Damped Oscillation Homework: Calculating b & Q for Lightly Damped System

[razor108]

Homework Statement

A damped oscillator of mass m=1,6 kg and spring constant s=20N/m has a damped frequency of \omega' that is 99% of the undamped frequency \omega.

As found out by me:
The damping constant b is 0.796 kg/s.
Q of the system is 7.1066 kg^-1.
Are the units here right?

The questions are:
a) Confirm that the system is lightly damped.
b) What new damping constant b_new is required to make the system critically damped?
c) Using b_new calculate the displacement of the mass at t=1,0s given that the displacement is zero and the velocity is 5,0 m/s at t=0.

Homework Equations

To calculate b I used
\omega' = ( \omega^2 - b/2m)^1/2


To calculate Q I used
Q = ((mass*spring constant)^1/2)/b

The Attempt at a Solution

I couldn't find any specific definition of when a system is lightly damped.
I found somewhere that if \omega' is about equal to \omega the system is lightly damped, which is the case here (\omega' = 99%\omega) but this can't be the answer since I have to find a new damping constant.

If I had the new damping constant I would just use the given data to make up a wave equation for x(t).

Thanks for helping.

 

[hage567]

To calculate b I used
\omega' = ( \omega^2 - b/2m)^1/2

I think this should be $$\omega' = \sqrt{\omega^2 - (\frac{b}{2m})^2}$$

I couldn't find any specific definition of when a system is lightly damped.

I think lightly damped means the same and underdamped. It just means the system is not critically damped or overdamped. So it is just damped enough that the amplitude decreases exponentially over time but still oscillates.

I found somewhere that if \omega' is about equal to \omega the system is lightly damped, which is the case here (\omega' = 99%\omega) but this can't be the answer since I have to find a new damping constant.

But in that part of the question you are supposed to find the new damping constant that would make the system critcally damped, that's a different situation.

Hope that helps.