Why is ro = 0.2255 in fluid dynamics vortices?

[yecko]

Homework Statement

A closed cylindrical tank of 4 m in height and 2 m in diameter contains water to a depth of 3 m. When the cylinder is rotated at a constant angular velocity of 40 rad/s, show that nearly 0.71 m2 of the bottom surface of the tank is uncovered.

Relevant Equations

pressure surface, Bernoulli's equation

why is ro = 0.2255?

Thank you

 

[haruspex]

Homework Statement:: A closed cylindrical tank of 4 m in height and 2 m in diameter contains water to a depth of 3 m. When the cylinder is rotated at a constant angular velocity of 40 rad/s, show that nearly 0.71 m2 of the bottom surface of the tank is uncovered.
Relevant Equations:: pressure surface, Bernoulli's equation

View attachment 273668
why is ro = 0.2255?
View attachment 273669

Thank you

I would have assumed the handwritten calculation was yours, but your question implies it is someone else's. Please clarify.

 

[yecko]

Sorry for being unclear.
The handwritten one is the answer written by prof. And the answer is same as final model answer in textbook.
However, when I substitute the number in the red bracket, I can only obtain 0.1519 from calculator instead of prof‘s answer 0.2255.
Have I done anything wrong? Thank you.

 

[haruspex]

Sorry for being unclear.
The handwritten one is the answer written by prof. And the answer is same as final model answer in textbook.
However, when I substitute the number in the red bracket, I can only obtain 0.1519 from calculator instead of prof‘s answer 0.2255.
Have I done anything wrong? Thank you.

OK, but you'll have to decipher the scrawl for me. It looks like $\frac{64g}{\zeta^2}+4r_0^2=1$. If so, what is $\zeta$?

 

[yecko]

From Kundu, by comparing the formula "Surfaces of constant pressure are given by ...", ζ written by Prof should be ω here, which implicates angular velocity. However, when I substite 40 rad/s, the answer is wrong. Am I having any misunderstanding with ζ?

 

[yecko]

for reference, this question is utilising the approach of "constant pressure surface"