Question

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

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

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)

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: \(\omega = 2\pi (2000\,\mathrm{Hz}) = 4000\pi\,\mathrm{rad/s}\)

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: \(a_{max} = 1807.92\,\mathrm{m/s^2}\)

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#

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: \(k = (0.050\,\mathrm{kg})(4000\pi\,\mathrm{rad/s})^2 = 31415.93\,\mathrm{N/m}\)

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 \(0.001015\,\mathrm{J}\).