The damping ratio and undamped natural frequency of a closed loop system as shown in the figure, are denoted as ξ and ω_n , respectively. The values of ξ and ω_n are

The correct option is ξ = 0.5 and ω_n = 10 rad/s.

Step-by-step Analysis of the Block Diagram:
By analyzing the provided block diagram, we can identify the following components and connections:
1. An input signal $R\left(s\right)$ and an output signal $Y\left(s\right)$.
2. A first block with a transfer function of $G_1\left(s\right)=\frac{10}{s}$.
3. A second block with a transfer function of $G_2\left(s\right)=\frac{10}{s}$.
4. An inner negative feedback loop around the first block $G_1\left(s\right)$ with unity feedback gain.
5. An outer negative feedback loop feeding back the final output $Y\left(s\right)$ to the first summing point with unity feedback gain.

Step 1: Simplify the Inner Feedback Loop
The inner loop consists of the forward block $G_1\left(s\right)=\frac{10}{s}$ and a unity negative feedback path. The transfer function of this closed inner loop, $H_{\text{inner}}\left(s\right)$, is calculated as:

$$H_{\text{inner}}\left(s\right)=\frac{G_1\left(s\right)}{1+G_1\left(s\right)}=\frac{\frac{10}{s}}{1+\frac{10}{s}=\frac{10}{s+10}}$$

Step 2: Determine the Open-Loop Transfer Function
The simplified inner loop is in series with the second forward block $G_2\left(s\right)=\frac{10}{s}$. Therefore, the combined forward path transfer function $G\left(s\right)$ is:

$$G\left(s\right)=H_{\text{inner}}\left(s\right)·G_2\left(s\right)=\frac{10}{s+10}·\frac{10}{s}=\frac{100}{s(s+10)}$$

Step 3: Determine the Closed-Loop Transfer Function
The entire system has a unity negative feedback loop, so the closed-loop transfer function $T\left(s\right)=\frac{Y\left(s\right)}{R\left(s\right)}$ is given by:

$$T\left(s\right)=\frac{G\left(s\right)}{1+G\left(s\right)}=\frac{\frac{100}{s(s+10)}}{1+\frac{100}{s(s+10)}}=\frac{100}{s^2+10s+100}$$

Step 4: Extract the Damping Ratio (ξ) and Natural Frequency (ω_n)
The standard second-order system transfer function is written as:

$$T\left(s\right)=\frac{{\omega }_n^2}{s^2+2\xi {\omega }_{n}s+{\omega }_n^2}$$

By comparing the denominators of our transfer function and the standard form:
1. Compare the constant term:

$${\omega }_n^2=100$$

Taking the square root gives the undamped natural frequency:

$${\omega }_n=10\text{rad/s}$$

2. Compare the coefficient of the $s$ term:

$$2\xi {\omega }_n=10$$

Substitute ${\omega }_n=10$ into the equation:

$$2\xi \left(10\right)=10$$

$$20\xi =10$$

$$\xi =\frac{10}{20}=0.5$$

Thus, the values of damping ratio and undamped natural frequency are $\xi =0.5$ and ${\omega }_n=10\text{rad/s}$ respectively.