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Math Help - Interesting problem on thirsty crow!

  1. #1
    Super Member fardeen_gen's Avatar
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    Interesting problem on thirsty crow!

    There is a cylindrical vessel whose height and radius is known. It is filled initially with water whose level is also known. A thirsty crow with hope of quenching his thirst approaches the cylindrical vessel. He throws small pebbles, which can be approximated as small spheres, into the water. However he observes that no matter how many pebbles are thrown the water level never reaches till the mouth of the vessel. Assume that there is no leakage anywhere. The radius of the pebbles is also known(i.e can be included in the calculations) Find the condition for this to happen?

    This was all that was given.
    Its really a good problem. Please state if you are making any assumptions in your solution.
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  2. #2
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    Quote Originally Posted by fardeen_gen View Post
    There is a cylindrical vessel whose height and radius is known. It is filled initially with water whose level is also known. A thirsty crow with hope of quenching his thirst approaches the cylindrical vessel. He throws small pebbles, which can be approximated as small spheres, into the water. However he observes that no matter how many pebbles are thrown the water level never reaches till the mouth of the vessel. Assume that there is no leakage anywhere. The radius of the pebbles is also known(i.e can be included in the calculations) Find the condition for this to happen?

    This was all that was given.
    Its really a good problem. Please state if you are making any assumptions in your solution.
    Perhaps the water is compressible and the crow is a supercrow who is chucking the rocks at Mach 0.7.
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  3. #3
    Super Member craig's Avatar
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    Or maybe the stones heat up during their fall from the crow's feet, evaporating a tiny bit of water everytime they the water, therefore no water rise.
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  4. #4
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    Quote Originally Posted by galactus View Post
    A submerged object displaces the same volume of water as its own volume.

    This is a twist on the old Aesop fable.

    Think about it. How many pebbles would it take to displace enough water to get the water to the top if the container were, say, half full?.

    How much volume would that be compared to the volume of the pebbles dropped in?.

    Draw out an example since the height and radius of the cylinder, as well as the radius of the spheres are known.

    Experiment a bit and you'll see it. Think about Buoyancy and Archimedes.
    So? The question states that NO MATTER HOW MANY PEBBLES ARE DROPPED the water level never rises to the mouth. Even if the pebbles wear near infinitesimally small, there would still be a certain amount of them for which the water would reach the mouth.
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  5. #5
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    Quote Originally Posted by craig View Post
    Or maybe the stones heat up during their fall from the crow's feet, evaporating a tiny bit of water everytime they the water, therefore no water rise.
    But yes, perhaps this is the case. Perhaps it's a hot sunny day and the water and crow are sweltering in the heat (hence his thirst). But the water evaporates faster than the rate at which he can displace the same volume with pebbles.
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  6. #6
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    The point here is that the pebbles are spherical, and so there will be space between them. If for example you make the assumption that the pebbles are all the same size, then they can only occupy at most something like 0.74 of the volume of the cylinder (see here). So if the cylinder was initially only one-quarter full of water, then even when the whole cylinder is filled with pebbles, the water will settle into the spaces between them and not reach the top of the cylinder.
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  7. #7
    Super Member fardeen_gen's Avatar
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    When the crow throws in the pebbles, let us assume, the pebbles get arranged in such a way that they are closest to each other much similar to the hexagonal type crystal structure. However, there remains a void in between the four spheres in contact.
    Volume of each small sphere is . If the crow throws in pebbles, the volume of the void is . Let initial level of water be . So the water never comes to the top if


    NOTE: The details will depend on how do you pack the spheres. More likely it will be a random packing and not a hexagonal one. I take for granted that the crow has good command on Chemistry.
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