Finding Center of Mass of Nonuniform Rod: Integrating Over dm

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To find the center of mass of a nonuniform rod, understanding how to integrate over dm is crucial, especially when density ρ varies with position x. The relationship dm = ρdx is fundamental, but direct integration over dm can be complex if ρ is not constant. It’s important to recognize when to switch variables and how to handle the varying density effectively. Approximating dV as dx and using the relationship dm/dV = ρ can simplify calculations. Properly applying these concepts allows for accurate determination of the center of mass in nonuniform materials.
BareFootKing
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When looking at example 3 in this pdf: http://www.physics.isu.edu/~hackmart/centerofmass.pdf where it shows how to find the Center of Mass of a Nonuniform Rod. I was wondering when or what you have to know in order to integrate over dm rather than changing the variables in terms of dx.
 
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The basic relationship is dm = ρdx, where ρ is the density. When ρ varies with x, it is very difficult to work directly with dm.
 
Thank you for the response. Do we approximate dV=dx dm/dv=p
 

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