It is shown from Newton's second law and the conservation of linear momentum, for arbitrary types of force, that the full "formula force" used in the second law is effective in exerting observable ponderomotive action on a test element only if the force-exerting element is infinitely massive or is otherwise immobilized in an inertial system to preclude its recoil. In the more realistic case of finite mass m of a force-exerting element free to recoil under mutual action-reaction, the recoil motion "steals" energy from the observable force action, so that the physically effective force exerted on a test element of mass M can be represented as the product of formula force and an inertial modulation factor Omega = m/(M + m) less than or equal to 1. Application of this Elementary result to classical electrodynamics shows that it can invalidate (as physics) many of the theorems universally taken for granted since the nineteenth century. For instance, the well-established mathematical theorem that a closed current loop external to a straight current-carrying test element necessarily exerts zero longitudinal ponderomotive force on the latter (for the Lorentz force law, the original Ampere force law, and all others differing from these only by an additive exact differential quantity) need not be valid physically, The reason is that the inertial factor Omega has been overlooked. Its presence as an extra factor (Green's function) multiplying the force differential can spoil the exact differential nature of the loop integrand. For instance, if the external loop is physically configured with a "weak link," namely, an "unanchored" section of conductor of relatively low mass m much less than M and high mobility (freedom to recoil), then along this portion of the circuit Omega approximate to 0, whereas for the remainder of the circuit, if anchored in the laboratory, Omega approximate to 1. Hence an integral around the whole circuit treats the low-mass portion of the circuit much as if it were an open gap despite the presence of current in it. This proposition is easily put to observational test. By using an electromagnetically driven tuning fork bearing straight segments of current-carrying conductor as sensor (test element), so arranged as to respond to longitudinal force, it has been confirmed through an observed alteration of fork resonance response under ac excitation that a suitably configured weak link in an external closed electrical circuit can, indeed, cause a readily detectable violation of the closed-loop-no-longitudinal-force theorem. In sum, we have exhibited "cross talk" between electromagnetic and inertial properties of conducting circuits that can limit the applicability of many of the classical electrodynamic theorems concerning "closed loops." Such theorems are assuredly valid only in the special case of completely immobilized force-exerting circuits, or circuits all parts of which are constrained to prescribed states of motion (not free to recoil). The ability to violate classical theorems that for a century have made empirical choices among candidate force laws "impossible" implies that now experiments can be designed to allow such choices to be made unambiguously.