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Topic: EASY: Anchor (Read 9210 times) |
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Kozo Morimoto
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Given the parameter of the riddle:
I assume that the total mass of the floating object at the beginning is M1 + M2. I assume that the 'person' who throws the anchor overboard is massless (or you can assume that the mass of the boat M1 includes the mass of the person but not that of the anchor M2)
Using basic physics, the mass of the water displaced by any floating obejct is equal to the mass of the floating object. So the mass of the water displaced is equal to M1 + M2 at the beginning. Now we separate M1 and M2, but ASSUME that the anchor is somehow floating (with aid of balloons or whatever). The mass of water displaced is still M1 + M2 and no change from before thus no change in height of water.
But we KNOW from the riddle that the anchor sinks. When an object sinks, its only displacing the same VOLUME of water as its own volume. From the fact that the anchor sinks, its DENSER (higher mass per equal volume) than water so the mass of water that now displaces is less than when it was floating. And since mass is proportional to volume...
h' (the new lake level) < h (original lake level)
So the water level drops when the anchor is thrown overboard and sunk to the bottom of the lake.
They ask a similar quetion like:
A scuba diver is on a raft in the middle of a swimming pool. The swimming pool is filled up so that its absolutely full. Does the pool overflow when the diver dives to the bottom of the pool? (ignore the splash and ripples and stuff).
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BigBadBert
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h=h' and here's why:
Assumption: M1 includes the weight of the person.
The water being displaced in is M1+M2, the ship with the anchor on it. But the amount of water being displaced is M1+M2, just in different places. By throwing the anchor overboard, the boat becomes more bouyant and rises as it loses the anchor.
To see this in action, fill a glass with water. Put a nail into a cork and place it in the glass. Mark the water level. Now separate the cork and nail and put both back in the water (the same effect as dropping anchor). The water level will stay the same.
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jmlyle
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Re: EASY: Anchor
« Reply #2 on: Jul 29th, 2002, 11:50am » |
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I'll try a common sense approach (which is almost assuredly a bad idea):
The person is not important, in my opinion. This would be the same with NO person onboard (assuming some elaborate remote controlled anchor tossing device, which is functionally part of the boat).
The boat floates with the anchor in it. (M1 + M2) in water h. Assume the lake is a bathtub, the boat is a Dixie cup, and the anchor is a tiny bit of a white dwarf.
The dixie cup displaces a volume of water equal to the volume of the Dixie cup (the white dwarf anchor is just heavy enough to keep the Dixie cup's rim just above water level). The elaborate apparatus is triggered, and the white dwarf is hurled over the side, into the tub. The volume of the anchor is only a few cubic nanometers. The cup then surges upwards to float virtually on top of the water. Now the displaced water is equal to a tiny bit of the volume of the Dixie cup, plus an almost infinetesimal amount from the anchor.
V1=Volume of Dixie cup below water at beginning.
V2=Volume of Dixie cup below water after ejecting anchor.
V3=Volume of anchor
The amount of water displaced by the Dixie cup after the auto-hurl would be the V1-V2. But the water is now also displaced by V3. So total water displacement afterwards is (V1-V2)+V3.
SA= Surface Area of bathtub
The water level would be reduced by the height of the volume of water displaced (V1-V2+V3)/SA
So, h'=h-((V1-V2+V3)/SA).
But this is probably insane.
--jmlyle
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Kozo Morimoto
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"To see this in action, fill a glass with water. Put a nail into a cork and place it in the glass. Mark the water level. Now separate the cork and nail and put both back in the water (the same effect as dropping anchor)."
Yes, if you actually do this, you will find that the water level will indeed drop. You are getting confused with the term 'amount' of water. Are you referring to 'volume' or 'mass'?
In your example, the cork will float slightly deeper in the water at the beginning of the experiment. At the end, it will float slightly higher, thus displacing less water, thus water level drop.
We may have to use hard numbers...
Boat is 11kg.
Anchor is 3kg. (I like to use 'odd' numbers with examples 'cause sometimes using 'round' numbers gets too confusing)
So the object, boat+anchor (with mass 14kg) is floating. This means that it is displacing 14kg of water, which is 14L. (1kg of water = 1L of volume)
Now, lets say the anchor is twice as dense as water (otherwise it won't sink), so its volume is 1.5L. (1kg of anchor only takes up 0.5L of volume)
So when the anchor is overboard and not longer floating, the total water displacement is:
11kg from the boat -> 11L
and 1.5L from the anchor
so in total, 12.5L is displaced, which is less than 14L originally. So the water level drops.
I used arbitrary numbers in the example, but it will work with any number as long as the anchor is denser than water (otherwise it will float and negate the riddle)
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Coren
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Re: EASY: Anchor
« Reply #4 on: Aug 6th, 2002, 11:09pm » |
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The issue is if the anchor sinks and lands on the bottom of the lake, then not all of it's mass needs to be displaced anymore (some of it is taken up by the container).
However, if the anchor stayed attached to the boat and didn't touch the bottom, then you would just be moving mass around and the water level wouldn't change.
I'm not sure exactly how it works if the anchor wasn't connected to the boat but wasn't heavy enough to reach the bottom (which is the case of a diver jumping off of a raft).
Although I think I would agree that the water level would decrease.
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Kozo Morimoto
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The last point you made is an interesting one...
So you want to know what the change in the water level would be if the object (ie anchor) had equal density as water? Say you threw out a bucket of water overboard? (it doesn't float and it doesn't sink)
The water level will remain the same.
If the object had less density than water (say the boat was attached to helium balloons) then the water level will rise.
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Coren
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Re: EASY: Anchor
« Reply #6 on: Aug 7th, 2002, 10:38am » |
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Actually if the object was less dense than water, it would stay on top, and again the water level wouldn't change. I think. For instance, if you had a piece of styrofoam on the boat and you dropped it in the water the water level would stay the same.
It all has to do with how much mass is beind displaced. If an object touches the bottom, then the water displaces less mass and the level drops. If the object doesn't touch the bottom the water has to displace the same amount of mass and doesn't change.
Another interesting observation is that as long as an object is falling, then not all of it's mass is being displaced so it counts as if it were on the bottom.
One more: If the object can change it's density (a submarine for instance) the water level will go down whenever the object dives, and go up whenever the object climbs. At least I think so. Because when climbing the effective mass is the total mass of the object + the "mass" for lifting the object.
I guess it really should be said that to be more precise we should use the term force (which is caused by gravity and mass), then it becomes much clearer why a sinking or raising object changes how much water is displaced.
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S. Owen
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Re: EASY: Anchor
« Reply #7 on: Aug 8th, 2002, 10:51am » |
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on Aug 7th, 2002, 10:38am, Coren wrote:Actually if the object was less dense than water, it would stay on top, and again the water level wouldn't change. I think. For instance, if you had a piece of styrofoam on the boat and you dropped it in the water the water level would stay the same.
It all has to do with how much mass is beind displaced. If an object touches the bottom, then the water displaces less mass and the level drops. If the object doesn't touch the bottom the water has to displace the same amount of mass and doesn't change.
Another interesting observation is that as long as an object is falling, then not all of it's mass is being displaced so it counts as if it were on the bottom.
...
I guess it really should be said that to be more precise we should use the term force (which is caused by gravity and mass), then it becomes much clearer why a sinking or raising object changes how much water is displaced. |
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It doesn't depend on whether the anchor touches bottom or not, or on force. Also, volume is displaced, not mass.
An object displaces a volume of water that is equal to its own mass - a 1g piece of styrofoam displaces a volume of water whose mass is 1g. Unless it sinks...
If the object is denser than water, and sinks, then the object displaces a volume of water equal to its own volume. A 1 liter Coke bottle full of lead (weighing 10kg) will sink and displace 1 liter of water (not a volume of water whose mass is 10kg)
But since this bottle is denser than water, its volume of 1L must be less than the volume of 10kg of water, of course. So this thing, submerged in the water (doesn't matter whether it is at the bottom or not) displaces less water than, say, a large 10kg styrofoam cup floating on the surface.
So before we have a floating ship weighing M1+M2. Afterwards we have a floating ship weighing M1 and a sunken anchor weighing M2. That anchor displaces less water when in the water, so the water level drops,
If the anchor were made of styrofoam, and floated, then indeed the water level would stay the same.
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Coren
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Re: EASY: Anchor
« Reply #8 on: Aug 8th, 2002, 6:22pm » |
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I realize that the first order of the system is clearly defined my mass and/or volume. However, if you look at WHY the object sinks or floats, if you extend the reaction to force, you can see that it is the same system, but you can use one parameter (force) instead of using two parameters (mass & volume).
Yes, an object on the surface displaces an equal amount of mass as long as it remains on the surface. However, this is equivalent to saying that the force produced on the object (mg) you have an equivalent amount of water moved (mg).
When the object is submerged, it is only because it's density is greater than that of water. So water can no longer provide enough pressure (force / unit area) to push back upwards.
Have you noticed that if you hold an object under water it "weighs" less? That is a force. Water innately "pushes back up" on the object. The rate that the object sinks will be determined by it's density vs. that of water. (Or effectively, the relative force between the water being pulled down and the object being pulled down.)
In fact, I would venture that one should consider the system as a system of force, because in the absence of a gravitational field water will no longer carry anything. So there will no longer be an amount of water displaced equal in mass to the mass of the object. It will only depend on how much volume the object is inside of the water. (No gravity, grab a piece of lead and put it into the water, it can go 1/2 way, 1/4 way, all the way, whatever you want, and it will stay there if you neglect the force of surface tension of the water.)
I'm not negating the idea of mass & volume displaced by an object. I'm just pointing out that you can also view it as a displacement by "force" and in one sense it is a simpler system.
I suppose my idea of how much "mass" is being displaced should have been explained better. I can make the statement how much mass is being displaced because the volume is linked to density, etc. Just a novel way to look at it, that probably wasn't explained very well. I hope I have done a better job this time.
If you add in the dynamic effect that the object feels a specific force when submerged (and moves at a specific rate accordingly), you can kind of see how it could be thought of as the weight that is being displaced and not just the mass. I think saying weight would have been more appropriate than mass.
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thx1138
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Now that we have all these theories for explaining things, lets see how well they treat a new question:
Suppose the anchor simply dangles off the boat, unable to reach the bottom of the lake? (ignore the chain)
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towr
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Re: EASY: Anchor
« Reply #10 on: Dec 15th, 2002, 8:37am » |
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If the anchor dangles the waterdisplacement will be the same as when the anchor was still in the boat. It still pulls the boat down, but less than when it's in the boat. On the otherhand it's volume now also displaces water, in the same amount it weighs less on the boat.
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| « Last Edit: Dec 15th, 2002, 8:38am by towr » |
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Kevin
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If the anchor touchs the bottom of the lake and the chain is pulling the boat into the water, the anchor is suspended at the bottom of the lake; in this case, the h'=h.
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WOUF
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Imagine a plastic cup floating in a sink with a piece of very dense metal (say uranium) which makes it sink all the way down to the rim. (see ...hum... diagram)
_________\ /___________
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L__0___|
Now remove the piece of uranium and throw it in the sink. The cup will rise up to the surface, and the uranium sink. Obviously the level of the sink will drop.
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| « Last Edit: Apr 12th, 2004, 3:35pm by Icarus » |
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Guest
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Is it ever possible for something to be thrown out of a boat and cause h' > h?
I'm thinking not since that means that the volume of the item (anchor) is > than it's displacement based on mass. A 1kg styrofoam anchor with volume of 10L would displace 1L of water. If we throw it overboard it will float since it's density is less than the water, so h = h'.
If we weight it to force submergence, then the dispalcement in the boat (before tossing overboard) would be greater than dispalcement from it's volume, so we're back to where we started, h'<h.
Is this right? Is there nothing you can throw overboard to make h'>h?
Thanks for your help. There is a reason I am in the easy section.
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towr
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Re: EASY: Anchor
« Reply #14 on: Apr 12th, 2004, 1:26pm » |
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under normal circumstances I think you're right.
Of course, suppose the lake is rather small, the water is 4 degrees Celcius, and the object you throw overboard has a slightly higher density that water but has enough energy to warm up the lake (an electric heater inside a solid sphere or something). The water will expand due to the heat from the object, which means the water level rises. (Of course it shouldn't be so hot that it boils away the water)
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Avrom Roy-Faderman
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If the thing you throw out of the boat is lighter than *air* (e.g., helium baloons), the overall water level will rise. The balloons will no longer be pulling the boat out of the water, and since they will float off into the sky, they will not displace any water on their own.
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Rejeev
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Re: EASY: Anchor
« Reply #16 on: Oct 7th, 2004, 11:42pm » |
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Heigth h' is lesser than h.
Let us define a parameter effective volume of the water system (effective volume (EV) means the volume of water + volume displaced by ship, and volume of any objects immersed in water, so that height h = EV/Area).
Initially EV = volume of water in litres (W) + weight of ship in kilograms (S) + weight of anchor (A)
Finally EV1 = W + S + volume of anchor in litres.
Since volume in litres of anchor is less than weight in kg for any object which sinks in water, EV1 is less and hence h'.
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Mayank
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Re: EASY: Anchor
« Reply #17 on: Oct 11th, 2004, 12:10pm » |
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Wow....Such a simple puzzle and so many explanations.
There are two things to remember
1. Water displaced by the boat.
2. Water displaced by the Anchor.
The water displaced by the boat is a constant here.
When the anchor is in the boat, its Mass acts as a force displacing water.
When the Anchor is in the water, its volume causes the diplacement.
Now Mass = Volume * Density
Volume for the anchor here is constant irrespective of the material it is made up off
So more density of Anchor material than Water, more pressure it exerts on the boat and more water the boat displaces.
Similarly, lesser the density of anchor than water, less pressure it exerts on the boat, less wate the boat displaces.
Since Anchors material density >>>> water (else it wont serve its purpose), it will displace more water when it is in the boat. Hence, water level will go down when it is put in water.
As per when the Anchor is tied to the boat, the water level will still go down but lesser. Since now the mass and volume both are acting to displace water. When in Boat, the forces acting on the boat are acting on a much bigger surface area relative to the anchor. When in water, the surface area reduces drastically for the anchor. So the forces acting on the Anchor to put it out of water will be lesser. Hence water level will still go down.
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TimK
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Re: EASY: Anchor
« Reply #18 on: Oct 11th, 2004, 12:38pm » |
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Mayank,
If the anchor is tied to the boat by a rope that doesn't allow it to reach the bottom, then the water level should not change at all - the anchor is still displacing its weight in water.
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Mayank
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Re: EASY: Anchor
« Reply #19 on: Oct 12th, 2004, 9:43am » |
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Hey TimK,
I agree that the anchor its displacing its 'weight' in the water, but notice that the effective weight of the anchor is much less in the water than its weight outside (as the gravity has to counter several new forces to pull it down in water). So when outside water it will apply more force and displace more water; inside water it will displace lesser water (using weight alone). Now depending on the the weight and volume it may displace more, less or no difference at all. In normal cases for an anchor it will displace less water.
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TimK
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Re: EASY: Anchor
« Reply #20 on: Oct 12th, 2004, 12:46pm » |
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Mayank,
I disagree  When the anchor is in the boat, it displaces its weight in water. When it's in the water, it displaces its volume plus its 'water weight'. The water weight is exactly its weight minus the weight of the water its volume displaces. So the level of the water stays exactly the same.
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Mayank
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Re: EASY: Anchor
« Reply #21 on: Oct 12th, 2004, 1:18pm » |
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Damn, I have forgotten my basic physics.. 
Is the buoyancy upthrust relate to the volume alone or does it relate to surface area and volume. (weight included by default). That will answer the question.
Also, we are assuming still water. In case of ocean, there will be ocean underwater currents, pressure differences and what not altering the effective 'weight' of the Anchor. So answering this will require a lot of assumptions or clarifications.
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towr
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Re: EASY: Anchor
« Reply #22 on: Oct 12th, 2004, 1:30pm » |
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on Oct 12th, 2004, 1:18pm, Mayank wrote:| Is the buoyancy upthrust relate to the volume alone or does it relate to surface area and volume. |
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Volume only.
(Well, I suppose, actually it would be
[int]Surface (upward component of) pressure * d surface
But for most intends and purposes that's in direct proportion to volume)
Quote:| Also, we are assuming still water. In case of ocean, there will be ocean underwater currents, pressure differences and what not altering the effective 'weight' of the Anchor. So answering this will require a lot of assumptions or clarifications. |
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I think we were dealing with a pond, not an ocean. But, true enough, it could make a difference. But that would make any puzzle of this type unanswerable.
So let's assume a still body of water of uniform density.
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| « Last Edit: Oct 12th, 2004, 1:31pm by towr » |
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