Cross products vanish Classical Mechanics

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Discussion Overview

The discussion revolves around the behavior of cross products in the context of classical mechanics, specifically related to an object moving in the x-y plane and the implications of the center of mass frame of reference. Participants are examining why certain cross terms vanish in the equations presented in the text.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions why the cross terms vanish in equation 8.9, seeking clarification on the concept.
  • Another participant suggests that the vanishing of the cross terms is due to the definition of the center of mass and encourages working through examples.
  • A third participant points out that understanding the center of mass in its own frame of reference is crucial and refers to another equation (5.56) for further insight.
  • A later reply acknowledges the zero value of r' in the center of mass frame but questions why the integral of the cross product of r' and v' does not also equal zero.

Areas of Agreement / Disagreement

Participants express differing views on the implications of the center of mass frame and the behavior of the cross products, indicating that the discussion remains unresolved with multiple competing perspectives.

Contextual Notes

There are references to specific equations and terms that may require further clarification, such as the definitions and implications of r', v', and the center of mass frame. The discussion does not resolve the mathematical steps involved in the integrals mentioned.

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As seen in the picture, this question is about an object moving in the x-y plane. But I don't get why in equation 8.9 the cross terms vanish? If anyone can help me, that would be really nice.
 

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... they vanish due to the definition of "center of mass".
The text explains with maths in the passage right below eq8.9.
Have you tried working out the cross terms for some example?
 
The book answers your question already. You need to think about the center of mass in the center of mass frame of reference. Also the author suggests hou look at equation 5.56
 
Equation 5.58 is just R_cm = sum m_i r_i / M.

I get why r' is zero in the frame of the center of mass. But if that's the case, why is the integral of the cross product of r' and v' not equal to zero also?
 

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