Which parameter determines whether a second-order system oscillates or decays smoothly?

• A. Damping Factor
• B. Natural Frequency
• C. Resistance
• D. Capacitance

Answer: (A)

Damping controls how energy is removed from the system each cycle. In a second-order system described by a form like x'' + 2ζω_n x' + ω_n^2 x = 0, the damping ratio ζ (the damping factor) determines the nature of the response. If ζ is less than 1, the system is underdamped and the motion oscillates with a gradually shrinking amplitude. If ζ equals 1, it’s critically damped and returns to equilibrium as quickly as possible without oscillating. If ζ is greater than 1, it’s overdamped and decays smoothly to equilibrium without oscillating. The natural frequency ω_n sets how fast the oscillations would occur if there were no damping, but it does not decide whether oscillations appear—the damping factor does. Resistance and capacitance can influence damping in circuits, but the key parameter that decides oscillatory versus smooth decay is the damping factor.