26. A frequently quoted rule of thumb in aircraft design is that wings should pr

Question

26. A frequently quoted rule of thumb in aircraft design is that
wings should produce about 1000 N of lift per square meter of
wing. (The fact that a wing has a top and bottom surface does
not double its area.) (a) At takeoff, an aircraft travels at 60.0
m/s, so that the air speed relative to the bottom of the wing is
60.0 m/s. Given the sea level density of air to be
1.29 kg/m3 , how fast must it move over the upper surface
to create the ideal lift? (b) How fast must air move over the
upper surface at a cruising speed of 245 m/s and at an
altitude where air density is one-fourth that at sea level?
(Note that this is not all of the aircraft’s lift—some comes from
the body of the plane, some from engine thrust, and so on.
Furthermore, Bernoulli’s principle gives an approximate
answer because flow over the wing creates turbulence.)

(please show all workings and solutions)

Explanation / Answer

here,

F = 1000 N/m^2

(a)

vb = 60 m/s

density of air , pa = 1.29 kg/m^3

let the speed over the upper surface be v

using bernouli's theorm

P = 0.5 * pa * ( v^2 - vb^2)

1000 = 0.5 * 1.29 * ( v^2 - 60^2)

v = 71.77 m/s

the speed of the upper surface is 71.77 m/s

(b)

air density , pa' = pa/4

let the speed over the upper surface be v'

using bernouli's theorm

P = 0.5 * pa' * ( v'^2 - vb'^2)

1000 = 0.5 * (1.29/4) * ( v'^2 - 245^2)

v = 257.35 m/s

the speed of the upper surface is 257.35 m/s

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