1) Consider a fluid whose mass density varies with height y above sea level. Use

Question

1) Consider a fluid whose mass density varies with height yabove sea level. Use the equation P1+gy1=P2+gy2. Part A:- Choose the correct expression that shows how dP, the change in pressure along a column of the fluid of infinitesimal height dy, varies with y

Part B:- Using the simplification that the mass density of air at any given height y above sea level is proportional to atmospheric pressure at that height, integrate the differential equation you found in the previous part to obtain an expression that shows how atmospheric pressure varies with height yabove sea level. Suppose that P0 is the atmospheric pressure at sea level, and 0 is the mass density of air at sea level. Express your answer in terms of the variables y, P0, 0, and the acceleration due to gravity g.

please I need the answer for this question as soon as possible. Thank you.

Explanation / Answer

1. given

mass densoty rho varies with height y above sea level

P1 + rho1*g*y1 = P2 + rho2*g*y2

for a column of infinitesmal height dy

dP = P2 - P1 = (rho1y1 - rho2y2)*g

but y2 = y1 + dy

dP = (rho1*y1 - rho2*y1 - rho2*dy)*g

now, rho2 = rho1 + d(rho)*dy/dy

hence

dP = (rho1*y1 - rho1*y1 - d(rho)dy*y1/dy - rho1*dy - d(rho)*dy*dy/dy)*g

consider d(rho)dy*dy/dy to be very small

dP = -(y1d(rho)/dy + rho1)*gdy

hence dP = -(rho*g + y*d(rho)/dy)*dy ( hence option 4)

2. assuming rho = k*P ( k is proportionality constant)

d(rho)/dy = k*dP/dy

hence

dP / dy= -(rho*g + y*k*dP/dy)

(1 - yk)dP/dy = -kP*g

dP/kP*g = dy/(yk - 1)

integrating

ln(P)/kg = ln(yk - 1)/k + C ( where C is constant of integration)

at y = yo, Patm

hence

ln(Patm)/kg = ln(yo*k - 1)/k + C

C = ln(Patm)/kg - ln(yo*k - 1)/k

hence

ln(P)/kg = ln(yk - 1)/k + ln(Patm)/kg - ln(yo*k - 1)/k

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