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\[

\bar{y} = \frac{M_x}{M}.

\]

In my calculus book, they are always finding the center of gravity of a flat object or objects spaced apart (like planets).

Suppose we have a block with length and width \(1\) and height \(a\). On top of this block is another with height \(a\), width \(1\), and length \(1 - a\). This block is flush with the first block except it is shorter by \(a\) so one side has that gap. It has uniform mass.

Suppose the density is then \(\rho\). The mass of the first block is \(V\cdot \rho = a\rho\) and the of the second object is \(V\cdot\rho = a(1 - a)\rho\). The total mass is then \(M_1 + M_2 = a\rho(2 - a) = M\).

What is \(M_x\)?