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Chapter 13: Problem 21
A hummingbird's wings vibrate at about \(45 \mathrm{Hz}\). What's the corresponding period?
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The human eye and muscles that hold it can be modeled as a mass-spring system with typical values \(m=7.5 \mathrm{g}\) and \(k=2.5 \mathrm{kN} / \mathrm{m} .\) What's the resonant frequency of this system? Shaking your head at this frequency blurs vision, as the eyeball undergoes resonant oscillations.
Two balls with the same unknown mass \(m\) are mounted on opposite ends of a 1.5 -m-long rod of mass \(850 \mathrm{g}\). The system is suspended from a wire attached to the center of the rod and set into torsional oscillations. If the wire has torsional constant \(0.63 \mathrm{N} \cdot \mathrm{m} / \mathrm{rad}\) and the period of the oscillations is \(5.6 \mathrm{s},\) what's the unknown mass \(m ?\)
How would the frequency of a horizontal mass-spring system change if it were taken to the Moon? Of a vertical mass-spring system? Of a simple pendulum?
A \(250-\mathrm{g}\) mass is mounted on a spring of constant \(k=3.3 \mathrm{N} / \mathrm{m}\) The damping constant for this system is \(b=8.4 \times 10^{-3} \mathrm{kg} / \mathrm{s}\) How many oscillations will the system undergo before the amplitude decays to \(1 / e\) of its original value?
Show that the potential energy of a simple pendulum is proportional to the square of the angular displacement in the small-amplitude limit.
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