Situation: I have a water tight canister that is 1.5 cubic feet in
interanl volume with one end that will extend to an infinite distance.
Inside the canister I have one cubic foot of lead weighing in around
707 pounds. I carry this canister to a depth of 100 feet below the
surface of a deep lake. I hold the canister so that the weight of the
lead causes the telescoping end of the canister to extend.

Question 1: How far will the canister extend?
Question 2: What is the formula and explanation of what is happening?

Please ask for clarifications if needed

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Request for Question Clarification by mathtalk-ga on 22 Jul 2003 22:58 PDT

As the canister end "telescopes", does it retain a cylindrical shape
with equal circular cross section?  I think to say how far the
canister will extend requires knowledge of its original length
(although one can say by what factor the length would be extended).

regards, mathtalk-ga

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Request for Question Clarification by andrewxmp-ga on 23 Jul 2003 11:53 PDT

I'm usually quite good at picturing these types of theoretical
situations, and I'm pretty good with general physics, but this is just
one of the most confusing questions I've ever heard.....could we get a
diagram of some sort?  Am I the only one with nfc whats going on?

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Clarification of Question by myxlplix-ga on 23 Jul 2003 15:31 PDT

I hope this clarification will help: Imagine that the cyclinder is a
telescope. When it extends, the telescoping section maintains its
shape. When I'm under the water I hold the telescope by the fat end
and let the weight of the lead extend the small end. Please disregard
the weight of the cylinder. I do want to take into consideration the
depth of the water which should be about 50 PSI at 100 feet.

For those that need the size of the original cylinder before it
extends.

radius: 3, diameter: 6
perimeter 18.85 
base: 28.274
height: 0.053
lateral area:1,
surface: 57.549
volume: 1.5

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Request for Question Clarification by mathtalk-ga on 23 Jul 2003 20:32 PDT

Your reference to a "fat end" and a "small end" of the telescoping
canister suggest that instead of modelling the extended shape as a
cylinder, a truncated cone might be more accurate.  Do you wish to
specify different diameters for the two ends?

regards, mathtalk-ga

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Clarification of Question by myxlplix-ga on 24 Jul 2003 08:22 PDT

I was using "fat" end and "small" end for illustrative purposes only.
Trying to help andrewxmp-ga visual the problem better...

If you assume stable equilibrium, constant cylinder diameter, and
negligible mass of the cylinder itself (not the contents obviously) I
think that the cylinder would telescope (remaining vertical) until the
buoyant force equalled 707 lbs. This would occur when the displacement
equals 707 lbs. If you further assume a unit height of 1 ft for the 1
ft3 original cylinder volume, the cylinder would have to extend to
about 11.3 ft.

Since the only forces we are considering are the weight of the
cylinder and the buoyant force, the cylinder should just hang there at
the depth you originally held it. When it has telescoped sufficiently
it will simply stop, because the buoyant force will counteract the
weight of the lead. However, it would not rise because the downward
force (weight) will equal the upward force (the buoyant force). That
is static equilibrium.

I don't think the depth is significant if we can ignore the stresses
due to pressure on the cylinder.

OOPS, I used 1 ft3 as the orginal volume. When you use 1.5 ft3, the
result is 7.6 ft.

I don't know a much about physics, but would like to offer a comment.

If you're holding the canister, would the cylinder stop extending when
it hits the bottom of the lake bed?  Assuming you're of average build
(and not 10 feet tall), your answer doesn't seem like it could be more
than a few feet (lets say the distance from your waist to the bottom
of the lake bed).

The assumption that I am standing on the bottom of the lake is is not accurate. :)

Sorry, I'm having a hard time picturing this.  You've stated twice
that you're holding the canister.  Where are you relative to the
bottom of the lake bed? I would imagine your feet are touching the
bottom (at the 100 foot level).  If not, wouldn't 700 pounds of lead
make you sink to the bottom?

As far as this question is concerned the lake is infinitely deep and I
am magical in that I am able to stop at only 100ft deep no matter how
much weight I am holding :). It appears that I may have figured out
the answer on my own though. The canister would extend until the water
pressure reached a high enough pressure to support 707 lbs of weight.
If I brought the canister to a depth of 100 feet then the canister
would not extend because the pressure on the bottom of the canister
would be over 40psi (.40 * 100ft) times the area of the bottom of the
canister (28.274)=1130 lbs. I would have to stop no later than 61.8
feet deep for the canister to extend. .404 * 61 * 28.274 = ~707lbs.
The bouyancy probably doesn't have anything to do with it just the
pressure caused by the depth of water.