The given transfer function is:
G(s) = s^6 / (Ks^2 + s^6)


The damping ratio, denoted by ζ (zeta), is a parameter that characterizes the behavior of a second-order linear system. It determines the response of the system to a step input and provides information about the system's stability and transient response.


The damping ratio is defined as the ratio of the actual damping coefficient to the critical damping coefficient. Mathematically, it is expressed as:
ζ = c / c_critical

where c is the actual damping coefficient and c_critical is the critical damping coefficient.


To find the damping ratio of the given transfer function G(s), we need to determine the actual damping coefficient c and the critical damping coefficient c_critical.

1. Actual Damping Coefficient (c):
The actual damping coefficient can be determined by examining the denominator of the transfer function. In this case, the denominator is Ks^2 + s^6. Since the denominator is a quadratic polynomial, we can compare it with the general form of a second-order system's characteristic equation:
s^2 + 2ζω_ns + ω_n^2

Comparing the coefficients, we can see that the actual damping coefficient c is equal to 2ζω_n, where ω_n is the natural frequency.

2. Critical Damping Coefficient (c_critical):
The critical damping coefficient corresponds to the case where the system is critically damped. In this case, the damping ratio is equal to 1. For a critically damped system, the damping coefficient c_critical is given by:
c_critical = 2√(K)


Now, we can equate the actual damping coefficient c to the critical damping coefficient c_critical and solve for the damping ratio ζ.

2ζω_n = 2√(K)

Dividing both sides by 2ω_n, we get:
ζ = √(K) / ω_n

In the given transfer function, the numerator is s^6 and the denominator is Ks^2 + s^6. The natural frequency ω_n can be determined by finding the square root of the coefficient of s^2 in the denominator.

Since the damping ratio is given as 0.5, we can substitute this value into the equation for ζ and solve for K.

0.5 = √(K) / ω_n

Squaring both sides, we get:
0.25 = K / (K + 1)

Simplifying the equation, we have:
0.25(K + 1) = K

Expanding the equation, we get:
0.25K + 0.25 = K

Rearranging the terms, we get:
0.75K = 0.25

Dividing both sides by 0.75, we get:
K = 1/3

Therefore, the value of K for which the damping ratio is 0.5 is 1/3, which corresponds to option C.