Resonance (Mechanical)
Q-factor controls peak sharpness.
Key Notes
Resonance: when driving frequency ω_d matches natural frequency ω₀, response amplitude is MAXIMUM.
Slightly DOWNSHIFTED by damping: ω_res = √(ω₀² − 2γ²) ≈ ω₀ for light damping.
Resonance amplitude A_res = F₀/(b·ω₀) — inversely proportional to damping.
Examples: tuning a radio (LC resonance), pushing a swing in time with its motion, microwave heating water (molecular resonance ~ 2.45 GHz), MRI scans.
Sharpness: Q-factor measures resonance sharpness; higher Q = sharper, larger amplitude at resonance.
Phase: at resonance, response lags drive by exactly π/2.
Resonance can be CONSTRUCTIVE (musical instruments, lasers) or DESTRUCTIVE (Tacoma Narrows, glass shattered by voice).
Mathematical analog applies to electromagnetic systems (LCR circuits), atoms (absorption spectra), nuclei (Mössbauer effect).
Formulas
Resonance frequency
Slightly less than natural for damped systems.
Amplitude at resonance
Larger for smaller damping.
Q-factor
Δω = full width at half maximum (FWHM) of resonance curve.
Bandwidth
Narrow for high-Q systems.
Important Points
Resonance can amplify small forces to large amplitudes — dangerous in structures, useful in sensors.
High Q-factor: sharp resonance peak, large amplification at exactly ω₀.
Low Q: broad, low peak. Damping spreads the response over a wider frequency range.
Tuning forks, organ pipes, lasers, antennas — all rely on resonance for selectivity.
MRI uses resonance of hydrogen nuclei in strong magnetic field at specific RF frequency.
Buildings, bridges have natural frequencies — earthquake design avoids these matching seismic spectra.
Resonance (Mechanical) 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.