Oscillator with and without damping - Need help please

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SUMMARY

The discussion focuses on an oscillator with natural frequency ω, featuring a mass on a spring on a frictionless table for x<0 and a frictional surface with an effective damping constant K for x>0. The equations of motion are defined as x=C*cos(ωt) for the undamped scenario and x=Ce-Kt cos(ωdt) for the under-damped case. The problem requires finding the frequency of the oscillator and the ratio of successive amplitudes, considering the distinct behaviors on either side of the table. The challenge lies in integrating the two equations of motion into a cohesive solution.

PREREQUISITES
  • Understanding of harmonic oscillators and natural frequency (ω)
  • Knowledge of damping concepts, specifically under-damping (K<ω)
  • Familiarity with equations of motion for oscillatory systems
  • Basic graphing skills to analyze piecewise functions
NEXT STEPS
  • Study the derivation of the equations of motion for damped oscillators
  • Learn about the effects of damping on oscillation frequency and amplitude
  • Explore graphical representations of piecewise functions in oscillatory motion
  • Investigate the mathematical techniques for solving differential equations related to oscillators
USEFUL FOR

Students in physics or engineering courses, particularly those studying mechanics and oscillatory systems, as well as educators seeking to clarify concepts of damping in oscillators.

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An oscillator with natural frequency ω consists of a mass on a spring positioned on a horizontal table. The table is frictionless for x<0 but has friction for x>0 and an effective damping constant K on that side of the table. Find the frequency of this oscillator and the ratio of successive amplitudes. Assume K<ω (Under-damping).
Relevant equations
x=C*cos(ωt) [Eqn of motion w/o damping] when x<0
x=Ce-Kt cos(ωdt) [Under-damping eqn of motion] when x>0Well it seems there will be two parts of the problem for each side of the table, which means it will probably be one quantity plus another. Any help getting started on this problem would be greatly appreciated. Thank you in advance.
 
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It won't be simply the sum of the two. It will consist of alternate pieces of the graphs of the two. We must consider the time intervals in which x>0 & x<0 separately.
 
Any others? I don't really think I need to use the graph on this one.
 
I've been stuck on this problem for a few days. I just can't see how to reduce it to one equation..
 
Any ideas? Class will be starting relatively soon and I've still yet to figure this one out. Any help much appreciated.. thanks.
 

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