Question

In the following figure, flow of a liquid through a horizontal pipe has been shown. Three tubes $A,B$ and $C$ are connected to the pipe. The radii of the tubes $A,B$ and $C$ at the junction are respectively $2\text{cm},1\text{cm}$ and $2\text{cm}$. It can be said that the

• A. height of the liquid in the tube $B$ is maximum
• B. height of the liquid in the tubes $A$ and $B$ is the same
• C. height of the liquid in all the three tubes is the same
• D. height of the liquid in the tubes $A$ and $C$ is the same.

Answer 1

The correct option is D height of the liquid in the tubes $A$ and $C$ is the same.

Cross-sectional areas of both the tubes $A$ and $C$ are the same and pipe is horizontal.
Hence, according to equation of continuity,
$Av=\text{constant}\Rightarrow v_A=v_C$
According to Bernoulli's theorem,
$P+\frac{1}{2}\rho v^2+\rho gh=\text{constant}...\left(i\right)$
Pipe is horizontal (No change in elevation of fluid) i.e $h=$ constant.
$\Rightarrow P_A=P_C$
So, we can conclude that pressure remains same at $A$ and $C$.
For liquid column in the vertical tube, the relation between height of liquid and pressure at the point is,
$P=\rho gh$
$P_A=P_C\Rightarrow h_A=h_C$
Hence, height of liquid will be the same in both the tubes $A$ and $C$.

Similarly, $v_B>v_A$, as the cross-sectional area at $B$ is lower.
According to Bernoulli's theorem,
$P+\frac{1}{2}\rho v^2+\rho gh=\text{constant}$
Since $K{E}_B>K{E}_A$, in accordance with Bernoulli's equation, we get:
$P_B<P_A$ [since $h$ is constant here also]
From $P=\rho gh$, $h_A>h_B$

$\Rightarrow$correct option is $\left(d\right)$