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Engineering Application: A suspension bridge oscillates with an effective force constant of $1.00 \times 10^{8} \mathrm{N} / \mathrm{m}$ . (a) How much energy is needed to make it oscillate with an amplitude of 0.100 $\mathrm{m} ?$ (b) If soldiers march across the bridge with a cadence equal to the bridge's natural frequency and impart $1.00 \times 10^{4} \mathrm{J}$ of energy each second, how long does it take for the bridge's oscillations to go from 0.100 $\mathrm{m}$ to 0.500 $\mathrm{m}$amplitude?

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Physics 101 Mechanics

Physics 103

Chapter 16

Oscillatory Motion and Waves

Periodic Motion

Wave Optics

Cornell University

Simon Fraser University

University of Sheffield

University of Winnipeg

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So we have the total energy for part A which would essentially be the maximum potential energy of the greatest displacement or the amplitude. And so this would be equal in 1/2 times k times X max squared X sub max. The maximum displacement is of course, equal to the magnitude. And so this would be equaling 1/2 times a spring constant of 1.0 times 10 to the eighth Newtons per meter. And this would be equaling to the maximum displacement 0.100 meters quantity squared. And we find that the maximum potential energy which in this case would be equal to the total energy 5.0 times 10 to the fifth. Jules, this would be our final answer. And this would be again here the amount of energy needed in order to oscillate the bridge with an amplitude of 0.100 meters. Now, for part B, we know that here, um, from part a, we found the energy of course toe oscillated 0.100 meters and we knew that this amount of energy was needed. So now we want to say that here the amount of energy needed to oscillate the bridge at amplitude of 0.500 meters. We could say that here, 1/2 times the spring constant multiplied by 0.500 meters squared equals 1/2 times again turned to the eighth Newton Mr Meter. This would be multiplied by 0.25 zero meters squared. And so we can essentially say that here, this would be equaling points 1 to 5 times 10 to the eighth. Jules. Now we can solve for the additional energy. This would essentially be point 125 times 10 to the eighth, minus 5.0 times 10 to the fifth. Jules, this is equal in 120 times, 10 to the fifth jewels. And so we know that this soldiers are important are imparting 10 to the fourth jewels each second. And so we can say that the time needed for the soldiers to make the obsolete to make the bridge oscillator. This would be equaling the total amount of energy or the total difference in energy divided by the power essentially of one soldier, we can say 1.0 times 10 to the fourth Jules per second and this is equaling 12 100 seconds. And so we can say that. Then it will take 1200 seconds for all of the soldiers to make the bridge. Also late to, um to make the bridge. Also to make the bridges oscillations go from 0.100 meters all the way to 0.500 meters. And so this would be our answer. 1200 seconds. That is the end of the solution. Thank you for

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