Question

Water flows through the tube shown in figure. The areas of cross section of the wide and the narrow portions of the tube are 5 $c{m}^2$ and 2 $c{m}^2$ respectively. The rate of flow of water through the tube is 500 $c{m}^3$ $s^{-1}$. Find the difference of mercury levels in the U-tube.

Answer 1

Given Rate of flow = 500 $c{m}^3/sec$

$\Rightarrow v_A=\left(\frac{500}{5}\right)=100cm/sec$

$V_A{a}_A=V_B{a}_B$

$\Rightarrow \frac{v_A}{v_B}=\frac{a_B}{a_A}=\frac{2}{5}$

$\Rightarrow 5v_A=2v_B$

$\Rightarrow v_B=\left(\frac{5}{2}\right)v_A$ ...(i)

$\Rightarrow \frac{1}{2}\rho v_A^2+\rho g{h}_A+p_A=\frac{1}{2}\rho v_B^2+\rho g{h}_B+p_B$

$\Rightarrow p_A-p_B=\frac{1}{2}\left[p(v_B^2-v_A^2)\right]$

$\Rightarrow h\times 13.6\times 980\frac{1}{2}\times 1\times \frac{21}{4}(100{)}^2$

[Using (i)]

[because $p_A-p_B$ = ?]

$\Rightarrow h=\frac{21\times (100{)}^2}{2\times 13.6\times 980\times 4}$

= 1.969 cm