A liquid of density 900 ` kg//m^3 ` is filled in a cylindrical tank open from top of upper radius 0.9 m and lower radius 0.3 m. A capillary tube of length l is attached at the bottom of the tank as shown in the figure. The capillary has outer radius 0.002 m and inner radius a.
When pressure p is applied at the top of the tank volume flow rate of the liquid is ` 8 xx 10 ^(-6 ) m^3// s ` and if capillary tube is detached, the liquid comes out from the tank with a velocity 10 m/s
[Given : ` pi a ^2 = 10 ^(-6) m^ 2 and a ^ 2//1 = 2 xx 10 ^(-6) m ]`
Determine the coefficient of viscosity in SI units when the capillary tube is attached :

Options

A. ` 10 ^5 Pa `
B. ` (2)/(3) xx 10 ^(5) Pa `
C. ` (4)/(9) xx 10^5 Pa `
D. ` (2)/(9) xx 10 ^(5) Pa `

Answer

C

Explanation

When the tube is not there , `p + p_0 + (1)/(2) rho v _ 1 ^ 2 + rho gH + (1)/(2) rho v _ 2 ^ 2 + p _ 0 `
` therefore p + rho gH = (1)/(2) (v_2 ^ 2 - v _ 1^2 ) `
Now, ` A _ 1 v _ 1 = A _ 2 v _ 2 or v_1 = (A_2v _ 2 )/(A_1 ) `
` therefore p + rho gH = (rho ) /(2) [v_2^2 - ((A_2)/(A_1) v _ 2 )^2] = (1)/(2) xx rho xx v _ 2 ^2[ 1 - ((pi (0.3)^2)/(pi (0.9)^2) ) ^2] = (1)/(2) xx rho xx (10) ^2 [ 1- (1)/(81)] = (4 x 10 ^3 rho ) /( 81 `
` = (4 xx 10 ^(3) xx 900)/(81) = (4)/(9) xx 10 ^5 N//m ^ 2 `.
This is also the excess pressure ` Delta p`.