What is the shape of an hour-glass with the following characteristics:
1)it will be used with water, not sand
2)two identical glass shapes are fused with a tiny hole connecting them and adequate venting to prevent any changes in air pressure from impeding water flow
3)the rate of flow through the hole between the top and bottom portion is proportional to water pressure
4)the horizontal cross section of the hourglass is a circle at every level, and it is symmetric around a vertical axis
5)the water level is proportional to the time elapsed

(A)  From 1) we know that the presssure at the hole between the sections ("exit hole") is proportional to the height of the water in the top section.

(B) From 3) we know that the flow velocity (and also volume flow rate due to constant area of the exit hole) thru the exit hole is proportional to pressure, and therefore, with (A), to height of the water.  I.e. the volume flow rate doubles as the height of water doubles.

(C) Given the above and 5) (change in height / change in time = constant), the cross sectional area of the top surface of the water in a horizontal plane at some height 2*h must be double of what it is at height h. 

(D) Given 4), the diameter, d, of a horizontal circular section at height 2h must therefore be ��2 times that at height h.

(E) Thefore the diameter d, of a horizontal circular section is proportional to ��h, if h= the height of water in the glass.

Errata:  The conditions of the Bernoulli equation must hold - particularly that in this case, the volume of water above and around the hole is large enough that it has no appreciable velocity.

Posted by Kenny M on 2010-03-15 20:35:02