Final answer:
The correct answer is option c. c = 15.
Step-by-step explanation:
To determine the value of c that leads to an underdamped system with a pseudo-period for the unforced damped harmonic oscillator, we need to consider the angular frequency of the undamped spring and the damping constant.
The equation of motion for the damped harmonic oscillator is given by:
m(d^2x/dt^2) + c(dx/dt) + kx = 0
where m is the mass, c is the damping constant, and k is the spring constant.
In this case, m = 1 and k = 20.
For an underdamped system with a pseudo-period, we have √k/m > c/2m.
Substituting the values, we have √20/1 > c/2*1.
Simplifying, √20 > c/2.
Multiplying both sides by 2, 2√20 > c.
Simplifying further, 4√5 > c.
The only value of c that satisfies this inequality is c = 15.