his own solution people insist on creating other solutions involving elasticity, air resistance etc.. It may be true that these factors will have affected the outcome however they are irrelevant to the matter at the heart of this discussion and I think it is clear even to those suggesting these solutions that such factors cannot account for the difference in position of the top of the tee and the bottom of the ball as the plank strikes the surface. Also if if such solutions where the real reason why has nobody been able to disprove Sascha's solution? However sascha, as to your point about people mis-trusting science I would argue it is more peoples perceptions of scientists and their fear of not understanding, and I think the arrogance and rudeness you have exhibited are interlinked with this problem.

    Your argument says: "All mass away from the tip also ‘wants’ to fall as freely as the tip,too... "

    No it doesn't. It wants to fall at g. That's the point of all this -once you've distributed some mass, the initial tip acceleration will be >g.

    But there's a problem with that. There will be a place somewhere on the plank which does fall at g. Mass added between the g-spot and the tip will destroy the monotonicity that your argument depends on because it will pull the tip up. Mass removed from it will destroy monotonicity because there's that much less mass "dragging the tip along". Seems like it's a forbidden zone.

    So it's not immediately clear that we can ever get any mass near the tip (apart from the stuff that was born there), let alone end up with correct uniform density for the plank. Your proof needs to be patched up with a demonstration that it is indeed possible. This isn't hard but your original statement was incorrect because it only applies before you've done a bit of smearing.

    Of course the solution is quite simple. You add an infinitesimal mass to the permitted zone (anywhere you like!). This moves the g-spot infinitesimally thus extending the forbidden zone infinitesimally which you top up with another infinitesimal amount of mass at the g-spot. Repeat until the whole of the mass is correctly distributed. But that's enough maths!

    Or maybe I missed it?

    vongehr
    Your argument says: "All mass away from the tip also ‘wants’ to fall as freely as the tip,too... "
    No it doesn't. It wants to fall at g.
    Sorry, I do not get it. "'wants' to fall as freely" means here that every bit just falls at g if we were to cut it from its constraints (the other bits which any bit is connected to). The tip also just "wants to fall at g".
    Monotonicity is not sufficient. You could be monotonically progressing towards a final acceleration which is less than g, not more. You need to show that it starts out in the right direction. I think that must be what you intend when you say: "all mass away from the tip also ‘wants’ to fall as freely as the tip, too, but cannot because it is connected to the tip, and so it must drag the tip along."

    Unfortunately, the mass is also holding other masses up and it is not at all obvious what the total effect is. Concentrating on just the sign of each contribution in order to avoid simple calculus leaves us with a huge double integral of sign functions instead.

    You can certainly argue that these contributions are second order infinitesimal and can be ignored but that would seem to be a bit cavalier given that there are an infinite number of them for every one that applies to the tip.

    vongehr
    Smearing is done at zero degrees and is the simplest smearing [think of a compressed spring that you let relax (no crazy relaxation of the spring either, just hold a homogeneous spring between the endpoints, one being the hinge) until it spans over the whole of the plank, then re-compress into the other limit (push the coil into the tip)] starting from one extreme limit to another extreme limit. Monotonicity between these two limits at zero degrees is obvious as long as you do not start to smear in ways nobody would ever smear. The limits are from infinity at one end down to g at the other. After that you start thinking about finite angles.
    Yes of course that scheme works! It is the same principle as the one I suggested - which you deleted :( That one also worked and was monotonic as it happens. However it is not *obvious* that either scheme is monotonic. You need an argument like mine to prove that. Or maybe I'm missing why it's obvious?

    Yes, a compressed spring is nice and clear if you start from a clean slate and are not distracted by monotonicity. But the adequacy of a scheme based on filling the plank from the tip to a dividing line is not what is suggested by the phrase "smearing mass uniformly along the plank". Not even now that you have produced your spring.

    The point is, the "bearing down" argument proves that the end point is > g without monotonicity being mentioned.
    My criticism of your argument is that it invokes monotonicity when it does not need it and then fails to provide proof.

    vongehr
    I deleted one post from you that you posted twice. The other posts that were deleted (including my answer) were not deleted by me. There is some sort of problem here with the Science2.0 software that we still have not figured out. I have almost every four or so days several comments just vanish into entropy. Very sorry about that, but there is little I can do about it.
    "smearing mass uniformly along the plank". Not even now that you have produced your spring.
    Sorry, but in my mind this was always the obvious simplest way to smear. What simpler and natural way than this is there?
    Oh, sorry, you're using both extremes. Well I don't see that monotonicity is obvious nor necessary. You just need continuity.

    Yes, I do have stamina. I acquired it through arguing with creationists and devout atheists!

    The problem begins with: "All mass away from the tip also wants to fall as freely as the tip, too, but cannot because it is connected to the tip, and so it must drag the tip along. Thus, the tip obtains additional acceleration."

    This strictly applies where the tip is falling at g. But "If [g] then [>g]" is a contradiction so you have four options, maybe more:

    1 Resort to mathematics instead. But that would be against the rules of the game.
    2 Use reductio ad absurdum: "Therefore not [g]". That could be a neat argument.
    3 Play fast and loose with infinitesimals. This is your argument, though you gloss over the details.
    4 Analyse the case where the tip stays at g. This is my argument, undoubtedly the best!

    In your argument, you:
    1 Make the initial smear infinitesimal.
    2 Say "All mass... additional acceleration".
    3 Assume that at [g+delta] the argument applies to enough of the plank to remain valid. 
    4 Conclude that the smearing starts off in the right direction.
    5 Invent a smearing scheme.
    6 Assert that it is monotonic.
    7 Therefore [>g]. QED.

    8 When 6 is challenged, assert that it is an obvious scheme and all obvious schemes are monotonic.
    9 When 8 is challenged, assert that all test cases are contrived and therefore not obvious.

    On the other hand, "my" argument is simple in the extreme. John got nearly there and, as you say, lost confidence. You got to the same point and went off to Lala land. That, I suppose has some meta-didactic interest. Anyway here's the argument:

    1 Force the tip to accelerate at [g] (NOT freely!!)
    2 Say "All mass ... but is restrained by the rigidity of the plank, so bears down on it".
    3 Release the tip.
    4 Therefore it accelerates faster. QED. 

    5 If 4 is challenged just expand it:
    Force a much higher acceleration [>>g]. Most of the mass that was pressing down now pulls up.
    Therefore there is some intermediate value [>g] where the two balance out. 

    Note that even if you insist on regarding it as a progression, all you need is continuity, not monotonicity, and continuity really is obvious.

    Surely you can see that this is simpler and better than all the flaffing about with limiting cases, Heath Robinson smearing schemes and shoddy arm-waving assertions of monotonicity that your argument relies on? You don't need any of it!

    I rest my case.
    ----------- 
    As for your challenge to create a better smearing, I was inclined not to bother as it's not relevant. However I confess, I was intrigued. So I ask my cat, "Where shall we put the mass at the start?" 
    "In my pooh-tray on the floor." she seems to say.
    "I can't do that, but I get your point, let's put it somewhere where it doesn't cause any problems. At the hinge."
    My cat agrees.
    "Or very close, so we don't have any infinities to worry about." I mutter under my breath.
    "What next, Puss?"
    "Smear it somewhere obvious", she purrs, "at the tip."
    "What, like one of Sacha's springs?"
    "Certainly not!" she spits, her fur bristling. "Talk about contrived! All the mass in a spring so the density keeps changing? Whatever next??!! No, no no! Keep it simple and natural. Just keep the density constant at the obvious value." 
    "Of the wood." she adds helpfully.
    "Like a spring that gains weight as it stretches?" I suggest. "Yes, Sacha will like that."
    But there is no answer. My cat has fallen asleep.
    vongehr
    Sorry, there is so much wrong and misunderstood with this, I do not even know where to start. But the most funny from a physicist's point of view got to be a smearing where the density dm/dx does not change. I am getting the feeling you want to troll. I am out of this discussion. Maybe somebody else likes to take you on, but I rest my case.
    vongehr
    "Like a spring that gains weight as it stretches?" I suggest. "Yes, Sacha will like that."
    But there is no answer. My cat has fallen asleep.
    Yeah - right - I like springs that gain weight as they stretch, because that is what springs do in wonderland and all my readers live down the rabbit hole or something, so they will love such wonderland everyday items, too, they can relate to them, so this is very pedagogical.
    Why is it that crazy people always have cats around? Is it that cats make people crazy or do crazy people like cats?
    Sorry Derek, but I will no longer engage you here. Keep digging if you like. Thanks for your interest.
    I am sorry you should think I am trolling.  I spent a lot of time trying to make a simple point clear. You should not assume that just because you don't understand someone, they are up to no good. 
     
    There's nothing funny about "dm/dx = constant".  I'm not expanding a constant mass, I'm taking mass from the hinge and expanding it at constant density.

    You are not stupid but if you can't be bothered to read what people write, why should anyone bother? You'll end up shouting into a void.

    vongehr
    expanding it at constant density
    This is either adding mass (NOT Smearing!) or it is trolling. I am out of here.
    SCHEMATIC EXPLANATION OF PLANK TIP ACCELERATION

    http://imageshack.us/photo/my-images/10/screenshot20120102at136.png/

    There are three huge mistakes on that diagram. 


    Derek: Thanks for taking the time to view my diagram. After reading this blog and all the comments, I felt it summarized what I had learned regarding what makes this apparatus work. It wasn't meant to be a scientific treatise, however, I felt it illustrated the basic principles at play. Please feel free to enumerate my errors.

     
     



    What is "Effective CG"?: That point on the hinge-plank system or its imaginary extension in space that will accelerate downwards at 1G when the plank is allowed to fall. Put a different way: no points on the plank can accelerate at 1G until a critical angle has been reached.

    What is arrow tip?: The tip of the upward-to-left sloping arrow represents the plank end.

    CG cannot move: Effectively it can in this system, because at some critical angle (which others have calculated above) the hinge counter-contribution no longer is a factor, at which point the effective CG would approach the bisect of the real CG as both merge at the midpoint of a plank of uniform density and dimensions when horizontal.

    What is the "Critical Angle": That angle at which we observe the effect seen in the video, namely the separation between tee and ball when the plank is allowed to fall.

    CG does not fall at 1G: It does when the hinge is no longer a factor.

    Force does not have to point along the beam [plank]: A diminishing amount would until the critical angle is reached.