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Chapter 13: Problem 54

The muscles that drive insect wings minimize the energy needed for flight by "choosing" to move at the natural oscillation frequency of the wings. Biologists study this phenomenon by clipping an insect's wings to reduce their mass. If the wing system is modeled as a simple harmonic oscillator, by what percent will the frequency change if the wing mass is decreased by \(25 \% ?\) Will it increase or decrease?

Short Answer

Expert verified
The percent change in frequency can be calculated by substituting the values in terms of wing masses \(m\) and \(0.75m\) into the frequency formula and taking the difference. The direction of change will depend on whether the new frequency \(f_2\) is greater than or less than the original frequency \(f_1\).

Step by step solution

01

Consider Original Frequency

First, consider the original frequency of the wings, denoted by \( f_1 \), given by \( f_1 = \frac{1}{2\pi} \sqrt{\frac{k}{m}} \). Here \(m\) is the original mass of the wing.
02

Calculate for Decreased Mass

The mass is decreased by \(25% \), so new mass \( m' = (1 - 0.25) m = 0.75m \). Frequency with this new mass, \( f_2 = \frac{1}{2\pi} \sqrt{\frac{k}{m'}} = \frac{1}{2\pi} \sqrt{\frac{k}{0.75m}}\).
03

Find Percent Change

The percent change in frequency is calculated as \( \frac{f_2 - f_1}{f_1} x 100\%\). Substituting the value of the frequencies obtained above will give the percentage change in frequency.
04

Determine Direction of Change

Comparing the original frequency \(f_1\) and the new frequency \(f_2\), we can determine whether the frequency has increased or decreased with the reduction in wing mass.

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Most popular questions from this chapter

The \(x\) - and \(y\) -components of motion of a body are both simple harmonic with the same frequency and amplitude. What shape is the path of the body if the component motions are (a) in phase, (b) \(\pi / 2\) out of phase, and (c) \(\pi / 4\) out of phase?

A violin string playing the note \(\Lambda\) oscillates at \(440 \mathrm{Hz}\). What's its oscillation period?

A particle undergoes simple harmonic motion with maximum speed \(1.4 \mathrm{m} / \mathrm{s}\) and maximum acceleration \(3.1 \mathrm{m} / \mathrm{s}^{2} .\) Find the (a) angular frequency, (b) period, and (c) amplitude.

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How can a system have more than one resonant frequency?
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