School-Safe Puzzle Games

A Tossed Anchor: A physics based math puzzle

A fisherman rowing his boat on a very small lake throws his anchor into the water. Does the water level of the lake rise, fall, or stay the same?

You can submit your answer below in the comment section. Will reveal answers in 1-2 days, thanks.

[from The Feynman Lectures on Physics]

23 Comments to “A Tossed Anchor: A physics based math puzzle”


  1. Falwan | Profile

    Water level will fall…
    because weight effect(on top) will be more than volume effect(bottom)..


  2. bilbao | Profile

    let’s make a try: lake level will fall


    explanation:
    let’s consider water density = 1 g/cm3 and anchor’s iron density = 7.8 g/cm3


    while the anchor is on the boat, due to its weight, it displaces a volume of water 7.8 times the volume of the anchor itself


    while the anchor is in the water it displaces a volume of water equal to 1 time its own volume


    comparing both situations lake level will fall due to a decrease in displaced water (7.8 to 1)


  3. bilbao | Profile

    there’s a variation of this puzzle where you must explain what happens with the water level when an ice cube in a glass of water melts


    Will reveal answers in 1-2 days :-)


  4. Eliot | Profile

    water level falls, assuming the anchor is made of a dense metal.


  5. deadskin | Profile

    The boat, fisherman, and anchor all have a total net weight that is equal to the weight of the water displaced by them while floating on the lakes surface.


    When the fisherman throws the anchor into the water we assume it is denser that the water and sinks.


    Assuming the anchor is attached via rope to the boat we can see one of two things can happen.


    1) The anchor hangs at the bottom or the rope because it does not reach the bottom of the lake and therefore create the same downward force in combination with the boat and fisherman to displace the same amount of water and the water level remains the same.


    2) The anchor has enough rope to settle on the bottom, since the anchor is denser than the surrounding water and may not at all times be providing its full downward force on the boat ( unless there is a strong cross wind ) the water level will rise slightly since there is now less water being displaced then in the previous state.


    So it depends


  6. suineg | PUZZLE MASTER | Profile

    For me the answer seems to be that the level of the lake FALL. Explanation( I guess):
    The anchor inside the boat trasladate his weight to the center of mass of the boat, so has a wider and longest surface to make contact with the water so the efect in level of the water of the lake is greater with the anchor “inside the boat”, when you trow the anchor, the anchor has less surface to make contact with the water getting less resistance.
    With an example you trow a marble inside a pool or a rectangular pice of wood that weight the same but has a wider and longest surface float getting more resistance from the water, on the contrary the marble get to the botton of the pool and has no longer resistance from the water, cool.


  7. deadskin | Profile

    Silly me it should fall since less water is displaced in outcome #2 , which is the more likely scenario since anchors work best when resting on the bottom of the body of water.


  8. michaelc | Profile

    An easier question would be, if the anchor is thrown in from the bank, does the water rise? To which the answer would be obvious, yes it does. The anchor would displace the water being more dense than the water falling to the bottom, and the water would rise ever so slightly.


    The anchor being on the boat, would displace the water equal in mass to the boat and all of it’s contents. So the level of the water, has already “risen” eqivalent to this mass before the anchor is thrown.


    Once the anchor is thrown, the boat’s total mass is less, and therefore displaces less water making the water level less than before. The boat displaces it’s weight in equal amounts to the weight of water, so the level rises/falls by the weight of the boat, but by the density of the water. The anchor entering the water would displace the volume of water equivalent to the volume of the anchor. But the density of the anchor is more than that of the water.


    So, it falls precisely by the amount of the anchor’s volume equivalent in the density of the water minus the anchor’s volume.


    Nothing you could measure with a yard stick, but theoretically. :)


  9. Mehmet Karatay | Profile

    It will stay the same. The boat has already displaced its weight in water.


    Mehmet


  10. aaronlau | Profile

    fall.


    the anchor is denser than water. the pressure exerted by the anchor while in the boat pushes up the water level higher compared to the amount of water displaced by the anchor when it is in the water.


  11. Shawn | PUZZLE GRANDMASTER | Profile

    The question is, does the displacement caused by the weight of the anchor in the boat equal the displacement caused by the volume of the anchor itself in the water.


    1. The heavier the anchor, the more water will be displaced by the boat/anchor combination while the anchor is in the boat


    2. When the anchor is in the boat, I think that it will displace water in an amount proportional to (equal to?) its weight.


    3. An iron anchor with a volume of 1 liter weighs about 7.87kg. An anchor with the same volume made of a heavier material would displace more water when in the boat, but the same amount when thrown overboard. A lighter anchor would behave conversely.


    4. The density of the anchor must be greater than the density of water, or we would be calling it a buoy!


    5. When the anchor is thrown overboard, it will displace its volume in the water. At the same time, the displacement caused by the boat will decrease.




    With 7.87kg of force being exerted by the anchor on the boat, versus 1kg of water being displaced when the anchor is thrown overboard, I would expect the water level in the pond to fall.


  12. DouggyD | Profile

    it would stay the same. the anchor is already in the boat, which is already in the water. Therefore, it would continue displacing the same amount of water whether it was inside or outside of the boat. I think…


  13. abeshua | Profile

    stays the same as the weight of the anchor already was on the water


  14. Ari | Profile

    It’s all about archimedes’ principle. The weight of the anchor while on the boat, displaces a volume of water which has a mass equal to the weight of the anchor. Water being less dense than the anchor’s material means it displaces more water than its own volume is.
    Now when the anchor hits rock bottom, it unloads the boat and only displaces its own volume of water.
    Resulting in the water level of the pond dropping.


    If the rope of the anchor is too short for it to hit the bottom, the water level I think stays the same. As the anchor displaces its volume of water, the boat feels this boyance and thus displacing the same volume less, so the systems net displacement remains the same.


  15. lograh | Profile

    Assuming that the anchor is initially contained in the boat, we know the boat was displacing water equivalent in mass to the mass of the anchor (being that’s how boats work). Picture the anchor was instead tossed to dry shore on the side of the lake. The boat would displace less water in that scenario, and thus the overall water level would drop.


    Now, when the anchor is tossed in the water, it only displaces water equivalent to its volume, which is less than the water required to match its mass (being that it is more dense than water, which is why it sinks). So if we take our anchor from dry shore and toss it back in to the lake it will now displace some water, causing the water level overall to rise, but not by as much as when the anchor was in the boat.


    So I’m going to go with the overall water level dropping ever so slightly, but probably not by an accurately measurable amount, given all the ripples caused by throwing the anchor in the water. A more important question is, how were the fish biting that day?


    Or I could be totally off.


  16. Obiwan | Profile

    I think the water level will rise (perhaps imperceptibly, though), assuming the anchor was in the boat, initially.


    The boat and anchor displace a certain mass of water, part of which is due to the weight of the anchor. When the anchor goes overboard, it then displaces a volume of water equal to its volume (rather than its weight). A useful anchor cannot be hypo- or iso-dense, compared to water–it would be hyper-dense, so that it would sink. In other words, it displaces less when in the water, than when in the boat.


    Mass versus density!


  17. joe | Profile

    I like this one, it is all about density.
    Density = Mass / Volume
    When an object is equally as dense as water it ill displace 1 gram of water per cubic centimeter of water (1 g/cm3). It will float about without sinking or floating. Less dense objects will float and displace its weight in water (so some of that object will stick out of the water). More dense objects will sink and will displace its volume of water.
    The anchor must be more dense as we know an anchor sinks to do its job properly. This means that it has a density greater than 1 gram per cm3.
    On the boat the anchor will displace that greater mass as it pushes the boat down, but when it is in the water it only displaces its volume (1 gram of water per cm3).
    Therefore on that basis the water level will be LOWER when thrown into the lake.
    Probably not the best explanation but…


  18. bizarette18 | PUZZLE MASTER | Profile

    It falls. The anchor displaces its weight in the boat and its volume in the lake.


  19. jordyf | Profile

    aha! In this scenario, the water level would, in all likelihood, LOWER; the jettisoned anchor would decrease the weight of the rowboat, causing it to rise and displace less water, causing the water level to go down. The only unknown variable would be whether the anchor pulls the boat lower into the water more than it did inside the boat.


  20. onfire | Profile

    Depends what the anchor is made of. (-;
    If it were, say, water, then the weight in the boat would already be pushing out (&up) as much water as it weighs. If it were really light (meaning not-dense – think of a giant balloon of an anchor), it will take up more volume than it displaced, if it’s denser than water (likely as it’s an anchor) then it will take up less room, thus lower the level.


    Note that if the tossing splashes water out of the lake that will also lower the level.


  21. Someone | Profile

    water level would decrease because the volume being deplaced by the boat from the weight of the anchor is greater than the volume of the anchor.


  22. RK | Founder | Profile

    Many think the answer is stay the same, but the official answer is water level would fall (although onfire & deadskin bring up interesting exceptions)


    Several very good explanations given above (Shawn, MichaelC, Bilbao, Suineg, Aaronlau, Ari, Joe, Someone, Lograh etc..)


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