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But since you postulate a force on a mass with some spatial extension,
opposed by an equal force; then I must consider the transmission time of
that force through the mass until an unyielding
force opposing this transmitted force appears: after which event, a
force is transmitted back to the originating force.
The initial unbalanced force gives some compression, the opposing force
appears after some delta t transmission time, and the initial force is
in turn balanced after a further delta t for the return traversal.
This does not appear entirely fanciful. One can propose instrumenting a
rod with a strain gage at each end.
The experiment avoids easy conclusions involving the CofM concept, as
it seems to me.