Cross products vanish Classical Mechanics

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SUMMARY

The discussion focuses on the vanishing of cross terms in the context of classical mechanics, specifically in the x-y plane as described in equation 8.9. The key reason for this phenomenon is the definition of the center of mass, which is elaborated in the text following equation 8.9. Participants emphasize the importance of analyzing the problem from the center of mass frame of reference, referencing equations 5.56 and 5.58 to clarify the relationship between position vectors and velocities in this context.

PREREQUISITES
  • Understanding of classical mechanics principles, particularly center of mass.
  • Familiarity with vector calculus, especially cross products.
  • Knowledge of equations of motion in the x-y plane.
  • Ability to interpret mathematical expressions in physics texts.
NEXT STEPS
  • Study the derivation and implications of the center of mass frame of reference.
  • Explore the mathematical properties of cross products in vector calculus.
  • Review equations 5.56 and 5.58 in the context of classical mechanics.
  • Practice solving problems involving cross products and center of mass in various reference frames.
USEFUL FOR

Students of physics, educators teaching classical mechanics, and anyone interested in the mathematical foundations of motion in the center of mass frame.

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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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