Calculating Q for a Driven Harmonic Oscillator at Resonance

[LBloom]

Homework Statement

The amplitude of a driven harmonic oscillator reaches a value of 25.0Fo /m at a resonant frequency of 380 Hz. What is Q?

Homework Equations

Q=m(wo)/b

Q=wo/(\Deltaw)

w=2\pif

wo= (k/m)^1/2

w'=(w0^2-y^2)^.5

y=b/2m

The Attempt at a Solution

Honestly, I don't know where exactly to start. They tell me that the resonant frequency, but is that wo (natural angular frequency) or w' (the frequency when damping is considered.) Also, I don't know what they mean by amplitude is 25 Fo/m. Force divided by mass isn't in units of distance (although in the textbook its Fo/k which makes more sense, but its another edition)

In an older version of the textbook, everything is the same except the resonant frequency is 383 and the amplitude is 23.7 F0/k. Somehow the value of Q is also 23.7. Can anyone help me start/decipher this problem? I understand damped and forced oscillations, (this is the last qu) but i just can't wrap my head around it

Homework Statement

Homework Equations

The Attempt at a Solution

 

[Delphi51]

I haven't answered for several hours because I don't know much about this. But it sure looks like the question is bad. To find Q, you need to know either the frequency range of the resonance as in Q = w/(delta w) = f/(delta f) or else the energy and energy loss. The amplitude might possible let you find the energy, but there is no clue to the energy loss.

 

[LBloom]

if its at resonance there shouldn't be energy loss (theoretically). I'm probably going to sleep on the problem and play with the equations later. I have maybe 2 and half hours to go to the physics help room before its due, so that's my last option if i can't get it this weekend. :)

 

[Delphi51]

If there is no energy loss, the Q is infinite.

 

[LBloom]

true, and i am looking for b (or y=b/2m or any possible relationship) so it is a damped forced oscillation.

 

[Delphi51]

Oh, do you know that b, y or m are somehow related to damping?
I assume m is the mass and have no idea what b and y are. Maybe there is hope after all.

 

[LBloom]

The equations is
x=Ae^(-y*t)*Cos(w't)
where w' is the angular frequency of the damped oscillator and y=b/2m
Force of damping is Fdamp=-bv, so b is a constant. However, I'm not sure how to find b or y, and the above equation is not for forced oscillations. Before when i had to find y, instead of finding b, i found a relation between w and Q that equaled b/2m and hence y. i was going to try the same thing, but i don't have that much information. I may be overthinking again, i dunno.

 

[Delphi51]

Check http://en.wikipedia.org/wiki/Q_factor. They have a formula relating Q to energy and energy loss per cycle.
Can you find the energy from your x= formula? Maybe differentiate it to get velocity and use Ek = .5m*v^2. Get the energy at its first and second peak values and you should be able to pull a Q value out of that formula.

I have to sign off for the night. Good luck!

 

[LBloom]

I found the equation i needed online (it wasn't mentioned once in my textbook so i don't know how i was expected to derive it)

Ao=(F/m)/[ ((w^2-wo^2)+b^2(w^2/m^2))^.5]

I actually learned a similar equation and how to derive it over the summer through mit open courseware and walter lewin, but the formula didnt include the last term with b (he only dealt w/ undamped oscillators)

When there's resonance, the first term cancels out because w=wo. The second term can be manipulated by using the equation wo/Q=b/m (derived from one of the original equations.
The problem simplifies out to
25*F/m=F*Q/ (m*wo^2). The f/m cancels out (which explains why they included that with amplitude, and all that's left is one unknown, Q.

Hurrah for the internet!

 

[Delphi51]

Excellent - congrats!