Example $5$:

A $1\mathrm{.}0cm$ diameter pipe widens to $2\mathrm{.}0cm,$ then narrows to $0\mathrm{.}50cm$ diameter. The speed in the first section is $4\mathrm{.}0\frac{m}{s}$ what is the speed in the other two sections? $V?$

The equation of continuity applies as long as the pipe is full. Note that $A_1,A_2,$ and $A_3$ and $($:$v_1\}$ and $v_3$ are the crosssectional areas and the speeds in the first, second, and third segments of the pipe.

$A=r^2$

$A=\frac{D^2}{4}$

$A_1{v}_1=A_2{v}_2=A_3{v}_3=>r_1^2{v}_1=r_2^2{v}_2=r_3^2{v}_3=>r_1^2{v}_1=r_2^2{v}_2=r_3^2{v}_3,A=$

$=>(0\mathrm{.}005m{)}^2(4\mathrm{.}0\frac{m}{s})=(0\mathrm{.}01m{)}^2{v}_2=(0\mathrm{.}0025m{)}^2{v}_3$

$=>,v_2=(\frac{0\mathrm{.}005(m)}{0\mathrm{.}01(m)}{)}^2(4\mathrm{.}0\frac{m}{s})=1\mathrm{.}0\frac{m}{s},v_3=(\frac{0\mathrm{.}005(m)}{0\mathrm{.}0025(m)}{)}^2(4\mathrm{.}0\frac{m}{s})=16\frac{m}{s}$

$($b$)$ The volume flow rate through the pipe is

$Q=A_1{v}_1=(0\mathrm{.}005m{)}^2(4\mathrm{.}0\frac{m}{s})=3\mathrm{.}1{10}^{-4}\frac{m^3}{s}$

If the density of the liquid in the pipe is $1000k\frac{g}{m^3},$ what is the mass flow rate and weight flow rate?

$m^{}=Q=1000**3\mathrm{.}1**{10}^{-4}=0\mathrm{.}31k\frac{g}{s},m^{}=\frac{m}{t}=\frac{V}{t}=Q$

$W^{}=Qg=1000**3\mathrm{.}1**{10}^{-4}**9\mathrm{.}81=3\mathrm{.}04\frac{N}{s},W^{}=\frac{W}{t}=\frac{mg}{t}=\frac{Vg}{t}=Q$