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Chapter 10: Problem 41

The diaphragm of a speaker has a mass of \(50.0 \mathrm{g}\) and responds to a signal of frequency \(2.0 \mathrm{kHz}\) by moving back and forth with an amplitude of \(1.8 \times 10^{-4} \mathrm{m}\) at that frequency. (a) What is the maximum force acting on the diaphragm? (b) What is the mechanical energy of the diaphragm?

Short Answer

Expert verified
Answer: The maximum force acting on the diaphragm is approximately 90.396 N, and the mechanical energy of the diaphragm is approximately 0.001015 J.

Step by step solution

01

Convert given values to appropriate units

We are given the mass, frequency and displacement, let's convert them to appropriate units: - Mass, \(m = 50.0\,\mathrm{g} = 0.050\,\mathrm{kg}\) (convert g to kg by dividing by 1000) - Frequency, \(f = 2.0\,\mathrm{kHz} = 2000\,\mathrm{Hz}\) (convert kHz to Hz by multiplying by 1000) - Amplitude, \(A = 1.8 \times 10^{-4}\mathrm{m}\) (already in appropriate units)
02

Calculate the angular frequency (\(\omega\))

We need to find the angular frequency using the relationship between frequency and angular frequency: \(\omega = 2\pi f\) Now we can plug in the values: \(\omega = 2\pi (2000\,\mathrm{Hz}) = 4000\pi\,\mathrm{rad/s}\)
03

Calculate the maximum acceleration (\(a_{max}\))

Maximum acceleration occurs when \(x = A\), so we can calculate the maximum acceleration using the equation for simple harmonic motion: \(a_{max} = -\omega^2A\) Plug in the values: \(a_{max} = -(4000\pi\,\mathrm{rad/s})^2(1.8 \times 10^{-4}\,\mathrm{m}) = -1807.92\,\mathrm{m/s^2}\) Note that the negative sign indicates the direction of the acceleration, but we are looking for the magnitude of maximum acceleration, so we take the absolute value: \(a_{max} = 1807.92\,\mathrm{m/s^2}\)
04

Calculate the maximum force (\(F_{max}\))

Now we can find the maximum force using the equation for force: \(F_{max} = ma_{max}\) Plug in the values: \(F_{max} = (0.050\,\mathrm{kg})(1807.92\,\mathrm{m/s^2}) = 90.396\,\mathrm{N}\) The maximum force acting on the diaphragm is \(90.396\,\mathrm{N}\). #b) Find the mechanical energy#
05

Calculate the spring constant (\(k\))

First, we need to find the spring constant using the equation relating mass and spring constant: \(k = m\omega^2\) Plug in the values: \(k = (0.050\,\mathrm{kg})(4000\pi\,\mathrm{rad/s})^2 = 31415.93\,\mathrm{N/m}\)
06

Calculate the mechanical energy (\(E\))

Now we can find the mechanical energy using the equation for mechanical energy in simple harmonic motion: \(E = \frac{1}{2}kA^2\) Plug in the values: \(E = \frac{1}{2}(31415.93\,\mathrm{N/m})(1.8 \times 10^{-4}\,\mathrm{m})^2 = 0.001015\,\mathrm{J}\) The mechanical energy of the diaphragm is \(0.001015\,\mathrm{J}\).

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Most popular questions from this chapter

A brass wire with Young's modulus of \(9.2 \times 10^{10} \mathrm{Pa}\) is $2.0 \mathrm{m}\( long and has a cross-sectional area of \)5.0 \mathrm{mm}^{2} .$ If a weight of \(5.0 \mathrm{kN}\) is hung from the wire, by how much does it stretch?
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