Simple Nature - Light and Matter

ight direction to add energy to the swing. At a driving frequencyvery different from the resonant frequency, we might get lucky andpush at the right time during one cycle, but our next push wouldcome at some random point in the next cycle, possibly having theeffect of slowing the swing down rather than speeding it up.The interpretation of the infinite amplitude at ω = ω o is thatthere really isn’t any steady state if we drive the system exactly atresonance — the amplitude will just keep on increasing indefinitely.In real life, the amplitude can’t be infinite both because there isalways some damping and because there will always be some difference,however small, between ω and ω o . Even though the infinity isunphysical, it has entered into the popular consciousness, startingwith the eccentric Serbian-American inventor and physicist NikolaTesla. Around 1912, the tabloid newspaper The World Today credulouslyreported a story which Tesla probably fabricated — or wildlyexaggerated — for the sake of publicity. Supposedly he created asteam-powered device “no larger than an alarm clock,” containinga piston that could be made to vibrate at a tunable and preciselycontrolled frequency. “He put his little vibrator in his coat-pocketand went out to hunt a half-erected steel building. Down in theWall Street district, he found one — ten stories of steel frameworkwithout a brick or a stone laid around it. He clamped the vibratorto one of the beams, and fussed with the adjustment [presumablyhunting for the building’s resonant frequency] until he got it. Teslasaid finally the structure began to creak and weave and the steelworkerscame to the ground panic-stricken, believing that there hadbeen an earthquake. Police were called out. Tesla put the vibratorin his pocket and went away. Ten minutes more and he could havelaid the building in the street. And, with the same vibrator he couldhave dropped the Brooklyn Bridge into the East River in less thanan hour.”The phase angle δ also exhibits surprising behavior. As the frequencyis tuned upward past resonance, the phase abruptly shifts sothat the phase of the response is opposite to that of the driving force.There is a simple interpretation for this. The system’s mechanicalenergy can only change due to work done by the driving force, sincethere is no damping to convert mechanical energy to heat. In thesteady state, then, the power transmitted by the driving force overa full cycle of motion must average out to zero. In general, the worktheorem dE = F dx can always be divided by dt on both sides togive the useful relation P = F v. If F v is to average out to zero,then F and v must be out of phase by ±π/2, and since v is ahead ofx by a phase angle of π/2, the phase angle between x and F mustbe zero or π.Given that these are the two possible phases, why is there adifference in behavior between ω < ω o and ω > ω o ? At the lowfrequencylimit, consider ω = 0, i.e., a constant force. A constant178 Chapter 3 Conservation of Momentum