How Does Friction Affect the Amplitude of a Harmonic Oscillator Over Time?

[belleamie]

Please help
1) The mechanism which powers the clock (using a harmonic oscillator to keep time with friction) Since the clock has friction the oscillation amplitude decreases in time. If the oscillation amp is unity at t=0 what will the amp be after 10 sec?

I believe we need to use the equation w=sq rt(k/m-(c/2m)^2)
But I'm not exactly sure what is the correct way to arrpoach this problem, Any hint would be appreciated! thanks!

 

[lightgrav]

They're not asking (at this point) for the new *angular frequency*,
they're asking for the *envelope* of the decaying cosine curve.
x(t) will be A_o exp(-t/tau) cos(wt + phi) ; the question is about
the exp(-t/tau) portion. You need to know how strong friction is,
or how much frictional work is done in a cycle with total Energy E.

 

[quicknote]

I understand your urgency in seeking help with your classical mechanics problem involving a harmonic oscillator. To answer your question, we can use the equation you mentioned, w=sqrt(k/m-(c/2m)^2), where w represents the angular frequency of the oscillator, k is the spring constant, m is the mass of the oscillator, and c is the damping coefficient.

To find the amplitude after 10 seconds, we can use the equation for the displacement of a harmonic oscillator, x(t)=A*cos(wt+phi), where A is the amplitude and phi is the phase angle. Since the amplitude decreases in time due to friction, we can use the equation A=A0*e^(-ct/2m), where A0 is the initial amplitude at t=0.

Plugging in the values for w, k, m, and c, we can solve for the amplitude A after 10 seconds. However, please note that the phase angle phi is not given in the problem, so we cannot accurately determine the exact amplitude at t=10 seconds without knowing it.

I hope this helps guide you in solving your problem. It is always important to clearly state all known variables in a problem and use relevant equations to find the solution. If you need further assistance, please do not hesitate to reach out for help from a professor or tutor. Good luck!