A certain vibrating system satisfies the equation u'' + γu' + u = 0. Find the value of the damping coefficient γ for which the quasi period of the damped motion is 90% greater than the period of the corresponding undamped motion.

The value of the damping coefficient γ is (6/19)√29, which is approximately 1.70058.

Step-by-step explanation:

Given that a certain vibrating system satisfies the equation

u'' + γu' + u = 0

We wish to find the value of the damping coefficient γ for which the quasi period of the damped motion is 90% greater than the period of the corresponding undamped motion.

The period of the undamped motion T is 2π

The quasi period T_d is (4π)/√(4 - γ²)

Since the quasi period is 90% greater than the period of the undamped motion, we have

T_d = 1.9T

So that we can finally obtain

γ as 1.70058.

THE STEP-BY-STEP WORKING IS SHOWN IN THE ATTACHMENTS.