This research paper explores the framework of Extended Classical Mechanics, with a focus on the Equivalence Principle, Mass, and Gravitational Dynamics. Volume 1 of this study re-examines the classical equivalence principle, which maintains the equivalence of inertial mass and gravitational mass (also referred to as gravitating mass) and extends this concept to incorporate new findings related to both illuminating (baryonic) matter and dark matter. The paper provides a comprehensive analysis of Matter Mass (Mᴍ) and Gravitating Mass (Mɢ) and their roles within the extended framework. It introduces the concept of Apparent Mass (Mᵃᵖᵖ), a negative mass component that influences Effective Mass (Mᵉᶠᶠ), in alignment with the research by A.D. Chernin et al. on dark energy and the structure of the Coma Cluster of Galaxies. Additionally, the paper reinterprets Newton's Law of Universal Gravitation by integrating Apparent Mass, leading to a revised understanding of gravitational potential. The study demonstrates how the interaction between Matter Mass and Negative Apparent Mass contributes to a redefined concept of Gravitating Mass. These extensions enhance classical mechanics by incorporating modern scientific insights, including the gravitational effects of dark matter and the alignment of Apparent Mass with the negative effective mass of dark energy.