Correct Answer  Option 4 : 3.0
Concept:
The general expression of the transfer function of the standard secondorder system is:
\(TF = \frac{{C\left( s \right)}}{{R\left( s \right)}} = \frac{{ω _n^2}}{{{s^2} + 2ζ {ω _n}s + ω _n^2}}\)
Where,
ζ is the damping ratio
ωn is the undamped natural frequency
Characteristic equation: \({s^2} + 2ζ {ω _n} + ω _n^2 = 0\)
The roots of the characteristic equation are:
\( ζ {ω _n} + j{ω _n}\sqrt {1  {ζ ^2}} =  \alpha \pm j{ω _d}\)
α is the damping factor
Calculation:
Given, the characteristic equation s2 + 6s + 25 = 0
Comparing this with the general expression of the transfer function of the secondorder control system, we can write:
2 ζ ω_{n} = 6
⇒ ζ ω_{n} = 3
The nature of the system is described by its ‘ζ’ value
ζ

Nature

ζ = 0

Undamped

0 < ζ < 1

Underdamped

ζ = 1

Critically damped

ζ > 1

Overdamped
