Consider a horizontal slab of air whose thickness (height) isdz. If this slab is

Question

Consider a horizontal slab of air whose thickness (height) isdz. If this slab is at rest, the pressure holding it up frombelow must balance both the pressure from above and the weight ofthe slab. Use this fact to find an expression for dP/dz, thevariation of pressure with altitude, in terms of density ofair.
This has been a confusing problem for me and severalclassmates.
This is what we have so far:
P = F/A m= Adz Sum of the forces: P2(A) + P1(A)+(A)(dz)(a) =0 P2(A)=-[P1(A)+(A)(dz)(a)] (eq a)
I guess I just don't know how to take the derivative of eq awith respect to z without getting a second derivative. Howcan we get a dP/dz solution when that would involve taking thederivative of dz (which would give us dP/dz^2 right?)
And if this whole slab is at rest, why is the altitudechanging?

Explanation / Answer

P2*A = P1*A + mg P1 =P(z)                 (z starts at z1) P2 = P(z + z)       z = z2 - z1 mg = A*g*z Sub into equation P(z + z)*A = P(z)*A + Ag*z                  P(z+z) - P(z) Ag = A* -----------------                          z Recall that the right side is the definition of the derivative whenz => 0 Ag = A*dP/dz dP/dz = *g EDIT: sorry about including the area. A proper force balance shouldrequire that P1*A + mg = P2*A P1*A gives units of force, as does m*g P1 and P2 are in units of pressure. It was wrong for me to say P1 +mg = P2. Hope that helps, and sorry for the confusion.

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