How Do Oscillator Configurations Affect Energy and Damping?

[razor108]

Homework Statement

For a mass m hung vertically from a spring with spring constant s, the restoring Force is F=-sx where x is the displacement from equilibrium.

a)What are the periods of the two harmonic oscillators i) and ii).
b)Assuming eaqch oscillator is moving with the same Amplitude A which has the greater total energy of oscillation and by what factor is it greater?

Homework Equations

F=-sx
omega=(s/m)^1/2

The Attempt at a Solution

I just don't really know if there is any difference between i) and ii) but there has to be since all the following questions build up on it. But how can I prove that mathematically?

 

[Doc Al]

The first thing you need to do is figure out the effective spring constant of those two spring configurations. Analyze how the force depends on the displacement: F = k' x.

 

[razor108]

Yes, I got it now.
The series one adds as in series therefore 1/stotal=1/s1+1/s2 and the other one like parallel therefore stotal=s1+s2.
Wouldn't have thought this because experience actually told me that it shouldn't be so much of a difference... I mean ii) has 4 times more total energy than i).

But now I'm stuck at a new problem.

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

a) What is the damping constant r?
Attempt:
As far as I know \omega ' = \sqrt{ \omega ^2 - \frac{b}{2m}}
But were to go from here?
edith: Ok, got that now.

therefore b = (0.99^2-1)*(s/m)*(2m)*(-1) which in this case is 0.796.

b)What is the Q of the system?
Attempt:
Q=\sqrt{mass * Spring constant} / r
But again I would have to find r from a) which I can't really figure out.
Ok, with a) answered I got Q to beeing 7.1066

c) Confirm that the system is lightly damped.
I think a system is lightly damped if omega' is about euqal to omega, but this can't really be the answer here. Because that's what's stated anyways since 99% is "about equal" isn't it?

d) What new damping constant r_{new} is required to make the system critically damped?
Again, I couldn't find any definition on the web or in my notes what critically damped means.

f)Using r_{new} calculate the displacement at t=1s given that the displacement is zero and the velocity is 5 ms^-1 at t=0.
No clue here.

Noteow can I insert Tex in this forum?