A horizontal tube has different cross sections at points `A` and `B`. The areas of cross section are `a_(1)` and `a_(2)` respectively, and pressures at these points are `p_(1)=rhogh_(1)` and `p_(2)=rhogh_(2)` where `rho` is the density of liquid flowing in the tube and `h_(1)` and `h_(2)` are heights of liquid columns in vertical tubes connected at `A` and `B`. If `h_(1)-h_(2)=h`, then the flow rate of the liquid in the horizontal tube is

Options

A. `a_(1)a_(2)sqrt((2gh)/(a_(1)^(2)-a_(2)^(2)))`
B. `a_(1)a_(2)sqrt((2g)/(h(a_(1)^(2)-a_(2)^(2))))`
C. `a_(1)a_(2)sqrt(((a_(1)^(2)+a_(2)^(2))h)/(2g(a_(1)^(2)-a_(2)^(2))))`
D. `(2a_(1)a_(2)gh)/sqrt(a_(1)^(2)-a_(2)^(2))`

Answer

A

Explanation

Let `a_(1)` and `a_(2)` be cross sectional area and `v_(1)` and `v_(2)` the velocities of liquid flow at `A` and `B`. If `p_(1)` and `p_(2)` are pressures of liquid recorded by manometer then
`p_(1)+1/2rhov_(1)^(2)=p_(2)+1/2rhov_(2)^(2)+1/2rhov_(2)^(2)impliesp_(1)-p_(2)=1/2rhov_(1)^(2)((v_(2)^(2))/(v_(1)^(2))-1)`
by equation of continuity `a_(1)v_(1)=a_(2)v_(2)`
Also `p_(1)-p_(2)=hrhog`
Substituting these value we have `v_(1)=sqrt((2gh)/((a_(1)^(2))/(a_(2)^(2))-1))`
Rate of flow of liquid is `a_(1)v_(1)=a_(1)a_(2)sqrt((2gh)/(a_(1)^(2)-a_(2)^(2)))`