According to Galileo's discovery the objects fall at the same rate due to the equality of inertial and gravitational masses. Where in our universe these two would not be equal? Following the question, if so does that mean that objects would have different rate of fall? Could someone give me sources of attempts to prove this equality wrong?
$\begingroup$ You will have better luck finding attempts to prove it is right. For example, see the Eötvös experiment $\endgroup$– mmesser314Jul 4, 2022 at 1:45
It is possible to design 3 experiments using several identical test pieces of the same material:
Monitoring the acceleration of one, two, or any other number of pieces joined together in free fall. We can check that they fall with the same acceleration, and we call that acceleration $g$.
Making them rotate with a given angular velocity $\omega$, changing the number of pieces being rotated. Between the center and the pieces, there is a spring to measure force. In that way, we check the second Newton's law: $F = m_i a$, where $a = \omega^2 R$ is the centripetal acceleration. We can choose $\omega$ so that $a = g \implies F = m_i g $.
Hanging different number of this pieces in a spring. In that way we check that $F = km_i$, for some constant $k$.
The fact that numerically $k = g$ is an empirical result. We could imagine without logical contradiction another outcome. For example, if $k = 3g$, we would postulate a gravitational mass $m_g = 3m_i$, so that $F = m_g g$