The problem involves a damped harmonic oscillator with the system's damping described by the damping ratio \( ζ \). In such systems, the damping ratio is defined as \( ζ = \frac{c}{c_c} \), where \( c \) is the damping coefficient, and \( c_c \) is the critical damping coefficient. It is given that the damping ratio is equal to the ratio of the damped frequency \( ω_d \) to the natural frequency \( ω_n \).
First, we compute the natural frequency \( ω_n \) as follows: \( ω_n = \sqrt{\frac{k}{m}} \), where \( k = 10 \) N/m and \( m = 1 \) kg. Substituting these values, we find:

\( ω_n = \sqrt{\frac{10}{1}} = \sqrt{10} \approx 3.16 \) rad/s.

The damped frequency \( ω_d \) for an underdamped system is given by \( ω_d = ω_n\sqrt{1-ζ^2} \). We have \( ζ = \frac{ω_d}{ω_n} \). Substituting this into the damped frequency formula, we get:

\( ω_d = ω_n\sqrt{1-ζ^2} = ζω_n \).

Dividing both sides by \( ω_n \),

\( ζ = \sqrt{1-ζ^2} \).

Squaring both sides, we have:

\( ζ^2 = 1-ζ^2 \).

Thus, \( 2ζ^2 = 1 \), giving \( ζ^2 = 0.5 \).

Hence, \( ζ = \sqrt{0.5} \approx 0.707 \).

The damping ratio rounded to two decimal places is \( ζ = 0.71 \). Verifying this within the provided range [0.7, 0.72], our calculated result \( 0.71 \) is indeed within the expected range.