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Mechanical Properties of Fluids

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Question : 16 of 31
Marks: +1, -0
The cylindrical tube of a spray pump has a cross-section of 8.0cm28.0\,\mathrm{cm}^2 on one end of which has 40 fine holes each of diameter 1.0 mm. If the liquid flow inside the tube is 1.5mmin11.5\,\mathrm{m}\,\mathrm{min}^{-1}, what is the speed of ejection of the liquid through the holes?
Solution:  
Here, cross-section of the tube, a1=8.0cm2=8.0×104m2a_{1}=8.0\,\mathrm{cm}^{2}=8.0\times10^{-4}\,\mathrm{m}^{2}
The speed of liquid in the tube, v1=1.5mmin1=1.560ms1v_{1}=1.5\,\mathrm{m}\,\mathrm{min}^{-1}=\frac{1.5}{60}\,\mathrm{m}\,\mathrm{s}^{-1}
=0.025ms1=0.025\,\mathrm{m}\,\mathrm{s}^{-1}
diameter of a hole, D=1.0mm=103mD=1.0\,\mathrm{mm}=10^{-3}\,\mathrm{m}
Therefore, cross-section of a hole, πD24=π4×(103)2\frac{\pi D^{2}}{4}=\frac{\pi}{4}\times(10^{-3})^{2}
=π4×106m2=\frac{\pi}{4}\times10^{-6}\,\mathrm{m}^{2}
Therefore, total cross-section of 40 holes, a2=π4×106×40m2a_{2}=\frac{\pi}{4}\times10^{-6}\times40\,\mathrm{m}^{2}
If v2v_2 is the speed of ejection of the liquid through the holes, then a1v1=a2v2a_1 v_1 = a_2 v_2
or v2=a1v1a2v_{2}=\frac{a_{1} v_{1}}{a_{2}}
=8.0×104×0.025π4×106×40=\frac{8.0\times10^{-4}\times0.025}{\frac{\pi}{4}\times10^{-6}\times40}
=0.637ms1=0.637\,\mathrm{m}\,\mathrm{s}^{-1}.
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