Calculating Q Value at Resonant Frequency - Harmonic Oscillator

[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/($$\Delta$$w)

w=2$$\pi$$f

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

 

[Jhero]

.
I understand your confusion with this problem. Let's break it down step by step.

First, we need to determine what is meant by "amplitude is 25.0Fo/m". This statement is referring to the maximum displacement of the oscillator from its equilibrium position. In other words, at the resonant frequency of 380 Hz, the oscillator is oscillating with an amplitude of 25.0 times the force applied (Fo) divided by the mass (m) of the oscillator. This means that the maximum displacement (A) can be calculated as A = 25.0Fo/m.

Next, we need to determine the value of the natural angular frequency (wo) at the resonant frequency of 380 Hz. We can use the equation w = 2πf to calculate this. Therefore, wo = 2π(380 Hz) = 2385.46 rad/s.

Now, let's look at the equation for Q. There are two forms, but they are essentially the same. Q = m(wo)/b is used when the damping coefficient (b) is known, and Q = wo/Δw is used when the damping ratio (Δ) is known. In this problem, we are not given the damping coefficient or the damping ratio directly. However, we can calculate them using the given information.

We know that at the resonant frequency, the maximum displacement (A) is 25.0 times the force applied (Fo) divided by the mass (m). We can express this in terms of the damping coefficient as A = (25.0Fo)/b. We also know that at resonance, the natural angular frequency (wo) is equal to the damped angular frequency (w'). Therefore, we can write the equation for w' as w' = (wo^2 - y^2)^0.5, where y = b/2m.

Using these equations, we can now solve for the damping coefficient (b) and the damping ratio (Δ). We can then substitute these values into the equation Q = m(wo)/b to find the value of Q.

I hope this helps you understand the problem better. Don't hesitate to ask for more clarification if needed. Good luck with your homework!