This paper derives equations for the relativistic proper period of oscillations of a pendulum driven by the electrical forces and for a pendulum driven by the gravitational forces. The derivations are based on the Einstein’s Special Relativity Theory and in particular on the Lorentz coordinate transformation, which has been experimentally verified many times and which is a well-recognized principle for all the modern physics. Since the pendulum proper period of oscillations is an absolute inertial motion invariant the derived formulas may be used to study the motion dependence of the inertial and gravitational masses. It is found that the well-publicized equivalence between these two masses, which is assumed independent of any inertial motion, cannot be sustained and a new mass equivalence principle must be considered where the equivalence of these two masses holds only at rest. INTRODUCTION The pendulum is an ages proven device that has attracted attention of many researchers in the past for its simplicity of operation, its accuracy to measure time, and for its ability to study the gravitational or electrical fields. One can only wonder why it was not studied in modern times in more detail, since it offers some clues for resolving the “mystery” of the inertial and gravitational mass equivalence, the so called Einstein’s week equivalence principle . Recently an interesting article was published [2] where the author derived relativistic equations of motion for the pendulum sta

-electromagnetic waves and the ether; the velocity of the earth through the ether; the michelson-morley experiment; explanations of the null result; coordinate systems; inertial systems; einstein's two postulates; the relativity postulate; the postulate of the constant velocity of light; time dilation; an experimental test of time dilation; relativity and simultaneity; the problem of synchronizing clocks; length contraction; perpendicular lengths; derivation of the lorentz transformation; properties of the lorentz transformation; the symmetry between coordinate systems; the twin paradox; transformation of velocities in one dimension; the exploding space-ship; the general transformation of angles; the doppler effect; a resolution of the twin paradox; accelerated motion; the inadequacy of newton's laws; the definition of momentum; relativistic mass and rest mass; transformation of momentum and relativistic mass; the invariant form of the transformation laws; total energy and E=mc 2 ; the equivalence of mass and energy; examples of the use of conservation laws; particles with zero rest mass; invariant masses; forces; the variation of relativistic mass with velocity; particle accelerators; particles, time dilation, and the second postulate; units of mass, energy, and momentum; experiments on the equivalence of mass and energy; particle and anti-particle production; the principle of equivalence; spae and general relativity; the gravitational red shift; time measurements in general

Equality of the Inertial and the Gravitational Masses for a Quantum Particle We investigate the interaction of the gravitational field with a quantum particle. We derive the wave equation in the curved galilean spacetime from the very broad Quantum mechanical assumptions and from covariance under the Milne group. The inertial and gravitational masses are equal in that equation. So, we give the proof of the equality for the non-relativistic quantum particle, without applying the equivalence principle to the Schr\"odinger equation and witout imposing any relation to the classical equations of motion. This result constitutes a substantial strengthening of the previous result obtained by Herdegen and the author. Published as: Acta Phys.Polon. B35 (2004) 613-624 arXiv categories: gr-qc