Question

The figure shows the commonly observed decrease in diameter of a water stream as it falls from a tap. The tap has internal diameter $D_0$ and is connected to a large tank of water. The surface of the water is at a height b above the end of the tap. By considering the dynamics of a thin "cylinder' of water in the stream answer the following: (ignore any resistance to the flow and any effects of surface tension, given ${\rho }_w$ density of water)
1. Which of the following equation expresses the fact that the flow rate at the tap is the same as at the stream point with diameter D and velocity v (i.e. D in terms of $D_0$, $v_0$ and v will be):

• A. $D=\frac{D_0{v}_0}{v}$
• B. $D=\frac{D_0{v_0}^2}{v^2}$
• C. $D=\frac{D_{0}v}{v_0}$
• D. $D=D_0\sqrt{\frac{v_0}{v}}$

Answer 1

The correct option is A $D=\frac{D_0{v}_0}{v}$

$1$ Flow rate will be same

i.e $A_1{V}_1=A_2{V}_2\frac{\pi D^2{V}^2}{4}=\frac{\pi D_0^2{V}_0^2}{4}\Rightarrow D^2{V}^2=D_0^2{V}_0^2\Rightarrow DV=D_0{V}_0\Rightarrow D=\frac{D_0{V}_0}{V}$