Final answer:
The damping coefficient must equal the critical damping coefficient to reach equilibrium quickly without overshooting or oscillating. Critical damping is achieved with c = √4mk, ensuring rapid equilibrium restoration. For no damping, the damping coefficient must be zero, resulting in persistent oscillation.
Step-by-step explanation:
To ensure that a system reaches equilibrium quickly, the damping coefficient constant (c), also known as the damping constant, must be critical. Critical damping occurs when the damping constant is such that it avoids any oscillation about the equilibrium and allows the system to return to equilibrium as swiftly as possible after a disturbance. If critical damping is desired, the damping constant c should be equal to the critical damping coefficient, which is given by c = √4mk, where m is the mass of the system and k is the spring constant. This provides the fastest return to equilibrium without overshoot.
For a system to experience no damping, the damping constant c must be zero. In a no-damping scenario, the system will continuously oscillate without energy loss. This can be seen in an ideal harmonic oscillator where there is no resistive force to oppose the motion.