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In classical mechanics, the total energy (E) of a system is defined as:
E = PE + KE
Potential Energy When v = 0:
When an object is at rest, all of its energy is stored as potential energy:
Eₜₒₜₐₗ = PE
which means the total mass equivalent remains simply m.
Energy Distribution When v > 0:
Once an object gains velocity, part of its potential energy is converted into kinetic energy:
PE − ΔPE = PE + KE
The change in potential energy (ΔPE) appears as kinetic energy:
KE = −ΔPE
Effective Mass Contribution:
Since kinetic energy arises from potential energy loss, the effective mass associated with kinetic energy follows:
|mᵉᶠᶠ| = −ΔPE
This ensures that total energy remains balanced, with kinetic energy representing a redistributed form of the system’s original mass-energy.
Given that potential energy (PE) corresponds to the inertial mass (m), and kinetic energy (KE) is linked to an effective mass (|mᵉᶠᶠ|), we express the force equation as:
F = |mᵉᶠᶠ|a
where the effective mass accounts for both the deformation-induced contribution from stiffness (k) and the inertial mass (m):
|mᵉᶠᶠ| = |kΔL|/a
Substituting this into the force equation:
F = (|kΔL|/a)⋅a
Expanding into the energy relation:
E = PE + KE ⇒ m + |mᵉᶠᶠ|
Since mᵉᶠᶠ contributes to balancing total energy, the correct formulation becomes:
E = m + |kΔL|/a
where kinetic energy is directly linked to effective mass (|mᵉᶠᶠ|), sometimes misinterpreted as relativistic mass (m′) and associated with time distortion (Δt′) in certain contexts.