Question

Hydrostatic pressure: derive the equation from pA+Pgdz-(p+dp)A=0 to Pgdz=dp (i think i know the rest)

Answer 1

P=density , guessing A=area, z=depth, p=pressure

Answer 2

i have : pA+Pgdz-pA+dpA=0 => (Pgdz-dp)A = 0 => Pgdz-dp=0 => Pgdz=dp

Answer 3

i'm guessing this is very very wrong

Answer 4

your eq has a glitch. The LHS of the 1st one must have PgdzA.

Answer 5

yea, i cheated, i was thinking of those kinematic equations where t=0 or t=.....

Answer 6

PgdzA = weight of the small vol of fluid considered. Mass = P x vol. Vol = dz x A , so weight = P x dz x A x g

Answer 7

uh, but my textbook gets the equation without A

Answer 8

it gets to dp/dz = Pg

Answer 9

if anyone can explain this to me, i will be very grateful!

Answer 10

\[F_1=mg+F_2 \]\[F_1=\rho g A \Delta y + F_2 /:A\]
\[\frac{ F_1 }{ A }=\rho g \Delta y+\frac{ F_2 }{ A }\]\[P(y)=\rho g \Delta y + P(y+ \Delta y)\]\[-\rho g \Delta y = P(y+ \Delta y)-P(y)\]
\[-\rho g =\frac{P(y+ \Delta y)-P(y)}{\Delta y }\]
\[-\rho g=\lim_{\Delta y \rightarrow 0}\frac{ P(y+\Delta y) - P(y) }{ \Delta y }\]
\[\frac{ dP }{ dy }=-\rho g\]

Answer 11

wow..i think there's an easier way than this..

Answer 12

you let P be a function of z(depth)?

Answer 13

i'm starting to get it..

Answer 14

There should be in your equation PgdzA where P is density not Pgdz

Answer 15

then your derivation is correct

Answer 16

i thought you used \[\rho\] as density and P as pressure

Answer 17

I used rho as density and P as pressure, you used P as density and p as pressure

Answer 18

oh, i wrote that wrong, i meant P to be rho=density

Answer 19

LOL, wait

Answer 20

yea, you're right

Answer 21

how'd you know the derivation..there's no way i could come up with this

Answer 22

\[\frac{ dP }{ dy }=-\rho g\]

Answer 23

shouldn't rho.g be positive??

Answer 24

this is a mass element of the liquid that I've drawn. I chose the increasing direction of y upwards that means if y goes up pressure decreases that's the meaning of that minus sign. But you are free to choose your direction of inc. val. of y so you can also say that y increases downwards, then you don't have minus sign because when y increases pressure also increases. But more often you see that equation with minus sign

Answer 25

and dy could be dz that's a matter of taste

Answer 26

F1 and F2 are the forces that acts on that mass element( according to Pascal's law)
All horizontal components eat each other up