The surface of a rotating liquid

Deriving the equation that defines the surface of a rotating liquid in a cylindrical container by considering the forces acting on a small element of the liquid.
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.
My website: benyelverton.com/
#physics #mathematics #centrifugalforce #centripetalforce #rotatingframes #rotation #force #liquid #fluid #parabola #paraboloid #surface #differentialequation #integration #fictitiousforces #maths #math #science #education

[@emrekt22]

I did my IB Physics HL internal assesment on this exact topic! Great video

[@DrBenYelverton]

Very nice, and thank you!

[@nerdy6600]

very nicely explained sir, can you do some on charge leak questions?🙏🏻

[@panagiotisathanasiou7918]

Try to solve it with navier stokes

[@onesky5570]

Is it a pure coincidence that the Y equation is basically the (tangential) velocity head if we use Bernoulli’s equation

[@nazperwar5518]

Great work sir

[@DrBenYelverton]

Thanks!

[@sandipdas7206]

Can we use Bernoulli's Equation to do this?

[@DrBenYelverton]

Good question - Bernoulli's principle won't help in this case as it only applies to points lying along a streamline. Here the streamlines are circles (i.e. lines of constant x) and Bernoulli's principle can't tell us how the y values of the different streamlines are related to each other. In fact, trying to apply it will lead to the (incorrect) conclusion that the surface is a paraboloid with a maximum at the centre, rather than a minimum.

[@leonardohenrique5391]

@@DrBenYelverton why does it gives the contrary answer?

[@DrBenYelverton]

Since the pressure is constant over the surface, trying to apply Bernoulli's principle to a point at the centre and another arbitrary point would give (xΩ)²/2 + gy = 0, which is inverted compared with the true solution. I'm not sure if there's any deep underlying reason why it happens to give the exact opposite answer, it's just not valid across different streamlines.

[@MissPiggyM976]

Very interesting!

[@Nirusaki]

We can also do it like this we take a general point x,y on the surface we equation atmospheric pressure giving us the answer