Vertical Spring (gravity)
x_eq = mg/k offset; period unchanged.
Key Notes
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.
Formulas
Static extension
Equilibrium stretch from natural length.
Period (vertical = horizontal)
Same as horizontal; gravity cancels out.
Angular frequency
Useful: measure δ to find ω.
Effective PE about new equilibrium
Gravity term absorbed into the equilibrium shift.
Important Points
Gravity shifts equilibrium but doesn't change ω, T, or frequency.
Measuring static extension δ → directly gives k = mg/δ. Useful in labs.
Frequency f = (1/2π)√(g/δ) — connects pendulum-like formula to spring oscillation.
If displacement is measured from natural length: equation has both spring and gravity. Switch to new origin for SHM clarity.
Many real systems (car suspensions, watches) work this way: gravity sets equilibrium; vibrations are SHM about it.
Adding mass: increases T (heavier slower) and changes δ.
Vertical Spring (gravity) notes from sciphylab (also known as SciPhy, SciPhy Lab, SciPhy Labs, Physics Lab). Class 11 physics revision for JEE Mains, JEE Advanced, NEET UG, AP Physics 1/2/C, SAT, and CUET-UG.