Finding frequency of spring and friction system

[JoeyBob]

Homework Statement

See attached

Relevant Equations

f=2(pi)/w

So first I made an equation representing the forces

Fnet=kx-12.8v

a=1/m(kx-12.8v).

Now I am not really sure how to get w from this. I could argue the mass is at its max amplitude when a=0, but that wouldn't help me find w. If I say x(t)=kx-12.8v, then 1/m would be w^2, but this isn't right. The answer is suppose to be f=0.98 Hz

 

[kuruman]

Homework Statement:: See attached
Relevant Equations:: f=2(pi)/w

could argue the mass is at its max amplitude when a=0, but that wouldn't help me find w.

You cannot even argue that. When a = 0, the mass has maximum speed for that part of the cycle. Have you studied damped harmonic motion? How does the amplitude depend on the driving force frequency when the motion is underdamped?

 

[JoeyBob]

You cannot even argue that. When a = 0, the mass has maximum speed for that part of the cycle. Have you studied damped harmonic motion? How does the amplitude depend on the driving force frequency when the motion is underdamped?

w=sqrt(k/m-(b/2m)^2) ? But then what is b?

Edit if B is the -12.8, I get 1.02 for f which is wrong.

 

[haruspex]

w=sqrt(k/m-(b/2m)^2) ? But then what is b?

Edit if B is the -12.8, I get 1.02 for f which is wrong.

You seem to be using a formula for underdamped oscillation of an unforced system. An unforced system has no resonance to consider, so no large amplitude swings develop.
This system is forced.
@kuruman, I get 1.14Hz. What do you get?

 

[kuruman]

@kuruman, I get 1.14Hz. What do you get?

I get 0.978 Hz. This is using the equation for the resonant frequency posted by OP and with which I agree $$f_r=\frac{1}{2\pi}\sqrt{\frac{k}{m} - \frac{b^2}{4m^2}}$$using the given values m = 1.76 kg, k = 89.8 N/m and b = 12.8 N⋅s/m. I get 1.14 Hz using the same equation with b = 0 (no damping).

 

[haruspex]

I get 0.978 Hz. This is using the equation for the resonant frequency posted by OP and with which I agree $$f_r=\frac{1}{2\pi}\sqrt{\frac{k}{m} - \frac{b^2}{4m^2}}$$using the given values m = 1.76 kg, k = 89.8 N/m and b = 12.8 N⋅s/m. I get 1.14 Hz using the same equation with b = 0 (no damping).

Seems to me that is the equation for the oscillation frequency of an unforced underdamped system. It is not the resonant frequency of a forced damped system, which is simply the natural (undamped) frequency.
See e.g. section 2.3 of https://ocw.mit.edu/courses/physics...-fall-2016/syllabus/MIT8_03SCF16_Text_Ch2.pdf and https://courses.lumenlearning.com/suny-osuniversityphysics/chapter/15-6-forced-oscillations/.

 

[kuruman]

Seems to me that is the equation for the oscillation frequency of an unforced underdamped system.

That is correct. Section 2.3 of the MIT reference talks about maximum transfer of power which indeed occurs at the resonant frequency, ##\omega_0=\sqrt{k/m}.## The question here is asking "for what frequency ##\dots~##will the mass undergo large amplitude oscillations?"

To answer that question one needs to maximize the amplitude which in the SUNY reference you provided should be $$A=\frac{F_0}{\sqrt{m^2 \left(\omega ^2-\omega _0^2\right)^2+b^2 \omega ^2}}.$$ Note the term ##m^2## in the denominator as opposed to just ##m## shown in the reference. The correction is needed because ##b## has dimensions of [m][ω]. I checked the corrected equation against my "Classical Mechanics by John R. Taylor" textbook and there is agreement. Setting the derivative equal to zero and solving for ##\omega## yields the frequency at which the amplitude is maximum.

The numerical answer is neither 1.14 Hz nor 0.98 Hz. This is still a "live" homework problem so I stop here. In my defense, I did remember that maximum amplitude is not at the natural frequency but at a frequency that involves the damping constant. However I was remiss in accepting the purported answer without further checking.

 

[haruspex]

That is correct. Section 2.3 of the MIT reference talks about maximum transfer of power which indeed occurs at the resonant frequency, ##\omega_0=\sqrt{k/m}.## The question here is asking "for what frequency ##\dots~##will the mass undergo large amplitude oscillations?"

To answer that question one needs to maximize the amplitude which in the SUNY reference you provided should be $$A=\frac{F_0}{\sqrt{m^2 \left(\omega ^2-\omega _0^2\right)^2+b^2 \omega ^2}}.$$ Note the term ##m^2## in the denominator as opposed to just ##m## shown in the reference. The correction is needed because ##b## has dimensions of [m][ω]. I checked the corrected equation against my "Classical Mechanics by John R. Taylor" textbook and there is agreement. Setting the derivative equal to zero and solving for ##\omega## yields the frequency at which the amplitude is maximum.

The numerical answer is neither 1.14 Hz nor 0.98 Hz. This is still a "live" homework problem so I stop here. In my defense, I did remember that maximum amplitude is not at the natural frequency but at a frequency that involves the damping constant. However I was remiss in accepting the purported answer without further checking.

Ah, yes, I confused tuning the driving frequency to maximise amplitude with tuning the natural frequency. It's less than 0.98, right?
In terms of the algebra at the link I posted (which writes Γ for b/m), it is a matter of maximising A²+B².

 

[kuruman]

Ah, yes, I confused tuning the driving frequency to maximise amplitude with tuning the natural frequency. It's less than 0.98, right?
In terms of the algebra at the link I posted (which writes Γ for b/m), it is a matter of maximising A²+B².

Yes, the answer I got is less than 0.98 Hz. Yes, in terms of the link you posted one ought to maximize A²+B² and the answer should be the same. I didn't do it that way, but if you do, we can compare numerical answers by PM.