The air pressure variations in a sound wave cause the eardrum to vibrate. Find the maximum velocity and acceleration of the eardrum for vibrations of amplitude 1.5 10-8 m at a frequency of 19.0 Hz.Repeat (b) for the same amplitude but a frequency of 19.0 kHz.
The maximum velocity is A*w and the maximum acceleration is A*w^2.
A is the amplitude (1.5*10^-8 cm) and w s the angular frequency, 2*pi*f
The same formulas apply to both of your questions.
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To find the maximum velocity and acceleration of the eardrum, we can use the formulas that relate amplitude, frequency, velocity, and acceleration in simple harmonic motion.
1. For a given amplitude and frequency, the maximum velocity (vmax) of an object undergoing simple harmonic motion can be found using the formula:
vmax = 2πfA,
where f is the frequency in Hz and A is the amplitude in meters.
2. The maximum acceleration (amax) can be found using the formula:
amax = (2πf)^2A,
where f is the frequency in Hz and A is the amplitude in meters.
Let's calculate the maximum velocity and acceleration for the given values:
For vibrations with an amplitude of 1.5 × 10^(-8) m and a frequency of 19.0 Hz:
- Maximum velocity (vmax) = 2π × 19.0 Hz × (1.5 × 10^(-8) m)
- Maximum acceleration (amax) = (2π × 19.0 Hz)^2 × (1.5 × 10^(-8) m)
For vibrations with the same amplitude but a frequency of 19.0 kHz:
- Maximum velocity (vmax) = 2π × 19.0 kHz × (1.5 × 10^(-8) m)
- Maximum acceleration (amax) = (2π × 19.0 kHz)^2 × (1.5 × 10^(-8) m)
Now, let's calculate them:
For vibrations with an amplitude of 1.5 × 10^(-8) m and a frequency of 19.0 Hz:
- Maximum velocity = 2π × 19.0 Hz × (1.5 × 10^(-8) m) ≈ 5.356 × 10^(-6) m/s
- Maximum acceleration = (2π × 19.0 Hz)^2 × (1.5 × 10^(-8) m) ≈ 9.596 × 10^5 m/s^2
For vibrations with the same amplitude but a frequency of 19.0 kHz:
- Maximum velocity = 2π × 19.0 kHz × (1.5 × 10^(-8) m) ≈ 5.356 × 10^(-2) m/s
- Maximum acceleration = (2π × 19.0 kHz)^2 × (1.5 × 10^(-8) m) ≈ 9.596 × 10^9 m/s^2
So, the maximum velocity and acceleration of the eardrum are different for vibrations of the same amplitude but different frequencies. For vibrations with a frequency of 19.0 Hz, the maximum velocity is approximately 5.356 × 10^(-6) m/s while the maximum acceleration is approximately 9.596 × 10^5 m/s^2. For vibrations with a frequency of 19.0 kHz, the maximum velocity is approximately 5.356 × 10^(-2) m/s and the maximum acceleration is approximately 9.596 × 10^9 m/s^2.