Simple Nature - Light and Matter

second derivatives are v = e −ct (−c sin ω) f t + ω cos ω f t) and a =e(c −ct 2 sin ω f t − 2ω f c cos ω f t − ωf 2 sin ω f t . Plugging these into theequation ma + bv + kx = 0 and setting the sine and cosine partsequal to zero gives, after some tedious algebra,andc =ω f =b2m√km −b24m 2 .Intuitively, we expect friction to “slow down” the motion, as whenwe ride a bike into a big patch of mud. “Slow down,” however, couldhave more than one meaning here. It could mean that the oscillatorwould take more time to complete each cycle, or it could mean thatas time went on, the oscillations would die out, thus giving smallervelocities.Our mathematical results show that both of these things happen.The first equation says that c, which indicates how quickly theoscillations damp out, is directly related to b, the strength of thedamping.The second equation, for the frequency, can be compared withthe result from page 116 of √ k/m for the undamped system. Let’srefer to this now as ω o , to distinguish it from the actual frequencyω f of the free oscillations when damping is present. The result forω f will be less than ω o , due to the presence of the b 2 /4m 2 term. Thistells us that the addition of friction to the system does increase thetime required for each cycle. However, it is very common for theb 2 /4m 2 term to be negligible, so that ω f ≈ ω o .Figure g shows an example. The damping here is quite strong:after only one cycle of oscillation, the amplitude has already beenreduced by a factor of 2, corresponding to a factor of 4 in energy.However, the frequency of the damped oscillator is only about 1%lower than that of the undamped one; after five periods, the accumulatedlag is just barely visible in the offsetting of the arrows.We can see that extremely strong damping — even stronger thanthis — would have been necessary in order to make ω f ≈ ω o a poorapproximation.3.3.2 The quality factorIt’s usually impractical to measure b directly and determine cfrom the equation c = b/2m. For a child on a swing, measuringb would require putting the child in a wind tunnel! It’s usuallymuch easier to characterize the amount of damping by observing theactual damped oscillations and seeing how many cycles it takes forthe mechanical energy to decrease by a certain factor. The unitlessg / A damped sine wave iscompared with an undampedone, with m and k kept the sameand only b changed.Section 3.3 Resonance 175