I am looking for a way to find the center mass of an object whose area is enclosed by [tex]x^{2}[/tex] and [tex]\sqrt{x}[/tex] without computing the x and y seperately (a great deal of paperwork).(adsbygoogle = window.adsbygoogle || []).push({});

So...

[tex]M\overrightarrow{R_{cm}} = \int \overrightarrow{r} dm[/tex]

[tex]\sigma = \frac{M}{A} = \frac{dm}{dA}[/tex]

[tex]\sigma A \overrightarrow{R_{cm}} = \int \sigma \overrightarrow{r} dA[/tex]

[tex]\overrightarrow{R_{cm}} = \frac{\int\int \sigma \overrightarrow{r} dy dx } {\int \sigma (f(x)-g(x))dx}[/tex]

Because they are constants, the sigmas cancel and I eventually end up with...

[tex]\overrightarrow{R_{cm}} = \frac{\int^1_0\int^{x^{2}}_{\sqrt{x}} (xi+yj) dydx}{\int^1_0 x^{2} - \sqrt{x} dx}[/tex]

(The answer is [tex]\frac{9}{20}i + \frac{9}{20}j[/tex])

But what happens when sigma/density is not constant, but is given a value say... xi or something like that? Obviously taking the dot product will not work, and I am unsure about the cross product (I haven't done a lot of vectors).

-Ataman

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# Center of Mass using vectors

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