<,> or =
 

A woman has gone fishing in a boat on a lake. She gets where she wants to be and throws out the anchor (do you use anchors in lakes? this one does). Is the water level of the lake higher, lower or the same as it was just before she threw the anchor? (the difference (if any) will be tiny, assume you can measure it)

Surprisingly and independently there is an iceberg floating in the lake which melts to nothing. Is the water level higher, lower or the same than before (assume all the ice in the 'berg becomes water in the lake and it's just the ice in the berg - no condensation from the atmosphere - it's not that sort of puzzle).


Graeme

Re: <,> or =
 

[quote="graeme"]A woman has gone fishing in a boat on a lake. She gets where she wants to be and throws out the anchor (do you use anchors in lakes? this one does). Is the water level of the lake higher, lower or the same as it was just before she threw the anchor? (the difference (if any) will be tiny, assume you can measure it)

Graeme[/quote]

It goes up and down repeatedly in a pattern that spreads out from the boat until the pattern reaches the side of the boat and the edge of the lake, whereupon it still goes up and down though in a markedly less simple pattern.

:)


Paul

P.S., I do know the "correct" answers to both original questions. It's all to do with the relationships between buoyancy, mass, volume and density. Once again, it's difficult to give clues without giving away the solution. Pity Archimedes isn't around to give a hand.

Re: <,> or =
 

[quote="graeme"]Surprisingly and independently there is an iceberg floating in the lake which melts to nothing. Is the water level higher, lower or the same than before (assume all the ice in the 'berg becomes water in the lake and it's just the ice in the berg - no condensation from the atmosphere - it's not that sort of puzzle).


Graeme[/quote]

A rather harder question: why does ice float in water, and I want an explanation with more detail than "because it's less dense".

An even harder question: why is liquid water less dense at 1C and 10C than it is at 4C?

Note, neither of these are puzzles in the sense we use the word here. Finding out the answer should teach you something valuable about chemistry and physics.


Paul

The anchor in the boat displaces an amount of water equal to its weight. When you toss it into the water it displaces an amount equal to its volume. My daughter was solving problems like that last year in Algebra and is getting another dose this year with chemistry.

Along the line of comments above, the temperature expansion coefficient of a material is usually non-linear. There are some specific temperatures for example at which water dramatically changes volume (and density) for a very small temperature change. There are quite a few materials with a negative coefficient; as the temperature rises, the volume shrinks or stays static through some temperature range.

I have my guesses. We assume that the anchor was in the boat before she threw it overbord, right?

I suppose this is also the sort of puzzle in which it is assumed that the anchor is attached to the boat via rope or chain of some sort and that the rope or chain of some sort is long enough to have some slack after the anchor sinks all the way to the bottom of the lake.

The water level will change as long as the anchor completely submerges but doesn't capsize the boat. It doesn't have to drop to the bottom.

If she is in a large boat, and drops the anchor from, say, 6 feet above the water - as soon as she lets go of the anchor, the water will lower by the volume of water weighing the same as the anchor. And as soon as the anchor drops in completely, the water will obviously go back up by the volume of the anchor.

So if she tosses the anchor onto the bank, or hands it to someone in a helicopter, the drop in water level will be greater. (But it is no longer a 'trick' question.)

There was a question on one of those aptitude tests, I don't remember the specifics, so I won't try to pose it here.
(besides, it's the same as this one except you have to go thru the bother of crunching the density/volume numbers.)
NOTE: Please do not encourage the original author of this evil construct by actually trying to solve it. This is posted as a warning only.

In summary ...

A barge hauling heavy sealed barrels is damaged while entering a lock. It starts to slowly sink at an increasing rate f(t). The lock is also damaged so the barge is trapped. The lock leaks so the water level starts to drop at decreasing rate g(t) (but it won't drain completely). Unloading barrels (with a max. rate r) raises the barge by some h1, but lowers the water by some h2. The task is to save as many barrels as possible while keeping the deck of the barge within +/-6 feet of the dock to allow unloading.

If that doesn't sound diabolical enough, :devil: a bonus solution involved dumping barrels over the side when the barge got too low - thus bypassing the max unload rate to raise the barge faster, and also negating some of the water drop.
:sick: :yucky: :sick:

The answers
 

Before this gets too side tracked here are the answers.

Strangely, though there was a reasonable amount of discussion no-one actually gave the answer to either question (or PM'd it). Obviously ther answers must have been obvious.

1/ The water level drops because of the reasons given by fusion_power above (using almost the same wording I would have used)

2/ The water level remains the same. The iceberg displaces an amount of water equal to its mass (as the anchor did) but when it melts it displaces an amount of water equal to the volume of water it has melted into. This new water has the same density as the lake (becasue it's now part of the lake) so the volume remains unchanged and the level of the lake is unchanged.

Graeme

[quote="Maybeso"]The water level will change as long as the anchor completely submerges but doesn't capsize the boat. It doesn't have to drop to the bottom.[/quote]

I think this is wrong. Interesting transitional effects were described, but after the transition effects, if the anchor doesn't reach bottom it becomes part of the weight of the boat again. It pulls the boat down into the water, displacing its weight, not its volume.

Re: The answers
 

[quote="graeme"]2/ The water level remains the same. The iceberg displaces an amount of water equal to its mass (as the anchor did) but when it melts it displaces an amount of water equal to the volume of water it has melted into. This new water has the same density as the lake (becasue it's now part of the lake) so the volume remains unchanged and the level of the lake is unchanged.

Graeme[/quote]

Except that I gave a hint to suggest that the real answer is much more complicated. You assume that the liquid in lake remains at the same temperature after melting the ice as before. I maintain that it will not. The latent heat of fusion of ice is significant. By hypothesis the lake is warmer than 0C (the ice melts after all) and so the liquid cools. Whether it expands, contracts or remains the same value depends very much on the initial conditions.

My wittering about the strange behaviour of the density of ice and its temperature dependency in liquid water was not entirely irrelevant. One could argue that the anchor-dropping part of the puzzle was a purely mechanical transformation and that temperature changes are irrelevant. My whimsical description of the dynamical mechanical processes made that assumption.

Melting, though, is very much a thermodynamical process and the ice/water system is a very unusual one. If you don't know how and why it's unusual you will find out when you do the research in order to answer my follow-up questions.

Note that I ignored all the other environmental considerations (solar heating, evaporation, etc) in both parts of the question. Those may be interesting nits to pick in their own right, but I find the thermodynamics of the water phase diagram vastly more interesting.

Paul