Question

Find the volume of the fluid flowing across the section of the tube per unit time.

• A. $V=\frac{\pi }{2}{R}^2{v}_0$
• B. $V=\frac{2\pi }{3}{R}^2{v}_0$
  
• C. $V=\frac{\pi }{2}{R}^3{v}_0$
  
• D. $V=\frac{\pi }{3}{R}^2{v}_0$
  

Answer 1

The correct option is A $V=\frac{\pi }{2}{R}^2{v}_0$

$Consideraringofradiusrandthicknessdrconcentricatwiththeaxis.Inunnittimethevolumeofliquidflowingthroughthisringis2\pi rvdr.Tofindtheflowrateweneedflowrate={\int }_0^{R}2\pi rvdr={\int }_0^{R}2\pi r{v}_0(1-\frac{r^2}{R^2})dr=2\pi v_0{\int }_0^{R}r(1-\frac{r^2}{R^2})dr=\frac{\pi }{2}{R}^2{v}_0$