Explanation:

When analyzing the behavior of a system, it is important to consider its damping ratio. The damping ratio is a dimensionless measure that describes how oscillations in a system decay after a disturbance. It is denoted by the symbol ζ (zeta).

Definition of Damping Ratio:
The damping ratio, ζ (zeta), is defined as the ratio of the actual damping coefficient of a system to its critical damping coefficient. The critical damping coefficient is the minimum amount of damping required to prevent oscillations.

Critical Damping:
A system is said to be critically damped when the damping ratio is equal to 1 (ζ = 1). In this case, the system returns to its equilibrium position in the shortest possible time without oscillating.

Effect of Gain on Damping Ratio:
When the gain of a system is doubled, it affects the damping ratio of the system.

- The gain of a system refers to the amplification factor applied to the system's input signal. It determines the response of the system to the input signal.
- Doubling the gain means increasing the amplification factor by a factor of 2.

Effect on Damping Ratio:
Doubling the gain of a system changes the damping ratio of the system. Specifically, it changes the damping ratio from being critically damped to being underdamped.

Underdamped System:
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Conclusion:
Therefore, when the gain of the system is doubled, the system changes from being critically damped to being underdamped. This is because the increase in gain affects the damping ratio, causing it to decrease below 1. Consequently, the system will exhibit oscillations before settling to its equilibrium position.