Center of mass and distance ratio

AI Thread Summary
The discussion focuses on demonstrating that the ratio of distances from two particles to their center of mass (CM) is inversely proportional to their masses. It begins with the definition of the center of mass, expressed as x_{cm} = (m_1ℓ_1 + m_2ℓ_2) / (m_1 + m_2). By setting the CM coordinate to zero, the origin is effectively shifted, allowing for the calculation of distance ratios. It's emphasized that one distance will be negative due to this shift, which is crucial for the correct interpretation of the ratio. This approach highlights the relationship between mass and distance in the context of the center of mass.
kthouz
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how to show that the ratio of the distance between two particles from their centers of mass is the inverse of their masses? I tried but i found a way which can be possible if only we assume that they are constituted by a same number of small particles.
 
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Start with the definition of CM:
x_{cm}={m_1\ell_1+ m_2\ell_2 \over m_1+m_2}
Then ask that the coordinate of the CM be zero. This is just a change in the position of the origin of x.
Then compute de ratio of distances to the CM. Don't forget that now one of the distances in negative. See why?
 

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