Gravitationally Equivalent Point Mass

There are two objects, a thin rod of mass \(M\) in the interval \([0,1]\) on the \(x\)-axis, and a point mass at \(x=2\) of the same mass \(M\), as shown above.

Suppose we wanted to replace the rod with a particle of mass \(M\), which would exert the same gravitational force on the point particle at \(x = 2\).

What should its approximate \(x\)-coordinate be?


Inspired by Steven Chase
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