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        riddles >> easy >> Toppling Pole
        
(Message started by: rmsgrey on Nov 26^th, 2003, 4:37am)

Title: Toppling Pole
Post by rmsgrey on Nov 26^th, 2003, 4:37am

Straight out of an interview:

You are perched on top of a light, rigid pole, which begins to topple. You have the choice of either letting go immediately or hanging on until the bitter end.

i) What should you do?

ii) What about when the pole has significant mass?

Title: Re: Toppling Pole
Post by towr on Nov 26^th, 2003, 5:51am

::[hide]
I'd suggest sliding down, but unfortunately that's not one of the two options..
[/hide]::

Title: Re: Toppling Pole
Post by LZJ on Nov 26^th, 2003, 7:50am

(i) Holding on seems better.
(ii) Holding on seems better...unless the pole crushes you.

Title: Re: Toppling Pole
Post by Kozo Morimoto on Nov 26^th, 2003, 3:40pm

Is holding on really any better? Is that because if you hang on, you 'fall' in an arc rather than straight down? But wouldn't the impact speed be the same either way?

Title: Re: Toppling Pole
Post by TenaliRaman on Nov 26^th, 2003, 6:04pm

Take a nice thick flexible straw and place a reasonably thick blob of clay on top of it.what would you see if the straw were topple?
The straws top since being heavywould bend first before completely falling down.(This would something like we see in the gymnastics [pole vaulting] but there we use this property to jump over the cross bar rather than the other way around)

however if the straw were thick and not flexible, then it don't matter you will die both ways.

Title: Re: Toppling Pole
Post by towr on Nov 27^th, 2003, 12:37am

the pole is rigid in both cases..

In the first case though, you are at the center of gravity, in the second case the center of gravity is halfway down the pole..
So in the first case it certainly doesn't matter if you hold on or not, since your endspeed is the same, the second case might be another story though..
(Note that is yet a whole other story again if you can let go halfway through falling, since horizontal speed is less detrimental. The floor won't stop you dead in your track in that direction, thus giving you more time to slow down and consequently less impact)

Title: Re: Toppling Pole
Post by rmsgrey on Nov 27^th, 2003, 5:08am

My answer (which may or may not be correct)
::[hide]
If you hold on, then there's a non-zero compression force in the pole, meaning your vertical acceleration is less. Of course, I've just realised that once you pass some point in your fall, you no longer compress the pole - tending to fly off at a tangent - but even so, with negligible mass, the pole will come sideways with you rather than developing tension. As I said, I may be wrong - I didn't get confirmation on my answers
[/hide]::

Title: Re: Toppling Pole
Post by TenaliRaman on Nov 27^th, 2003, 5:11am

on 11/27/03 at 00:37:03, towr wrote:

the pole is rigid in both cases..

that's the problem. That's why i agree with your idea of sliding down with which the person has some hope of surviving.

Title: Re: Toppling Pole
Post by towr on Nov 27^th, 2003, 5:20am

There is no mention of how long the pole is, so the height may well not be life threatening, but just crippling ;)

I think that with the heavy pole your endspeed is greater if you hold on, because the CoG is closer to the ground, and will (thus) also be at the ground faster (meaning you also get there with a greater speed). But I haven't done to math to back it up..

Title: Re: Toppling Pole
Post by phobos on Nov 29^th, 2003, 4:43pm

For the light-pole case, if we assume that the part of the pole on the ground is more or less stationary, it should be better to hold on to it right? Since as long as the pole makes an angle to the ground there'll be an upward force. (or am i imagining things..)

Title: Re: Toppling Pole
Post by rmsgrey on Nov 30^th, 2003, 4:22am

It's not quite that simple. If you consider the situation near the beginning of your fall, then you have a reasonable horizontal velocity. If the bottom of the pole were fixed (to a hinge say) and you stayed at the top of the pole throughout, then just before landing, you would have negligible horizontal velocity. Therefore, at some point, the pole must stop being compressed and start being under tension. With a light pole whose end isn't fixed, this isn't a problem since you drag the pole sideways with you, but with a heavy pole, the effect is reduced. Even then, you should still have some sideways velocity when you hit the ground, but for a sufficiently heavy pole, the effect is negligible, and you end up hitting the ground just as hard.

Title: Re: Toppling Pole
Post by phobos on Nov 30^th, 2003, 4:55am

Hm...yeah. But come to think of it, it's like sliding down an arc-shaped plane or free fall. Is there going to be an upward thrust?
If the pole is compressed in the beginning for a certain period, that will actually contribute to an upward thrust isn't it? So it's going to be slower than free fall.

Title: Re: Toppling Pole
Post by rmsgrey on Nov 30^th, 2003, 5:22am

on 11/30/03 at 04:55:01, phobos wrote:

If the pole is compressed in the beginning for a certain period, that will actually contribute to an upward thrust isn't it? So it's going to be slower than free fall.

It also contributes a sideways thrust, so you end up moving at the same speed when you hit the ground (ignoring friction/air resistance) - you have to end up at the same speed because you fall through the same height so convert the same amount of gravitational potential energy which must end up as kinetic energy since there's no processes going on that would convert it to anything else (to a first approximation - if you want to allow for air resistance, ground friction, elastic deformation of your own body, etc, then things get even more complicated)

Effectively what you should be doing is attempting to maximise your horizontal velocity since your total speed at impact is fixed, and (generally) horizontal impact is less painful than vertical.

Title: Re: Toppling Pole
Post by phobos on Nov 30^th, 2003, 7:12am

Essentially the impact is only vertical, so the best deal is to maximise the horizontal velocity like you said.

I'm not too sure about your claim that it'll end up at the same speed because we fall through the same height so convert the same amount of gravitational potential energy which must end up as kinetic energy. As the pole will have different impact depending on whether or not we hold on to it while we fall, part of our kinectic energy has to contribute to that difference.

Title: Re: Toppling Pole
Post by rmsgrey on Dec 1^st, 2003, 3:19am

With a light pole, the kinetic and gravitational potential energy of the pole are both negligible so short of getting the pole up to relativistic speeds, you're not losing significant amounts of energy that way.

With a heavy pole, yes, you could probably dump energy into the pole. I'd have to actually work through numerical examples to figure it out, but I suspect it doesn't make much difference.

Title: Re: Toppling Pole
Post by aero_guy on Dec 1^st, 2003, 12:30pm

A little bit of mathematics shows that if your mass is many times the mass of the pole than your end vertical velocity is sqrt(2*g*h) whereas if the pole is many times your weight than it will be sqrt(3*g*h). A graph between those two extremes shows that it is monotonic. Thus, if the pole has no mass it does not matter if you hold on or not. If it does have mass, than you better let go.
The reason is thus: The potential energy is converted to kinetic and rotational energy when you fall. Consider a pole with a very heavy point mass in the center and no other mass. That mass converts its potential energy to kinetic energy when it falls irregardless of you being there (so long as you are significantly lighter). You are on an extra moment arm from that fall, so you will end up falling at twice the speed of the point mass. The pole will pull you along and slam you into the ground.

This of course assumed no air resistance and no slipping of the base of the pole. It is possible that you could jump at the last second in a variation on the falling elevator problem to lessen your impact. If the pole is heavy enough you may be able to do something. The best bet is to push against the pole to jump out for some horizontal speed and roll it out as has been suggested. Unless the pole buckles or slips you will definitely be able to do that.

Title: Re: Toppling Pole
Post by Zeke the Geke on Dec 4^th, 2003, 2:21pm

Man, I'm not sure I'd want to give up that easily and resign myself to falling at all! Being suitably muscular and flexible ::), I should be able to flip myself over into a hand-stand on the top of the pole (the light one, that is). Then, grasping the pole firmly, I should be able to get it and myself to jump up and down (like a pogo stick). While doing this, I will exert a gentle lateral force on the pole, so that when the pole lands it is directly under my centre of gravity, and I simply remain balanced on the top. At this point I may choose to either slide down safely or continue my clown routine to the delight of onlookers and the increase of my pocket change (provided I first throw my hat on the ground to collect it). ;)