prexam1a - 1 When air of density 1.0 kg/m3 flows past the...

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Unformatted text preview: 1. When air of density 1.0 kg/m3 flows past the top of the tube shown, water (density 103 kg/m3 ) rises in the tube to a height of 1 cm. Compute the air speed, v. Take g = 10 m/s2 . v 1.0 cm Solution: Let p be the pressure at the top of the water column (with height h, density w ) in the tube and p0 be atmospheric pressure at the water-air interface below the tube. The weight of the column of water is supported by the upward force associated with the pressure difference. If we denote the crosssectional area of the cylindrical tube, A, then this balance of forces is given as, (p0 - p)A = w gAh p0 - p = w gh Let y measure vertical distance with y = 0 at the water-air interface below the tube where the pressure is p0 . The Bernouilli equation applied to the air relates p to p0 as: 1 p0 = p + a gy + a v 2 2 Combining this with the first relation yields, 1 a gy + a v 2 = w gh 2 w v 2 = 2gh - 2gy a w y a v 2 = 2gh 1- . a h w We assume that the top of the tube (at height y) where the air moves at speed v is a distance comparable to h; thus the second term in the parentheses is negligible since it is comparable to (a /w ) 10-3 . Ignoring this term (i.e. ignoring the weight of the air) we get the final result, w = 2 10 10-2 103 m/s v = 2gh a = 10 2 m/s. ...
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