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10 cards from this deck
SHM with resistive force proportional to velocity
Oscillation about a moving equilibrium position
d2xdt2+kdxdt+ω2x=0\frac{d^2x}{dt^2} + k\frac{dx}{dt} + \omega^2 x = 0dt2d2x+kdtdx+ω2x=0
Resistance strength (larger kkk = stronger resistance)
Discriminant Δ=k2−4ω2\Delta = k^2 - 4\omega^2Δ=k2−4ω2 of auxiliary eq.
k2−4ω2>0k^2 - 4\omega^2 > 0k2−4ω2>0 (positive)
k2−4ω2<0k^2 - 4\omega^2 < 0k2−4ω2<0 (negative)
Fastest approach to equilibrium without oscillating
CF + PI (Complementary Function + Particular Integral)
Opposes motion (opposite to velocity)
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