Aula teórica 11
Rotação do tipo corpo sólido.
Princípios de Conservação
(massa, momentum e energia)
Rotating Fluid
http://www.youtube.com/watch?v=gCZL3mZAQos&feature=related
http://www.youtube.com/watch?v=YNC2_vtNtyk
Rotação do tipo corpo sólido
Angular velocity:
Aceleration:
Pressure Gradient must
balance the inertia
force:
Integração da Equação
• Integrating fist in r (in an horizontal plane)
• Differentiating in “z”
• Where p0 is the pressure at the free surface. In
fact at r=0 and z=0
The free surface is a parabola
• The position of the free surface can be found
using volume conservation or knowing that:
p  p0  gz 

2
2r 2
The over-elevation
• Applied to the point located on the glass
surface, at the same vertical coordinate as the
free surface at the rotation axis gives:
1
p0  p0  gh   2 r 2
2
2 R 2
h
2g
• The over-elevation above the still level is h/2.
The Tea Cup paradox
• Is the effect that moves tea leaves laying on the
cup bottom to the cup axis when one rotates the
tea inside the cup, using a spoon.
• This case differs from the “solid rotation” because
the velocity of the tea on the cup bottom is zero.
As a consequence the pressure gradient
necessary for balancing the centrifugal force is
higher at the surface and lower at the bottom.
See details at
http://en.wikipedia.org/wiki/Tea_leaf_paradox
Conservation of momentum
(Quantidade de movimento)
• The force exerted by the jet on
the ball is equal to the force
exerted by the ball on the jet
(contrary sign) and balances
ball weight.