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.
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.
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.