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Understanding Rotational Dynamics in Solid Bodies: Conservation Principles

Explore rotational dynamics in solid bodies involving mass, momentum, and energy conservation. Learn about angular velocity, acceleration, pressure gradients, and free surface behavior. Dive into the Tea Cup paradox and conservation of momentum.

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Understanding Rotational Dynamics in Solid Bodies: Conservation Principles

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  1. Aulateórica 11 Rotação do tipocorposólido. Princípios de Conservação (massa, momentum e energia)

  2. Rotating Fluid http://www.youtube.com/watch?v=gCZL3mZAQos&feature=related http://www.youtube.com/watch?v=YNC2_vtNtyk

  3. Rotação do tipocorposólido Angular velocity: Aceleration: Pressure Gradient must balance the inertia force:

  4. 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

  5. The free surface is a parabola • The position of the free surface can be found using volume conservation or knowing that:

  6. 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: • The over-elevation above the still level is h/2.

  7. 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

  8. 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.

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