How Do You Calculate the Q Value of a Driven Harmonic Oscillator at Resonance?

[fantexihong]

Homework Statement

The amplitude of a driven harmonic oscillator reaches a value of 21.0 F0/k at a resonant frequency of 390Hz . What is the Q value of this system?

Homework Equations

w0=2*pi*f0=(k/m)^(1/2)
Q=(mvo)/b
A = (Fo/m)/ (sqrt( (wo^2-w^2)^2+(b^2*w^2)/m^2) )

The Attempt at a Solution

First I did k=mw^2, 21F0/K=21F0/mw^2. Because w0=w, so the first term of the amplitude equation is 0, so I got A=F0/mbw, then I substitute Q in, I got A=QFo/mw^2. So 21F0/mw^2=QF0/mw^2. Q=21. But this is not the answer, could anybody help me please?

 

[anathema]

Hi there,

I think you may have made a mistake in your calculation. Let's go through it step by step.

First, we know that the amplitude of a driven harmonic oscillator is given by:

A = (F0/m) / sqrt((w0^2 - w^2)^2 + (b^2*w^2)/m^2)

We are given that the amplitude reaches a value of 21F0/k at a resonant frequency of 390Hz. This means that at w = w0 (resonant frequency), the amplitude is equal to 21F0/k.

Substituting this into the equation, we get:

21F0/k = (F0/m) / sqrt((w0^2 - w0^2)^2 + (b^2*w0^2)/m^2) = F0/m

Simplifying, we get:

21k = m

Next, we know that w0 = 2*pi*f0 = sqrt(k/m). Substituting this into the equation above, we get:

21k = m = (2*pi*f0)^2 / k

Solving for f0, we get:

f0 = sqrt(21k^2 / (4*pi^2*k)) = sqrt(21 / (4*pi^2)) = 0.369Hz

Finally, we can calculate the Q value using the equation Q = m*w0/b. Substituting in the values we know, we get:

Q = m*w0/b = (21k) * (2*pi*f0) / b = (21*2*pi*0.369Hz) / b = 24.57 / b

So the Q value of this system is 24.57 / b. I hope this helps clarify the process for you! Let me know if you have any further questions.