The level of liquid in a cylindrical vessel is kept constant at 30cm. It has three identical horizontal tubes A, B and C of length 40 cm each coming out at heights 0, 5 and 10 cm respectively. Calculate the length of a single overflow tube of the same radius as that of identical tubes which can replace the three when placed horizontally at the bottom of the cylinder.

Answer

16 cm

Explanation

Pressure head for tube A,
`h_(1)=30` cm of water column
Pressure head for tube `B,h_(2)=30-5=25 cm`
Pressure head for tube `C,h_(3)=30-10=20 cm`
Length of each tube, `l=40 cm`
Rate of flow of lliquid through different tubes is
`Q_(1)=(pi P_(1) r_(1)^(4))/(8 eta l)=(pi h_(1) rho gr^(4))/(8 eta l) , Q_(2) =(pi h_(2) rho g r^(4))/(8 eta l)
Q^(3)=(pih_(3) rho g r^(4))/(8 eta l)`
Total volume of the liquid flowing `Q=Q_(1)+Q_(2)+Q_(3)`
`=(pi rho gr ^(4))/(8 eta l) [h_(1)+h_(2)+h_(3)]`
Let l' be the length of the signal tube through which the same volue of liquid be flowing out. Then
`Q=(pi h_(1) rho g r_(4))/(8 eta l')`
From (i) and (ii), we have
`pih_(1) (rho g r^(4))/(8 eta l')=(pi rho g r^(4))/(8 eta l)[h_(1)+h_(2)+h_(3)]`
or `(h_(1))/(l')=(1)/(l)[h_(1)+h_(2)+h_(3)`
or `(30)/(l')=(1)/(40)[30+25+20]=(75)/(40)=(15)/(8)`
or `l'=(30xx8)/(15)=16 cm`