Densities & volumes

[james ozzie]

Not directly aviation but most aviation people seem clued up on densities. Here is a simple brain teaser with a not-so-simple answer. I am not sure of the answer myself but would appreciate a reasoned logical explanation from some smart thinker out there.
" A boat is floating in a pond. Someone throws an anchor over the side. Does the water level in the pond go up, down or stay the same?"
(yes, the effects are negligible, we all know that, but what is the theoretical answer?)

[HotDog]

Displacement of a vessel is determined by its length, width and draft which is a function of load (weight). Therefore if you throw the anchor overboard, you reduce the load, ergo the draft. The vessel rises, water level drops. I think I've got that right?

[Loose rivets]

If the anchor was lying on the bottom, gravity would no longer be playing a part in the equation...not for that component anyway. The anchor would however displace a given volume of water, causing the level to rise.

Now, due to the weight of the anchor having been deducted from the boat, the displacement will be less, causing the water level to decrease.

Without knowing the ratio of weight to volume, we can't even start to calculate the net effect. :confused:

[Nubboy]

Have a go though.

If theanchor sinks, then it must have an SD of greater than that of the water in the lake (1 for practical purposes).
Therefore the mass of the anchor divided by its volume must be more than 1, that is it will displace less volume of water when immersed, than water displaced to support it when in the boat.

I think that the water level will go down. I hope.:uhoh:

[vapilot2004]

I think it depends upon the relative size of the boat and pond.

Anchor density and weight must also play into the above equation.

[PIGDOG]

Yup! I agree with the above posts. The water level will appear to drop slightly. I find using hypothetical numbers can help.

Here's some I made earlier:

Boat: weight = 500kg
Anchor: weight = 6kg, volume = 1000cm3
Water: 1kg = 1000cm3 (not made up)

The weight of the boat will displace an equal amount of water. So the boat with the anchor on board will displace 506kg of water. When the anchor is thrown into the water, and isn't being supported by the boat any more, the boat will only displace 500kg of water.
However, the volume of the anchor will displace some water now. 1000cm3 of anchor in; 1000cm3 of a water rise. (Not much in a small pond, but the displacement of a boat in a body of water it has any busines in being in, will be negligible anyway.)

So the overall drop in my little experiment is by 5000cm3. Basically, it's the difference between the density of the anchor and the density of water.

Of course, the anchor doesn't have a chain in my little world, and I hope to god we're not talking about salt water!!!!! ;)

[james ozzie]

Some erudite Archimedeans out there!
One part of the riddle was left out - did the anchor reach the bottom of the pond or not? Rivets picked up on this in his posting, where he assumed the anchor was on the bottom i.e. rope slack. I think this gives two solutions but it is beyond me to reason out.
This question was apparently used by one of the GEC managers in the 1960s when interviewing young prospective engineers for jobs - a rather tough one to encounter at a job interview.

[john_tullamarine]

... a rather tough one to encounter at a job interview

.. only for the folks who hadn't done much fluid mech and aerody ...

[HotDog]

Pretty poor seamanship to drop anchor that won't reach bottom!

[ok1]

Another way to think about it:

Take the boat and anchor together as one object. When the anchor is on the boat or if it doesn't reach the bottom, the the whole weight of the 'object' will be supported by water, so there'll be no difference in water level between these 2 cases. When the anchor reaches the bottom, part of the weight of the 'object' will be be supported by the ground, so it will have a smaller displacement hence the water level will decrease.

[matt_hooks]

I think we have sorted this one out. All you have to remember is that the displacement of a floating objects is defined by its mass, whereas the displacement of an object which is completely submerged is governed by it's volume.

Ignore the mass of the boat, and think of just the anchor.

When in the boat it is effectively floating. Hence the displacement is the mass.

Now assuming it sinks (and it wouldn't be much use as an anchor if it didn't) then it has a density higher than that of water.

Thus when it is submerged it will displace a smaller volume of water. Hence the water level should fall, albeit by a small amount.

Simple when you know how! :D

[forget]

Hmmm. What about the volume of the anchor rope - now in the water?

[ok1]

If my explanation is right, then it doesn't matter. As long as the anchor doesn't touch the bottom, the whole thing is supported by water, so it has the same displacement regardless of the shape. When the anchor rests on the bottom, A PART of the weight of the anchor (and perhaps of the rope as well, depending on its density) will push on the ground and won't be supported by water anymore.
Just done my fluids exam :)

[forget]

Now now Rainboe. Just because you forgot about the bloody rope don't get uppity with me. :p

[enicalyth]

And to think that old Greek solved it all in his bath! Anything in the boat is floating and therefore has to displace a volume of water whose upthrust equals the weight of the boat and everything in it. Water is 62.4lbs/cu ft fresh and 64lbs/cu ft salty whereas stone, iron etc are much more dense by a factor of two to three. Once anything is in the water whether it is touching bottom or not it merely displaces its own volume. Thus the floating anchor displaces more water than the sunken anchor. M'lud chuck the anchor overboard, less water is displaced overall and therefore the water level falls. Whether or not the anchor touches bottom is irrelevant.

[Jaguar Pilot]

The water level will in fact go up, since the anchor is more dense that water. Basic Archimedes stuff. Before the anchor was dropped the floating boat experienced an upthrust equal to the weight of water displaced.

And if you all like this sort of thing, try this:

A jet transport is in the cruise. Autothrottle and autopilot are not engaged and the aeroplane is in straight and level flight at constant speed and trimmed.

This aeroplane has no seats installed. Instead it has 1000 bird perches fixed to the floor and on each perch sits a parrot. A human being claps hands loudly and all 1000 parrots get airborne and fly around the cabin.

Question:

Does the aeroplane stay level, climb, or descend?

This question was part of one of my Central Flying School oral examinations.

[PIGDOG]

Jaguar pilot said:

The water level will in fact go up, since the anchor is more dense that water. Basic Archimedes stuff. Before the anchor was dropped the floating boat experienced an upthrust equal to the weight of water displaced.


Come on lad. The water level will drop. It's been unanimous since you got here. Either you didn't read any of the above posts or you're just a stirrer!!:=


As for the birds flying in a plane (and this has been done before) the plane wouldn't stay straight and level. 'Every action...' and all that. The act of the birds jumping into the air at the same time (they can't fly from a stand-still, just like us :}) will cause the plane to momentarily weigh more. Try standing on a scales. It'll read your weight. Now try jumping from the scales. The needle will swing massively. Of course if the plane was trimmed properly it should get back to straight and level after everything settled down, I should think.

[PIGDOG]

...and why are they parrots? :suspect:

[Jaguar Pilot]

Any bird will do - I just used parrots. Budgies or eagles will do......

Now think again about your answer.....

JP

[PIGDOG]

How about you use some reason to show me the error of my ways.


Remember now, we're not talking about the water level on the side of the boat. We're talking about the water level in the pond as a whole!!!

[Jaguar Pilot]

OK Pigdog.

You have rumbled me as a "stirrer".

I just like to see what happens with spoofs.

I watch the "parrots" bit with interest and will submit the answer when there have been more replies.

In the meantime, here's another:

The equation of continuity and Bernoullis' Theorem tell us that in incompressible fluid flow an increase in velocity results in a decrease in pressure and vice versa.

Consider a windsock where the entrance is of larger area than the exit.
This represents a convergent duct. In a convergent duct (such as the first part of a carburettor venturi) there is an increase in velocity and a drop in pressure.

Question:

Why therefore does the windsock not collapse.....?

JP;)

[PIGDOG]

Ok, I'll have a go, but I'm not 100% sure.

I would guess that the dynamic pressure of the wind hitting the inside of the sock is greater than the drop in static pressure.

Like when an aircraft flies upside down. The increase in pressure would still be produced on the underside of the wing, and the decrease would be on the topside. The only way level flight can be maintained is with a greater angle of attack. (By this I mean the pilot would push the control column forward.) The greater AoA uses dynamic pressure to produce lift.

Does that pass your test Mr. Jaguar? As I said, I'm not 100% sure, but... :8

[Jaguar Pilot]

Not a bad effort Pigdog.

Most people are wrong-footed by the word "venturi".

A venturi can only be formed by a tube with rigid construction.
A windsock is not rigid and therefore the pressure inside is higher than the static outside, so it is inflated.

JP

[PIGDOG]

I see you're an ATPL lecturer. You must really enjoy your job since you're happy to teach on bank holidays.:ok: Good for you.

Forgive me if I'm a little bit happy with myself... I haven't even started my ATPLs yet, and probably won't for another six months.

Our paths may yet cross on this holiest of Forums.:)

[Jaguar Pilot]

Pigdog:

I do not teach on bank holidays - I just sometimes get bored.
I have in the past also worked for the CAA on the question database.

Our paths will undoubtedly cross again on this forum, and if you go to the right school they will cross in a classroom.

By the way, my specialist subject is Principles of Flight, but at RAF Central Flying School it was more aerodynamics. The two subjects are not the same.

Best of luck with your studies - it's a worthy profession.

JP