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I know how to calculate centroids in single integrals and double integrals and derive the formula for center of mass, but I don't know why the center of mass is defined by $\text{mass}_1 \times \text{distance}_1 = \text{mass}_2 \times \text{distance}_2$. I can't find any site that can explain this.

Vince
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    Assuming the one dimensional object takes the form of a rod then this equation comes from equating the turning moments on either sides of the rod.$$m_1g\cdot d_1=m_2g\cdot d_2\iff m_1d_1=m_2d_2$$ –  Jul 20 '20 at 06:48

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If $x_1$ is a point of mass $m_1$ and $x_2$ is a point of mass $m_2$ (assume $x_1<x_2$ to fix ideas) , the center of mass is $$ y=\frac{m_1x_1+m_2x_2}{m_1+m_2} $$ (note that $x_1<y<x_2$).

The equation you asking for is but $$ m_1(y-x_1)=m_2(x_2-y) $$ which is very straightforward to check.

This is the algebra, for the physical reason see Peter Foreman's comment.

Andrea Mori
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