Surprising Physics: Falling Flat Boards & Chimneys

[Integral]

Take a flat board of uniform density. Mass M and Length L.
Now hold the board horizontally at the edge of a table. So you hold one end of the board at distance L from the edge of the table, while the other end is resting on the table.

Now release the board. Gravity will exert a torque about the axis where the board touches the table. The gravitational force will act on the center of gravity of the board, so:
$$\tau = \frac{L}{2} Mg$$
The moment of inertia of this board about an axis at the edge is
$$I=\frac{1}{3}ML^2$$
So the board will rotate with an angular acceleration of:
$$\alpha = \frac{\tau}{I}=\frac{3}{2}\frac{g}{L}$$
That means for the part of the board at the loose end at the moment of release an acceleration of:
$$a=\frac{3}{2}\frac{g}{L}L=\frac{3}{2}g$$

Faster than freefall!

Some of you may not be surprised by it, but I was.
I realize that the normal force at the end is responsible, but still. It's kinda counterintuitive. I didn't expect it.

Does anyone else got funny and surpising physics about which you say "I didn't expect that?".

EDIT: Fixed a typo. Thanks Brad.

If you have ever seen a tall chimney fall, you will have observed that they break up on the way down. This is because of the effect you have found.

What is even more interesting a cylinder falling in that manner will break into 3 large pieces and some crumbs (about .14 ) so the chimney breaks into $\pi$ pieces.

 

[Doc Al]

What is even more interesting a cylinder falling in that manner will break into 3 large pieces and some crumbs (about .14 ) so the chimney breaks into $\pi$ pieces.

Say what?

 

[Integral]

Ok, Ok, the chimney is a streach... Drop a full stick of chalk it will usually break into $\pi$ I have always felt that it was somehow related to the falling chimney

 

[yourdadonapogostick]

how can a chimney break into an irrational number of pieces?

 

[Integral]

how can a chimney break into an irrational number of pieces?

Sigh... Let me explain again...

Drop a full piece of chalk, I have done this experiment many times... With amazing regularity, it will break into 3 large pieces and several smaller fragments + dust, or .14 so it breaks into $\pi$ pieces.

Ok, do you get it? It is an approximation.

 

[LeonhardEuler]

If it breaks into 3 large pieces and, say 10 small fragments, and, say, 1000 dust particles, then doesn't it break into 1013 piecies, not pi?

 

[Integral]

Good lord, do you people have no sense of approximations?

If the 3 larger pieces are each a unit (ie 1) then the chuncks and dust compromise about .14 of a unit... is that clear enough?

 

[LeonhardEuler]

Sigh... Let me explain again...

Drop a full piece of chalk, I have done this experiment many times... With amazing regularity, it will break into 3 large pieces and several smaller fragments + dust, or .14 so it breaks into $\pi$ pieces.

Ok, do you get it? It is an approximation.

Then what do you mean by "breaks into $\pi$ pieces"? Aren't each of the smaller fragments pieces? How can you have .14 of a piece? When you talk about units, it seems like your defining a unit as (1/pi)*total length, so then it's no surprise that the fragment lengths sum to pi.

 

[Integral]

Then what do you mean by "breaks into $\pi$ pieces"? Aren't each of the smaller fragments pieces? How can you have .14 of a piece? When you talk about units, it seems like your defining a unit as (1/pi)*total length, so then it's no surprise that the fragment lengths sum to pi.

No, I am saying that a new piece of chalk will break into 3 large sections and a number of much smaller fragments + dust. Envision the fragments as being a small fraction (approximately of course) ~ .14 of one of the (not exactly equal sized, but of the same magnitude ) larger sections.

Don't over think this, just find a box of chalk and start dropping chalk from several feet. See for yourself.

 

[rachmaninoff]

I've done this accidentally before, it broke into 2 (not 3) big pieces, ~5 small-ish pieces and some very fine dust... but I suppose it matters on the kind of chalk? For example, whether or not it's very thick or pencil-like, or the 'dustless' kind, etc.?

 

[PerennialII]

hmmm, is chalk 'ideally brittle' ? If not there would be a size and length effect (the thing of course, different perceptions) + a neat distribution of characteristic parameters.

 

[russ_watters]

Ok, Ok, the chimney is a streach... Drop a full stick of chalk it will usually break into $\pi$ I have always felt that it was somehow related to the falling chimney

Chalk breaks when it hits the ground due to the impact - a chimney breaks due to gravitational acceleration causing large internal stresses. They are utterly unrelated.

Chalk will break into any number of pieces based on the specifics of the impact. Try a higher energy impact...

Chimneys/towers break into 2 or more pieces based on their geometry and strength. HERE is one that broke into two pieces.

 

[Integral]

Chalk breaks when it hits the ground due to the impact - a chimney breaks due to gravitational acceleration causing large internal stresses. They are utterly unrelated.

Are they indeed? That is a pretty strong statement, much stronger then my hunch that they may be related.

Only if you drop the chalk very carefully can you make it hit anyway other then one end first. As soon as one end hits first you have a "falling chimney"

I think your statement is to strong. I would certainly buy " there is some doubt" but not a "utterly unrelated"

EDIT:
I just looked at the pictures in your link. I disagree with you. It appears to me that the break in the 5th pic is a DIFFERENT break then the one in the 6th pic. If you ask me it broke into 3 pieces _+ change! PI!

 

[pattylou]

No, I am saying that a new piece of chalk will break into 3 large sections and a number of much smaller fragments + dust. Envision the fragments as being a small fraction (approximately of course) ~ .14 of one of the (not exactly equal sized, but of the same magnitude ) larger sections.

Don't over think this, just find a box of chalk and start dropping chalk from several feet. See for yourself.

Is "several feet" important?

If so, then doesn't this seem like simple coincidence?

Does the length or the width of the chalk matter?

 

[brewnog]

I can't say I'm convinced.

A piece of chalk will fail when it hits the floor, as a result of shock waves traveling through the chalk at the speed of sound, which cause the brittle chalk to shatter.

A chimney will fail once its foundations removed, by collapse under gravitational fall in whatever manner the steeplejack has intended. I suspect the failure is due to local stress concentrations within the chimney walls exceeding some critical value, where it either snaps (if a 'toppling' fall has been planned), or just collapses in on itself (in a kind of 'telescopic' collapse).

I do think the scenarios are rather different.

In any case, if the chalk thing is true, it's quite a nice little discovery Integral!

 

[Integral]

Is "several feet" important?

If so, then doesn't this seem like simple coincidence?

Does the length or the width of the chalk matter?

Good questions, perhaps the fact that I made my observations some years ago at a single university, it may have been typical of that brand and size of chalk. ...

Actually it was one of profs. can't really remember which, who pointed this out.

 

[Integral]

I can't say I'm convinced.

A piece of chalk will fail when it hits the floor, as a result of shock waves traveling through the chalk at the speed of sound, which cause the brittle chalk to shatter.

A chimney will fail once its foundations removed, by collapse under gravitational fall in whatever manner the steeplejack has intended. I suspect the failure is due to local stress concentrations within the chimney walls exceeding some critical value, where it either snaps (if a 'toppling' fall has been planned), or just collapses in on itself (in a kind of 'telescopic' collapse).

I do think the scenarios are rather different.

In any case, if the chalk thing is true, it's quite a nice little discovery Integral!

Defiantly a fall with rotation is much different from a collapse, that is a very different scenario. Even a hollow chimney would be different from a solid cylinder such as a stick of chalk.

 

[Curious3141]

Integral, I presume you meant to claim that a piece of chalk tends to break in a ratio of lengths/mass equal to $1:1:1:(\pi-3)$. The last proportion tends to further fragment into an indefinite number of pieces. Correct ?

Without a theoretical reason why this should be so (and I can't see one), I cannot accept it. Your assertion in fact reminded me of Buffon's needle, but then there really is a theoretical reason why the probability of an idealised thin needle of unit length intersecting an infinitesimally thin line on lined paper with unit spacing is $\frac{2}{\pi}$. It's trivial to see that, and I've proved it myself. Your assertion, alas, I cannot see my way to proving.

 

[russ_watters]

Are they indeed? That is a pretty strong statement, much stronger then my hunch that they may be related.

Yes, I know.

Only if you drop the chalk very carefully can you make it hit anyway other then one end first. As soon as one end hits first you have a "falling chimney"

Ahh, misunderstood what you were saying in the first post. The problem is still that it depends an awful lot on the energy of the impact. If the force is large enough, it'll still shatter the entire piece of chalk.

I just looked at the pictures in your link. I disagree with you. It appears to me that the break in the 5th pic is a DIFFERENT break then the one in the 6th pic. If you ask me it broke into 3 pieces _+ change! PI!

Hmm... I think that's an optical illusion due to the tower falling away from us, but I'm not sure. I'll look for more pics...

A chimney will fail once its foundations removed, by collapse under gravitational fall in whatever manner the steeplejack has intended. I suspect the failure is due to local stress concentrations within the chimney walls exceeding some critical value, where it either snaps (if a 'toppling' fall has been planned), or just collapses in on itself (in a kind of 'telescopic' collapse).

I do think the scenarios are rather different.

Yes, the specifics of the structure make a big difference. http://www.southcoasttoday.com/daily/05-99/05-30-99/c01lo082.htm is one that looks to me like a hybrid between a "toppling" and "telescopic" collapse. It does not appear to break anywhere along the structure, but disintegrates from the bottom up as it falls over. A large building would do the same thing (but less toppling and more telescoping) because structurally they are much weaker than a masonary chimney - they would not be able to support their own weight leaning over even a little bit.

This is actually a fairly common college level dynamics problem (in fact, there is a thread open in college help about it...)

edit: http://www.randyspier61.com/photos1.html is another one that appears to fall in 2 pieces.

http://www.www.dykon-explosivedemolition.com/Archives/AthensOhio/AthensOhio.htm is a cool video clip of one that also appears to fall in two pieces. Click the index for more videos - most seem to break in two pieces, but there is one that doesn't break and another that crumbles (the 6 smokestacks clip).

 

[Moonbear]

Hmm...I wonder if it would depend on the brand of chalk. Dustless chalk is pretty dense and uniform, but more "traditional" chalk is a lot more porous and non-uniform. I wouldn't know how it breaks when you drop it because I'm not sure that in my entire life I've ever gotten a piece that wasn't already broken in two while still in the box, or pre-shattered by the person teaching in the lecture ahead of me, or that I didn't snap in half in my hand after the first few words (I learned to properly press hard on the board with the chalk so it is dark enough for the students to read, but then I end up snapping chalk...nobody was ever nice enough to buy me one of those nifty chalk holders). But, there must be a length requirement, because I'm pretty sure when I've dropped those short pieces, they didn't shatter into three more pieces (+ change). I think more often, just an edge chipped off, but it has been so long since I've had to used chalkboards instead of whiteboards that I'm not sure any more.

 

[εllipse]

It must be a conspiracy. The chalk Integral uses was specifically designed by the government to break into 3.14159 pieces so as to keep us busy talking about things that are unrelated to their other, more dastardly cover-ups.

I've done this accidentally before, it broke into 2 (not 3) big pieces, ~5 small-ish pieces and some very fine dust... but I suppose it matters on the kind of chalk? For example, whether or not it's very thick or pencil-like, or the 'dustless' kind, etc.?

Unless I am wrong--and I am never wrong--that must have been one of the boxes of chalk designed by the Soviets during the Cold War, which normally breaks into 2.7182818 pieces.

 

[Moonbear]

It must be a conspiracy. The chalk Integral uses was specifically designed by the government to break into 3.14159 pieces so as to keep us busy talking about things that are unrelated to their other, more dastardly cover-ups.

...

Unless I am wrong--and I am never wrong--that must have been one of the boxes of chalk designed by the Soviets during the Cold War, which normally breaks into 2.7182818 pieces.

 

[cronxeh]

well we got one camp of engineers with statics & dynamics in mind, and in another camp we have mathematician with analytical-probability in mind - oh boy, oh boy! who will win?


We need a physicist here who will conduct a few dozen controlled experiments in the vacuum as well as different density air environment to determine the true cause and probability of chalk break. Is this too much? No way! Bring over material scientists over here and get them to make different kinds of chalks of a unit lenght! No time to waste people, this is for science!

 

[εllipse]

well we got one camp of engineers with statics & dynamics in mind, and in another camp we have mathematician with analytical-probability in mind - oh boy, oh boy! who will win?


We need a physicist here who will conduct a few dozen controlled experiments in the vacuum as well as different density air environment to determine the true cause and probability of chalk break. Is this too much? No way! Bring over material scientists over here and get them to make different kinds of chalks of a unit lenght! No time to waste people, this is for science!

I'm writing to the MythBusters.

 

[cronxeh]

Can anyone actually write a computer simulation for this in Java for example? We should have some AE grad students here somewhere..

 

[Moonbear]

Can anyone actually write a computer simulation for this in Java for example? We should have some AE grad students here somewhere..

Will you settle for a biologist studying the little critters who make the chalk? Who knows, maybe things like nutrient availability affect their shell quality. We can't leave these variables to chance on such an important issue as broken chalk!

 

[PerennialII]

Can anyone actually write a computer simulation for this in Java for example? We should have some AE grad students here somewhere..

Not a problem ... you want it in quantum scale or will traditional nano/micro scale suffice ?

 

[cronxeh]

I'm pretty sure there is an optimized scale with respect to air resistance and modulus of elasticity, unless you want to only treat this problem as a random area/constant force hit and go from there, which should simplify to a finite difference method..

I'm not sure how this would work exactly. Does the chalk break into pi pieces in vacuum or only in room environment? Does the radiation affect the chalk's integrity? Is temperature a variable? How about humidity? Barometric pressure? How about g? I think this is indeed a lucky coincidence that it 'happen' to break into 'pi' pieces for Integral. I mean what mathematician can't dream after all

 

[PerennialII]

... think we need to include all of those, we could start doing some DFT to derive a basis for a macroscopic model, continuum damage mechanics with some discrete & stochastic elements would probably do the trick from thereon... we can continue the thread in about a year or so and get a couple of PhDs in the process.

 

[Integral]

I've done this accidentally before, it broke into 2 (not 3) big pieces, ~5 small-ish pieces and some very fine dust... but I suppose it matters on the kind of chalk? For example, whether or not it's very thick or pencil-like, or the 'dustless' kind, etc.?

Was it a brand new, just out of the box, stick or a used stick? Used sticks break differently... I like that e theory!

 

[Integral]

Russ let me point out that early in this thread I conceded chimneys, see post #3 . The title of the thread is “breaking chalk” . I have further specified; dropped from “several feet” I will refine that to 36” +/- 12”(1m+/- 30cm with zero initial velocity, I believe that this falls into the “several feet “ category. So that should rule out chimneys right there.


Beyond that I claim that your photos do NOT disprove my conjecture. and indeed even support my claim. Note that in the first 3 pictures the top of the chimney is at the level of the top wires, the first break occurs in frame 5, in frame 6 the chimney tip has dropped significantly look at the power lines. It is not a rotation but a direct vertical fall. We see piece 1 in frame 5, in frame 6 we see piece 2. Frames 7 and 8 show the last major unit and the .14 This is not exactly high speed video and, by the way, your video shows nothing of interest, it is impossible to tell how the chimney breaks. Your other photos are inconclusive. I say that your second video the chimney is fall directly in line with the camera, it is impossible at this angle to judge when and where the chimney breaks. Rather then make a claim, I will simply say, I can't tell.

Note in post #9 that I did allow that difference in length was not ruled out.

Moonbear has brought in density as a valid parameter, that, with the radius to length ratio will be some of the critical factors.


Indeed the chalk will shatter due to impact, with the shock wave traveling though the clock at speed of sound in chalk. What is the speed of sound in chalk?

 

[zanazzi78]

I can't really be bothered to read all the comments in this thread (since i`ve only just got here!) but it did give me a dam good reason to break a whole box of chalk and realse some of the days tention!

and yes they all broke into 3 pieces and some dust!

 

[BobG]

Is "several feet" important?

If so, then doesn't this seem like simple coincidence?

Does the length or the width of the chalk matter?

I think it has to be dropped from about π feet.

I've done this accidentally before, it broke into 2 (not 3) big pieces, ~5 small-ish pieces and some very fine dust... but I suppose it matters on the kind of chalk? For example, whether or not it's very thick or pencil-like, or the 'dustless' kind, etc.?

Unless I am wrong--and I am never wrong--that must have been one of the boxes of chalk designed by the Soviets during the Cold War, which normally breaks into 2.7182818 pieces.

I think it's more likely that rachmaninoff dropped the chalk during a time when he was shorter.

I stood on my counter and threw colored pieces of chalk from pie feet high and found they broke into about 8.54 pieces. That's an approximation since I had to try to count them while my crying daughter was scooping up as many pieces as she could, while the pet cat chased and batted stray pieces around the floor, and while my wife shrieked in horror when she saw me standing on the counter next to her pie. I would do more testing, but, unfortunately, I'll living in a car for a few days.

 

[Integral]

I think it has to be dropped from about π feet.


I think it's more likely that rachmaninoff dropped the chalk during a time when he was shorter.

I stood on my counter and threw colored pieces of chalk from pie feet high and found they broke into about 8.54 pieces.

You had to stand on a counter to get $\pi$ feet! You must be very vertically challenged!

That's an approximation since I had to try to count them while my crying daughter was scooping up as many pieces as she could, while the pet cat chased and batted stray pieces around the floor, and while my wife shrieked in horror when she saw me standing on the counter next to her pie. I would do more testing, but, unfortunately, I'll living in a car for a few days.

I know that the price of science is high.. I appreciate your efforts.

Humm... I did not specify color.. wow.. yet another parameter!

 

[BobG]

You had to stand on a counter to get $\pi$ feet! You must be very vertically challenged!

No, not pi feet high. pie feet high!

pie should always be rounded and never squared.