144Chapt. 15Fluidsthe water coming out of the primary. The volume of waterVaccumulated in the catchmentmeasures time:V0−Vis the volume of water remaining in the primary.a) Show that the differential bit of water volume sent to the catchment isdV=Ahvhdt,wherevhis the water velocity at the hole anddtis a time differential.HINT:Actually,this is one of those obviously it’s...things.b) Using the continuity equationAv= Constantand Bernoulli’s equationP+12ρv2+ρgy= Constantshow thatAhvh=kradicalbigV0−V ,wherek=Ah√A0radicalBigg2g1−(Ah/A0)2.c) Solve the differential equationdVdt=Ahvh=kradicalbigV0−Vwith the initial conditionV= 0 att= 0: i.e., findVexplicitly as a function of time. Atwhat timetmaxdoesVreach a maximum?What isVmax?At what timetemptyis theprimary empty? If one drops the 2nd order term in theV(t) solution, what wouldVbe attimetempty?
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Escapement, Astronomical clock, water clock, Ctesibius, Su Sung