Final answer:
The values of x that result in underdamping, overdamping, or critical damping depend on system parameters and are determined by comparing the system's natural frequency to its damping ratio. For a mass-spring system, it's based on the relationship √k/m and b/2m, and for an RLC circuit, it's √1/LC relative to R/2L.
Step-by-step explanation:
When analyzing a damped harmonic oscillator, such as a mass-spring system or an RLC circuit, the behavior of the system is dependent on its damping ratio. The three types of damping are underdamping, overdamping, and critical damping. In underdamped systems, the system oscillates about the equilibrium point and decays over time. For an overdamped system, the approach to equilibrium is slower than critically damped and does not oscillate. A critically damped system moves toward equilibrium as quickly as possible without any oscillations, which is ideal for systems like automotive suspensions where quick settling is desired.
The specific values of x (or system parameters) that result in each type of damping depend on the system's characteristic equation, which relates physical constants such as mass (m), damping coefficient (b), and spring constant (k) for a mass-spring system, or inductance (L), resistance (R), and capacitance (C) for an RLC circuit. The general rule is that a system is underdamped if √k/m > b/2m, critically damped if √k/m = b/2m, and overdamped if √k/m < b/2m. By analogy, an RLC circuit is underdamped if √1/LC > R/2L, critically damped if √1/LC = R/2L, and overdamped if √1/LC < R/2L.