The air pressure variations in a sound wave cause the eardrum to vibrate. (a) For a given vibration amplitude, are the maximum velocity and acceleration of the eardrum greatest for high-frequency sounds or lowfrequency sounds? (b) Find the maximum velocity and acceleration of the eardrum for vibrations of amplitude \(1.0 \times 10^{-8} \mathrm{m}\) at a frequency of $20.0 \mathrm{Hz}\(. (c) Repeat (b) for the same amplitude but a frequency of \)20.0 \mathrm{kHz}$.

To find the velocity of the eardrum, take the first derivative of the displacement function with respect to time: \(v(t) = \frac{dx(t)}{dt} = -Aω\sin(ωt)\)

To find the acceleration of the eardrum, take the second derivative of the displacement function with respect to time: \(a(t) = \frac{d^2x(t)}{dt^2} = -Aω^2\cos(ωt)\)

The maximum values of velocity and acceleration can be found by determining their absolute maximum values when taking into account the sine and cosine functions in these expressions: - The maximum velocity is given by \(|v(t)| = | - Aω\sin(ωt)| \leq Aω\). Thus, the maximum velocity depends on the angular frequency \(ω\), and so the maximum velocity is higher for high-frequency sounds. - The maximum acceleration is given by \(|a(t)| = | - Aω^2\cos(ωt)| \leq Aω^2\). Thus, the maximum acceleration also depends on the angular frequency \(ω\), and so the maximum acceleration is higher for high-frequency sounds.

Given a vibration amplitude of \(1.0 \times 10^{-8} \mathrm{m}\) and a frequency of 20 Hz, we can calculate the maximum eardrum velocity and acceleration as follows: 1. Calculate the angular frequency: \(ω = 2πf = 2π(20 \mathrm{Hz}) = 40π\) rad/s 2. Calculate the maximum velocity: \(v_\text{max} = Aω = (1.0 \times 10^{-8} \mathrm{m})(40π \text{ rad/s}) = 4.0π \times 10^{-7}\) m/s 3. Calculate the maximum acceleration: \(a_\text{max} = Aω^2 = (1.0 \times 10^{-8} \mathrm{m})(40π \text{ rad/s})^2 = 1.6π^2 \times 10^{-5}\) m/s²

Given the same vibration amplitude and a frequency of 20 kHz, we can calculate the maximum eardrum velocity and acceleration as follows: 1. Calculate the angular frequency: \(ω = 2πf = 2π(20,000 \mathrm{Hz}) = 40,000π\) rad/s 2. Calculate the maximum velocity: \(v_\text{max} = Aω = (1.0 \times 10^{-8} \mathrm{m})(40,000π \text{ rad/s}) = 4.0π \times 10^{-4}\) m/s 3. Calculate the maximum acceleration: \(a_\text{max} = Aω^2 = (1.0 \times 10^{-8} \mathrm{m})(40,000π \text{ rad/s})^2 = 1.6π^2 \times 10^{-1}\) m/s² So, for part (a), the greatest maximum velocity and acceleration are for high-frequency sounds. In part (b), the maximum velocity and acceleration for a 20 Hz sound wave are \(4.0π \times 10^{-7}\) m/s and \(1.6π^2 \times 10^{-5}\) m/s², respectively. In part (c), the maximum velocity and acceleration for a 20 kHz sound wave are \(4.0π \times 10^{-4}\) m/s and \(1.6π^2 \times 10^{-1}\) m/s², respectively.