Vertical Spring (gravity)

Vertical spring-mass: gravity adds an extra constant force, shifting the equilibrium position from natural length to L₀ + mg/k.

About the NEW equilibrium, the system undergoes SHM with same ω = √(k/m) — gravity doesn't affect period.

At new equilibrium: spring force kδ exactly balances mg. δ = mg/k = static extension.

Total energy in vertical spring: includes gravitational PE in addition to spring PE.

If we measure displacement y from the new equilibrium: F = −ky and energy = ½ky² (gravity absorbed into the equilibrium shift).

Period unchanged by gravity (in the small-displacement, linear regime).

Static stretch gives a quick way to measure k: k = mg/δ.

Used in: weighing scales, vehicle suspensions, oscillators, accelerometers.