Spring-Mass Oscillator
ω = √(k/m), animated zigzag spring.
Key Notes
Spring-mass system: a mass m attached to a spring of constant k. Displacement from equilibrium produces a restoring force F = −kx.
Equation of motion: m·d²x/dt² = −kx ⇒ SHM with ω = √(k/m).
Period: T = 2π·√(m/k). Independent of amplitude.
Heavier mass → larger T (slower). Stiffer spring → smaller T (faster).
Total energy: E = ½kA². Oscillates between KE and PE.
Horizontal spring: no gravity to consider; equilibrium at natural length.
Vertical spring: gravity stretches it; new equilibrium x₀ = mg/k. SHM about x₀, with same ω = √(k/m).
Damping (friction) reduces amplitude over time; ω stays approximately the same for light damping.
Formulas
Equation of motion
F = ma applied to spring force.
Angular frequency
Independent of amplitude.
Period
Standard formula. Doubling m ⇒ T × √2.
Total energy
Stored when stretched fully.
Important Points
T depends on m and k only — NOT on amplitude or gravity (for ideal Hookean spring).
Doubling m increases T by √2 (~1.41×). Doubling k decreases T by √2.
Vertical spring: gravity just shifts equilibrium; SHM about new equilibrium has same ω.
Springs in series: 1/k_eq = 1/k₁ + 1/k₂. T = 2π·√(m/k_eq) — larger T (softer).
Springs in parallel: k_eq = k₁ + k₂. Smaller T (stiffer).
Common pitfall: writing T in terms of √k (instead of 1/√k). T ∝ 1/√k.
Spring-Mass Oscillator 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.