# Breaking chalk

#### Integral

Staff Emeritus
Gold Member
Galileo said:
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!!!
:surprised
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 in to $\pi$ pieces.

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#### Doc Al

Mentor
Integral said:
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 in to $\pi$ pieces.
Say what?

#### Integral

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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

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

#### Integral

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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

Gold Member
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

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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?

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#### LeonhardEuler

Gold Member
Integral said:
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

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LeonhardEuler said:
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

Gold Member
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

Mentor
Integral said:
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

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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!

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#### pattylou

Integral said:
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

Gold Member
I can't say I'm convinced.

A piece of chalk will fail when it hits the floor, as a result of shock waves travelling 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

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pattylou said:
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

Staff Emeritus
Gold Member
brewnog said:
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

Homework Helper
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

Mentor
Integral said:
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....
brewdog said:
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 [Broken] 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 [Broken] is another one that appears to fall in 2 pieces.

http://www.www.dykon-explosivedemolition.com/Archives/AthensOhio/AthensOhio.htm [Broken] 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).

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#### Moonbear

Staff Emeritus
Gold Member
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.

rachmaninoff said:
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

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Gold Member
?llipse said:
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.

:rofl: :rofl: :rofl: :rofl:

#### cronxeh

Gold Member
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

cronxeh said:
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

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

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