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Try It Yourself #8 A certain damped harmonic oscillator loses 5% of its energy in each full cycle of oscillation. By what factor must the damping constant be changed in order to damp it critically? Picture: The critical damping constant is related to the mass and period of the motion. The systems Q factor is also related to these parameters, so could be used to solve for b. Take the ratio to find the factor asked for. Solve: Find an algebraic expression for the critical damping constant Find an algebraic expression for the Q factor in terms of the energy los. Find an algebraic expression for the factor in terms of the damping constant, the mass, and the resonant frequency IG-mu-
14.5. DRIVEN OSCILLATIONS AND RESONANCE 255 Equate the two expressions for the Q factor and re-arrange to solve for the current damping constant. Find the ratio of the critical damping factor to the current damping factor. bc/b = 251 Check: If the system only loses 5% of its energy in each full cycle, the system must be under- damped, so the damping will have to increase to critically damp the oscillator. Our result agrees with that assessment. Taking It Further: Once the system is critically damped, how many oscillations occur before the oscillator approaches its equilibrium configuration?
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