Hydrostatic equilibrium: force-based derivation

Deriving the relationship between pressure gradient, density and gravitational field for a fluid in hydrostatic equilibrium, by considering the balance of forces acting on a small Cartesian fluid element.
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About me: I studied Physics at the University of Cambridge, then stayed on to get a PhD in Astronomy. During my PhD, I also spent four years teaching Physics undergraduates at the university. Now, I'm working as a private tutor, teaching Physics & Maths up to A Level standard.
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[@kimsgeorgspavlovs4286]

Nicely explained

[@DrBenYelverton]

Thanks!

[@press2701]

Nicely done! As a Canadian, I appreciate the "x, y, zed". :) 30yrs engineering in the USA, I still spell colour incorrectly, enunciate "oout" the way it should be, and refuse to call it "zee".
ps: when might you show us a new solution to the Navier Stokes equation (approximate is ok!). I still remember my fluids (Dr. J. Ottino) prof "if you have a new solution, you'll get a Nobel prize".

[@DrBenYelverton]

Excellent, I didn't know you pronounced z the same way in Canada! As for the Navier-Stokes equation, I'll have to get back to you on that one...

[@AdonizedeckAckahBlayMiezah]

Nice video. But please, what method did you use to do limit?

[@mingmiao364]

Do you mean 6:45? That’s the definition of partial derivative

[@DrBenYelverton]

Thanks for watching. ∂p/∂x is defined as the limit of [p(x+dx, y, z) - p(x, y, z)]/dx as dx -> 0. In the video the p terms on the numerator were switched around, hence taking the limit gives -∂p/∂x.

[@danilonascimentorj]

what if that weight vector was pointing downwards?

[@DrBenYelverton]

In that case gₓ = 0, so there's no pressure gradient in the x direction, and likewise for the z direction, so the pressure only changes with y. This is just a special case of the result in the video - as we didn't assume anything about the direction of g in the derivation, it's valid for all possible fields.