When responding to sound, the human eardrum vibrates about its equilibrium position. Suppose an eardrum is vibrating with an amplitude of 6.7 x10-7 m and a maximum speed of 3.9 x10-3 m/s. (a) What is the frequency (in Hz) of the eardrum's vibrations? (b) What is the maximum acceleration of the eardrum?
d=Asin(2PIf)t
v= 2PIf*A cos(2PIft)
you are given A, and v, and A, solve for f.
a= (2PIf)^2 A sin(2PIft)
now solve for acceleration a
To find the frequency of the eardrum's vibrations, we can use the formula:
f = v / λ
where f is the frequency, v is the speed of sound, and λ is the wavelength.
In this case, we are given the speed of the eardrum's vibrations as 3.9 x 10^(-3) m/s. However, we still need to find the wavelength in order to calculate the frequency.
The definition of wavelength (λ) is the distance a wave travels during one complete vibration. In this case, since we know the amplitude (A) of the vibration, we can use the formula:
λ = 4A
where A is the amplitude.
Substituting the given amplitude of 6.7 x 10^(-7) m, we get:
λ = 4 × 6.7 x 10^(-7) m
= 2.68 x 10^(-6) m
Now we can calculate the frequency using the formula:
f = v / λ
Substituting the given speed v of 3.9 x 10^(-3) m/s and the calculated wavelength λ of 2.68 x 10^(-6) m, we get:
f = (3.9 x 10^(-3) m/s) / (2.68 x 10^(-6) m)
≈ 1455 Hz
Therefore, the frequency of the eardrum's vibrations is approximately 1455 Hz.
To find the maximum acceleration of the eardrum, we can use the formula:
a = w^2 A
where a is the acceleration, w is the angular frequency, and A is the amplitude.
The angular frequency (w) is related to the frequency (f) by the equation:
w = 2πf
Substituting the calculated frequency f of 1455 Hz, we get:
w = 2π × 1455 Hz
≈ 9137.75 rad/s
Now we can calculate the maximum acceleration using the formula:
a = w^2 A
Substituting the calculated angular frequency w of 9137.75 rad/s and the given amplitude A of 6.7 x 10^(-7) m, we get:
a = (9137.75 rad/s)^2 × (6.7 x 10^(-7) m)
≈ 518.79 m/s^2
Therefore, the maximum acceleration of the eardrum is approximately 518.79 m/s^2.