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The cross sectional area of a horizontal...

The cross sectional area of a horizontal pipe is decreasing from `5 xx 10^(-2) m^2" to "1 xx 10^(-2) m^2?.` Water is flowing with speed of 5 m/s through this pipe with a pressure of `5 xx 10^5 Pa` through the larger cross section. The pressure at the smaller cross section of the pipe is `n xx 10^(4) Pa.` Find the value of n.

Text Solution

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`a_(1)=5 xx 10^(-2) m^(2)`
`a_(2)=10^(-2) m^(2)`
`v_(1)=5m//s`
`p_(1)=5 xx 10^(5) pa`
According to equation of continuity
`a_(1)v_(1)=a_(2)v_(2)`
`v_(2)=(a_(1)v_(1))/(a_(2))`
`=(5 xx 10^(-2) xx 5)/(10^(-2)) =25m//s`
Accoring to Bernoulli.s theorem
`p_(1)+1/2 pv_(1)^(2)=p_(2)+1/pv_(2)^(2)`
`p_(2)=p_(1)+1/2 pv(1)^(2)-1/2pv_(2)^(2)`
`p_(2)=p_(2)+1/2p(v_(1)^(2)-v_(2)^(2))`
`=5 xx 10^(5) +1/2 xx 1000 ((5)^(2)-(25)^(2))`
`=5 xx 10^(5) +500 (25-625)`
`=5 xx 10^(5) +500 (-600)`
=500000-300000=200000 Pa
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