Solution:

Given: m = 50 kg, K = 5,000 N/m, C = Cc/2

Damped natural frequency is given by:

ωn = √((K/m) - (C^2/4m^2))

Substituting the given values, we get:

ωn = √((5000/50) - ((Cc/2)^2/4(50)^2))

ωn = √(100 - Cc^2/160000)

We know that the critical damping coefficient is given by:

Cc = 2√(mK)

Substituting the given values, we get:

Cc = 2√(50 x 5000)

Cc = 1000 Ns/m

Substituting this value in the expression for ωn, we get:

ωn = √(100 - (1000/160)^2)

ωn = √(100 - 3.125)

ωn = √96.875

ωn = 9.84 rad/s

Therefore, the damped natural frequency when C=Cc/2 is 9.84 rad/s.

Explanation:

Spring mass damper system is a mechanical system that is widely used in engineering applications. It consists of a mass connected to a spring and a damper. The behavior of the system is determined by the mass, spring constant, and damping coefficient. The damped natural frequency of the system is an important parameter that determines the response of the system to external stimuli.

The damped natural frequency is the frequency at which the system oscillates when subjected to damping forces. It is given by an expression that depends on the mass, spring constant, and damping coefficient. In this problem, we are given the values of mass and spring constant, and we need to find the damped natural frequency when the damping coefficient is half of the critical damping coefficient.

The critical damping coefficient is the minimum damping coefficient required to prevent the system from oscillating indefinitely. It is given by an expression that depends on the mass and spring constant. In this problem, we are also given the values of mass and spring constant, and we can use them to calculate the critical damping coefficient.

Once we have the value of the critical damping coefficient, we can use it to calculate the damped natural frequency by substituting it in the expression for ωn. Finally, we simplify the expression to get the numerical value of the damped natural frequency, which is approximately 9.84 rad/s.