Question

What will be the total resistance offered to the flow of water by the system of pipe $1$, pipe $2$, pipe $3$ and pipe ignoring the joints? If all the pipes are of length $3\text{m}$ and radius $2\text{m}$ and '$\eta$' is viscosity of water. (Assume $\pi =3$)

• A. $1.25\eta$
• B. $3\eta$
• C. $4.5\eta$
• D. $9\eta$

Answer 1

The correct option is A $1.25\eta$

Given, radius and length of the pipes are $r=0.6\text{m}$ and $l=3\text{m}$.
Divide the four pipes into three section as show in figure.


Section $1:$
Resistance offered to flow of water by pipe $1$.
$R_1=\frac{8\eta l}{\pi r^4}$
Section $2:$
Pipe $2$ and pipe $3$ are in parallel connection. Resistance offered to flow of water by pipe $2$ and pipe $3$ is given by $\left(R_2\right)$ $\frac{1}{R_2}=\frac{1}{R}+\frac{1}{R}[\text{where}R=\frac{8\eta l}{\pi r^4}]$
$\Rightarrow \frac{1}{R_2}=\frac{2}{R}$
$\Rightarrow R_2=\frac{R}{2}=\frac{8\eta l}{2\times \pi r^4}$
Section $3:$
Resistance offered by the pipe $4$ to the flow of water $\left(R_4\right)$
$R_4=\frac{8\eta l}{\pi r^4}$
Total Resistance offered by the pipe $1$, pipe $2$, pipe $3$ and pipe $4$ are
$R=R_1+R_2+R_4$
$\Rightarrow R=\frac{8\eta l}{\pi r^4}+\frac{8\eta l}{2\pi r^4}+\frac{8\eta l}{\pi r^4}$
$\Rightarrow R=\frac{20\eta l}{\pi r^4}=\frac{20\eta \times 3}{3\times (2{)}^4}=1.25\eta$