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I was answering a question on proving the parallel axis thereom for angular momentum and came across this: $$\int Yy'dm=Y\int y' dm=0$$ Where the position of the center of mass of an object is given by $(X,Y,Z)$, $(x',y',z')$ is a position relative to the centre of mass and m is the mass of the object.

My text book (Introduction to classical mechanics) says that this is due to the definition of the centre of mass. There are two things that I don't understand firstly why is $Y$ independent of mass whilst $y'$ is not? and secondly please can you explain what definition of the centre of mass they are using to get the reslut above? I really have no idea to the answer for either of these questions?

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The center of mass, $\vec R$, is just a vector. It's not a function of anything, it is just the result of the integral $\frac{1}{m}\int \vec r dm = \frac{1}{m}\int \rho(\vec r)\vec r dV$ ($\rho(\vec r)$ is the volume density) over all the region delimited by the body, it's a definite integral, that gives you a vector.

The fact that the integral you posted is zero is simply because you are measuring distances relative to the CM. What is the position of the CM relative to the CM? Zero. So: $\frac{1}{m}\int \vec r' dm = 0$, this is the integral of a vector, so the integral of every component must be zero. So, $\frac{1}{m}\int y' dm = 0$.

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