question_answer

The cylindrical tube of a spray pump has radius R, one end of which has n fine holes, each of radius r. If the speed of the liquid in the tube is v, the speed of the ejection of the liquid through the holes is

A)  \[\frac{v{{R}^{2}}}{{{n}^{2}}{{r}^{2}}}\]

B)  \[\frac{v{{R}^{2}}}{n{{r}^{2}}}\]               

C)  \[\frac{v{{R}^{2}}}{{{n}^{3}}{{r}^{2}}}\]

D)  \[\frac{{{v}^{2}}R}{nr}\]

Correct Answer: B

Solution :

Key Concept During the streamline flow of viscous and incompressible fluid through a pipe varying cross-section, the product of area of cross section and normal fluid velocity (Ay) remains constant throughout the flow. Consider a cylindrical tube of a spray pump has radius R, one end having n fine holes, each of radius r and speed of liquid in the tube is v as shown in figure. According to equation of continuity. Av = constant where, A is a cylindrical tube and v is velocity of liquid in a tube. Volume in flow rate = volume an out flow rate \[\pi {{R}^{2}}v=n\pi {{r}^{2}}v'\] \[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,v'=\frac{{{R}^{2}}v}{n{{r}^{2}}}\] Thus, speed of the ejection of the liquid through the holes is \[\frac{{{R}^{2}}v}{n{{r}^{2}}}.\]