Natural Frequency:
The natural frequency of a system is the frequency at which it oscillates when there are no external forces acting on it. In this case, the system has a natural frequency of 5 Hz.

Given Data:
- Mass (m) = 10 kg
- Moment of Inertia (J0) = 5 kg-m^2
- Distance to first center of mass (r1) = 100 mm = 0.1 m
- Distance to second center of mass (r2) = 250 mm = 0.25 m

Amplitude of Free Vibration:
When the system is disturbed by giving it an initial displacement, it undergoes free vibration. The amplitude of free vibration is the maximum displacement from the equilibrium position. In this case, the amplitude of free vibration is reduced by 80 percent in 10 cycles.

Damping Ratio:
The damping ratio (ζ) is a measure of how quickly the vibrations of a system decay after being disturbed. It is given by the equation ζ = c / (2√(mk)), where c is the damping coefficient.

Decay Ratio:
The decay ratio (DR) is the ratio of the amplitude of one vibration cycle to the amplitude of the previous cycle. In this case, the decay ratio is 0.8, which means the amplitude is reduced by 80 percent in each cycle.

Number of Cycles:
We are given that the amplitude is reduced by 80 percent in 10 cycles. This means that the decay ratio is 0.8 and the number of cycles is 10.

Calculating Damping Coefficient (c):
We can use the decay ratio and the number of cycles to calculate the damping coefficient (c). The formula to calculate c is c = -2πfn / (log(DR)), where f is the natural frequency and n is the number of cycles.

Substituting the given values, c = -2π(5)(10) / (log(0.8)) ≈ 26.12 Ns/m

Calculating Stiffness (k):
The stiffness (k) of the system can be calculated using the formula k = 4π²mfn² - J0ω², where ω is the angular frequency given by ω = 2πf.

Substituting the given values, ω = 2π(5) = 10π rad/s, k = 4π²(10)(5)² - 5(10π)² ≈ 1960 N/m

Final Values:
- Damping Coefficient (c) ≈ 26.12 Ns/m
- Stiffness (k) ≈ 1960 N/m

Explanation:
- The natural frequency of the system is given as 5 Hz.
- The amplitude of free vibration is reduced by 80 percent in 10 cycles.
- Using the decay ratio and the number of cycles, we can calculate the damping coefficient (c) to be approximately 26.12 Ns/m.
- The stiffness (k) of the system can be calculated using the given formula to be approximately 1960 N/m.
- These values of c and k represent the damping