Keskustelujen arkisto

S?ren:

> a) The yoyo-effect continues forever

This (and your other descriptions) exactly match the story that Dani?l mentioned before: the Dutch Geradts/Verhagen story.

--Harry.

Harry Fluks -- TNO Telecom -- Delft -- Nederland
h.w.fluks at telecom.tno.nl -- http://dd50.inducks.org
"Wie ben ik? Wat doe ik hier? Waar is mijn Roddelflop?"

I'm afraid the ontopicness of this message tends now to become rather
farfetched, so for those who are not physics nerds as some of us, you
can skip the following.

Anyhow: According to Newton's shell theorem #2 A uniformly dense spherical
shell exerts no gravitational force on a particle (or duck in these
circumstances, the size of a duck compared to the earth is more or less
that of a particle...) located anywhere inside it.
(There's a proof for it also, but I don't have time/energy to write it
down. see e. g. Physics vol 1 by Halliday, Resnick and Crane if you're
interested.) (there's a shell theorem #1 as well: A uniformly dense
spherical shell attracts an external particle (duck) as if all the mass of
the shell were concentrated at its centre)

ANd then to the 'problem':
Suppose a hole is created through the Earth from one side (Duckburg) to
the other (I'm assuming now that the hole goes through so our poor
test-duck has somewhere to get out in case his speed is grave when he
reached the middle...). A duck of mass m is is dropped into the tunnel
from rest at the surface. a) what is the force exerted on the duck when it
is a distance r from the centre? b) What is the speed of the duck when at
distance r from centre? and c) speed at r=0? Neglect all frictional forces
and assume that the Earth has a uniform density.

Solution: From the theorem #2 above, we conclude that the gravitational
force on the particle only comes from the portion of the Earth that lies
inside the sphere of radius r, and we can also consider this mass to be
concentrated in the centre of the Earth (shell theorem #1). Let the mass
inside r be M, and total mass of the Earth M_E (radius R). Then the
fraction of the mass inside radius r is the same as the fraction of the
volume inside radius r (true only if density is uniform):
M/M_E=(4/3Pi*r^3)/(4/3Pi*R^3) or M=M_E*r^3/R^3
Regarding this mass as concentrated at the centre, we find the
gravitational force on mass m to be proportional to r:
F=G*mM/r^2=GmM_Er^3/(r^2R^3)=GmM_Er/R^3

Now it will be hard, I have to introduce vectors in ascii... Hm. Let rr be
a vector r from the center of the earth to the duck m, then we note that
the force F (now vector F=FF) acting on the duck is opposite in direction
to rr (r in vector form) -> FF=-(GmM_E/R^3)*rr (answer to question a).
Since we got a minus here, the form of the force looks very much like that
of a spring force, F=-kx!
Ok, so given the similarity with the spring force we can represent the
potential energy U of the system (earth and falling duck) as 1/2*kr^2,
taking U=0 in the center (NB k here is the constant in the force law:
k=GmM_E/R^3). App?lying conservation of energy at the surface and at
radius r we have K_s+U_s=K_r+U_r or 0+1/2*kR^2=1/2*mv^2+1/2*kr^2. Solve
for v (velocity) yields:
{ANSWER}
v=SQRT(k/m*(R^2-r^2))=SQRT(GM_e/R^3*(R2-r^2)) and at the center (r=0) the
value becomes v=SQRT(GM_E/R) (look in tables for the values) = 7910km/s
So... it is a good idea to dig the hole out on the other side as well to
avoid flat ducks. (probably worse than flat, but let's not go into gross
details...).

Now, it's definitely time to stop procrastinating, no matter how fun it
is...
/Theresa

Theresa wrote:
> It's not 7910 km/s it's m/s! of course... units...

Soren wrote:
> The velocity at the center of Earth is about 8134 m/s
> (i.e. 2260 km/h)

8134 m/s = 29.282 km/h

> So... it is a good idea to dig the hole out on the other side as well
to
> avoid flat ducks.

So it is really _not_ necessary to dig the hole out on the other side as
well: Even at the "moderate" speed of 8 km/sec the ducks would long have
been burned to ashes by air friction before they could be flattened at
the center of the earth. Roasted, flattened duck. Must think happy
thoughts...

Cord

From: <H.W.Fluks at telecom.tno.nl>
Subject: RE: old castle's new secrets
>>>One disadvantage of having the title on page 4: I don't know on which
page to look for the D.U.C.K. As a result I still haven't found it! 8-)

It's always in the splash-panel, first panel of the story... meaning not on
the prologue page. Keep looking...

>>>Don has the habit of drawing his text balloons over several panels.
Probably to get the effect like in the movies, where you still hear a
previous scene while already looking at the next scene, or vice versa. But
to me, it simply looks *ugly*. A page would look *so* much better with all
the panel borders being intact.

I'm not sure why panel borders should be "intact" -- often I and all other
cartoonists omit the borders on a panel or two on a page altogether... and
surely my page designs are the most plain and unimaginative as could be
possible (quite purposely, because I don't believe that artsy-fartsy page
lay-outs serve any purpose other than to make a story use up more pages than
needed... whee -- extra page fees for nothing! -- my only mission is to cram
as much of everything onto each page and into each story as I can). But look
back at any spot where my balloons overlap panel borders -- can't you see
why I do it? No, not to suggest any cinematic technique (but that's a cool
idea! Thanks!), it's just so I can fit all that @#$%& dialogue onto the
page! You might have noticed that my stories contain lots more yakkety-yak
than others -- it's the only way I can see to fit it all in there. And the
idea to do it that way struck me as a simple solution for too-many-balloons
when I noticed that Jean Giraud does it that way... so I figgered if it's
good enough for him, I should try it.

>>>Anyway, I liked reading this looong story. In the middle there was a bit
too much treasure hunting to my taste. (A treasure even bigger than the
biggest treasures from the past stories, which were already bigger than the
biggest treasures from the previous stories...)

If there's a problem here, it's with the previous too-big treasures, not
this one. If there is a single lost treasure that would be bigger than all
others, what could it be? Not the treasure of a single Pharaoh. Not the
treasure of a single pirate. It would have to be the treasury of a
continent-wide society that existed for centuries and was richer and more
powerful than any entire nation -- the Lost Treasure of the Knights Templar.
(If it was ever really "lost" or just looted by the Knights when they
scattered to the four corners of the world when the King chased them from
France.)

>>>But when I almost got bored, the story went into a different direction.

Oh, hey, I get *bored to death* with ALL of my stories such that I wanna
scream! Can you imagine putting every ounce of concentration and meager
effort into a *single* story every day, every minute 8 hours a day, for 3 or
4 months, non stop!? Aaaaaaaaaaggggghhhhhhhhh!!!!!!!!!

>>>>I also happened to read about the Templars on a totally different place.
They state that Philip IV had forbidden the Templars because of their sexual
outrages. I guess all history is coloured by the history teller...

Look at the Templar Seal -- notice there are *two* Knights riding the *same*
horse... in a very intimate position. What's *that* all about?
King Philip wanted the Templar wealth. So he created imaginary threats they
posed to the world for which to blame the Templars in an international court
and tried to take over their entire society for the profit of himself and
his cronies. He called them "Tempulars" because he wasn't really very bright
(he'd only managed to become King because his father had been King before
him). Yes, apparently he was a Republican.
Somebody remind France that their own unelected leader once tried to pull
this crap, too. Philip became NONking in less than two years after that. We
can but hope...

>>>>I loved the stuff about combinational maths. I don't know how the nephew
calculated 362,880 that fast, but the subject was very educational. (Aw
well, the nephew had a whole week time to calculate it: the time between
parts 2 and 3 of the story...)

During that week, the Nephew apparently consulted DCML member
statisticalmathmastermind Nils Lid Hjort, like I did, for that calculation.

By the way... and I don't want to get out my old college physics and
thermodynamics and mechanics of deformable solids textbooks... or my old
slide rule... (actually, it might be fun to get out the old slide rule and
show it to a twenty-something engineering student and try to convince him
what it is and what we did before pocket calculators... he'd split a
gut)(then I could put it on eBay!)...
But all this talk about a yo-yo effect of a Duck falling through the center
of the earth seems to be ignoring something very important... actually LOTS
of very important things, those being vectors of force in an infinite number
of directions other than straight up and straight down. You're all thinking
very one-dimensionally. Forget the idea of how this would need to be done in
a vacuum due to air friction (I dealt with a falling body pulled by gravity
into the atmosphere in "The Duck Who Fell to Earth"), forget about air
pressure, forget everything else...
What about the fact that a body at or anywhere near the earth's core is NOT
only being pulled straight toward the exact center of the earth until it
crosses that precise point. It's also being pulled UP as more and more of
the earth's mass is above it. But get this -- it's also being pulled equally
hard in ALL directions to the right, left, front, back, etc., etc. I should
think that all of these forces would be like puttin' the *brakes* on the
falling Duck (or whatever), and there would likely be no yo-yo effect at
all... the Duck would fall slower and slower and finally just drift very
slowly to the center of the earth's gravitational mass, being pulled in
every possible direction. That's why a Duck at the earth's core would be
weightless... NOT because it's being pulled equally up and down... but
because it's being pulled every whichaway. (Sorry to show off and use my old
engineering terms like "every whichaway" on you laymen, but I am, after all,
a civil engineer in my former life.)

> From: Cord Wiljes [mailto:cord at wiljes.de]
> Subject: Yo Yo it is
> Forces in all directions except up and down cancel out because
> the opposing forces are of equal value - as masses on both sides
> are equal. Only in the vertical direction is the distribution of
> mass not equal (except at the center of the earth). So the only
> force which can be measured is the gravitational force pointing
> down.
> At the center of the earth there the acceleration has dropped to
> Zero - but by then you have gained the speed of 30,000 km/h.
> If there is
> no friction (in vacuum) then this YoYo effect will continue until
> the end of the universe.

I can understand the fun of trying to figure this out (it's the sort of fun
I base lots of my stories on)... but why is everyone considering it only in
a vacuum environment? Because we're referring to people doing the falling
and they would be, as I've said, squished by air pressure? Why omit air
friction from the equation? Consider a falling bowling ball. (But even it
would be squished to dust by the titanic air pressure of an 8000 mile tall
pile of air!)

> To sum it up:
> Hole only to the center of the earth: duck would crash
> Hole all the way to the other side: duck would yoyo

But only in a vacuum which is impossible if the hole is open at the ends.
Wait. I'll just test these theories with a practical demonstration. I'll get
a shovel. Be back shortly.

CORD:

> Theresa wrote:
>
>>It's not 7910 km/s it's m/s! of course... units...
>
>
> Soren wrote:
>
>>The velocity at the center of Earth is about 8134 m/s
>>(i.e. 2260 km/h)
>
>
> 8134 m/s = 29.282 km/h

Yes, you're right. Just typical (for me) to make mistakes with simple
things. Luckily I didn't use km/h during my simulation. However, as a
silly excuse, your period also confused me. Normally you would use a
comma, e.g. 29,282 km/h.

S?ren