Question bank — Damped oscillations — underdamped, critically damped, overdamped

Partly true aur partly ek trap: ek underdamped system ${\omega }_d<{\omega }_0$ pe oscillate karta hai, toh frequency sach mein kam hoti hai — lekin jab $\gamma \ge {\omega }_0$ ho toh koi oscillation frequency hi nahi hoti, isliye "frequency kam karna" wahan meaningless ho jaata hai.

False. Overdamped ka matlab hai do real negative roots, isliye dono terms pure decaying exponentials hain — mass monotonically equilibrium ki taraf jaata hai (zyada se zyada ek baar zero cross kar sakta hai) aur bilkul bhi wobble nahi hota.

False. Ye bina overshoot ke sabse tezi se wapas aana deta hai, lekin phir bhi finite time lagta hai — solution $(A+Bt)e^{-\gamma t}$ sirf $t\to \infty$ pe zero tak pahunchta hai.

False. $b$ badhane se settling tab tak tezi hoti hai jab tak critical damping na aa jaye; uske baad (overdamped) slow root ${\lambda }_1=-\gamma +\sqrt{{\gamma }^2-{\omega }_0^2}$ zero ke kareeb aa jaata hai aur system phir se dhima ho jaata hai.

False. Amplitude $e^{-\gamma t}$ ki tarah decay karta hai, lekin energy $\propto$ amplitude${}^2$ hoti hai, isliye energy $e^{-2\gamma t}$ ki tarah decay karti hai — exponent mein do guna tezi se.

True. Kyunki $Q={\omega }_0/2\gamma$ hai, $Q=\frac{1}{2}$ set karne se $\gamma ={\omega }_0$ milta hai, jo exactly critical damping hai — discriminant ki borderline.

False. Kyunki ye $b>0$ ke saath $-\dot{x}$ ke proportional hai, ye hamesha current velocity ko oppose karti hai, isliye ye kabhi energy add nahi kar sakti.

False. Ek repeated root sirf ek exponential deta hai; required second independent solution $t{e}^{-\gamma t}$ hai, isliye general form $(A+Bt)e^{-\gamma t}$ hai.

Use drop karna galat hai: underdamped ke liye root imaginary hoti hai, $\sqrt{{\gamma }^2-{\omega }_0^2}=i{\omega }_d$, aur wahi imaginary part precisely oscillation produce karta hai — aap use $\cos ({\omega }_{d}t+\varphi )$ ke roop mein rakhte ho.

Error: ${\omega }_d=0$ ka matlab hai koi oscillation nahi, koi motion nahi nahi. Mass phir bhi $(A+Bt)e^{-\gamma t}$ ke zariye equilibrium mein wapas jaata hai; sirf periodic part gayab hota hai.

Sign galat hai: overdamped ke liye ${\gamma }^2-{\omega }_0^2>0$ chahiye (discriminant positive) taaki square root real ho; negative toh underdamped case ke imaginary roots deta.

Zyada ho gaya: overdamping bounce toh rokti hai lekin sluggishly wapas aati hai, bumps pe mushy feel hoti hai. Engineers critical damping ke paas aim karte hain bounce-free recovery ke liye.

Sirf zero damping ke liye: damping present hone par aap actually ${\omega }_d=\sqrt{{\omega }_0^2-{\gamma }^2}<{\omega }_0$ measure karte ho; ${\omega }_0$ idealized undamped value hai.

Ulta hai: high $Q={\omega }_0/2\gamma$ ka matlab hai chhota $\gamma$, yani light damping — fork kaafi cycles tak bajta rehta hai aur energy dheere khoता है.

Ye sirf cosmetic nahi hai: $m$ se divide karne se physically meaningful $\gamma =b/2m$ aur ${\omega }_0=\sqrt{k/m}$ saamne aate hain, jo quantities hain jis par poori classification depend karti hai.

Kyunki low speed par viscous fluid/air drag sach mein speed ke saath scale karta hai — jitna tezi se medium mein push karo, utna zyada resist karta hai — isliye $-b\dot{x}$ sabse simple physically realistic loss term hai.

Kyunki $e^{\lambda t}$ ko differentiate karne se bas use $\lambda$ se multiply hota hai, differential equation ko $\lambda$ mein ek ordinary algebraic (quadratic) equation mein convert kar deta hai jise hum directly solve kar sakte hain.

Euler's relation $e^{i{\omega }_{d}t}=\cos {\omega }_{d}t+i\sin {\omega }_{d}t$ se, ek imaginary exponent sines aur cosines generate karta hai — back-and-forth motion ka mathematical fingerprint.

Pure convenience: ye quadratic roots ko $-\gamma \pm \sqrt{{\gamma }^2-{\omega }_0^2}$ ke roop mein clean coefficients ke saath deta hai bjaaye $\frac{1}{2}$ ke trailing factors ke.

Heavier (overdamped) damping ek slow root ${\lambda }_1$ introduce karta hai jo zero ke paas lingering rehta hai; exactly critical par, decay exponent $-\gamma$ magnitude mein utna bada hota hai jitna possible hai overshoot avoid karte hue.

Amplitude $e^{-\gamma t}$ ki tarah girta hai, aur stored energy $\sim \frac{1}{2}k{A}^2$ amplitude ke square par depend karti hai, isliye $E\propto (e^{-\gamma t}{)}^2=e^{-2\gamma t}$.

Kyunki dono roots real hain, solution do decaying exponentials ka sum hai different rates ke saath — koi periodic term nahi hai, isliye koi frequency quote karne ko hai hi nahi.

Roots $\lambda =\pm i{\omega }_0$ ban jaate hain, ${\omega }_0$ par pure Simple Harmonic Motion dete hain constant-amplitude, kabhi na khatam hone wali oscillation ke saath.

Discriminant zero hai, isliye ek repeated root $\lambda =-\gamma$ hai aur solution $(A+Bt)e^{-\gamma t}$ hai — ek single, possibly-once-crossing, non-oscillatory return.

Fast root ${\lambda }_2\approx -2\gamma$ almost instantly mar jaata hai, jabki slow root ${\lambda }_1\approx -{\omega }_0^2/2\gamma \to 0^{-}$ aur bhi slower hota jaata hai, toh system bahut sluggishly wapas creep karta hai.

Haan — ye shrinking envelope $\pm A{e}^{-\gamma t}$ ke andar oscillate karta hai, kaafi baar equilibrium cross karta hai, har swing pehle se chhoti.

Zyada se zyada ek baar: initial velocity ke hisaab se ye equilibrium ko ek baar overshoot kar sakta hai, lekin ye periodically usse wapas nahi guzar sakta kyunki koi oscillatory term nahi hai.

Bahut chhota aur real, isliye system bahut low frequency par oscillate karta hai lambe period ke saath, jabki envelope $e^{-\gamma t}$ comparatively tezi se decay karta hai — aap barely ek ya do faint wobbles dekhte ho.

Humne system ko loss-free idealize kar liya; real oscillator phir bhi energy khoyega, isliye genuinely $b=0$ (undamped SHM) ek mathematical idealisation hai, jo Forced Oscillations & Resonance se connect karta hai jahan energy resupplied hoti hai.

Identically: $R$ ka role $b$ jaisa hai, isliye $R^2<4L/C$ underdamped hai, $R^2=4L/C$ critical, aur $R^2>4L/C$ overdamped — same discriminant dono ko Second-order linear ODEs ke zariye govern karta hai.