A cylindrical bucket of inner radius m contains water and is spun steadily about its…

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Kinematics, conservation laws and ideal flow · Kinematics, conservation laws and ideal flow

Fluids and Continuum Physics, Ch Euler's equation and Bernoulli's theorem

Problem

A cylindrical bucket of inner radius

0.12

m contains water and is spun steadily about its vertical axis at

3.0

revolutions per second. The water has long since taken up the motion of the bucket and rotates with it as a rigid body, so the flow is steady and the speed at radius

r

\Omega r

. The free surface is open to the atmosphere everywhere. Water density

1000

kg m

^{-3}

g = 9.8

m s

^{-2}

; take

z

upward. A student compares the point on the free surface at the axis with the point on the free surface at the wall, writing

p + \tfrac12\rho|\mathbf{u}|^2 + \rho g z = \text{constant}

for the two points because both are in the same steady flow of the same incompressible liquid and viscosity plays no part once the water turns as a solid body. ((a)) Carry out the student's calculation: what height of the rim point above the axis point does it give? (Give a signed value, positive meaning the rim is higher.) ((b)) Using the free-surface shape for a liquid in solid-body rotation, find the true height of the rim point above the axis point. ((c)) Which of the four conditions attached to Bernoulli's theorem has the student broken?

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