As Section 27.5 discusses, sound waves diffract or bend around the edges of a doorway. Larger wavelengths diffract more than smaller wavelengths. (a) The speed of sound is 343 m/s. With what speed would a 57.0 kg person have to move through a doorway to diffract to the same extent as a 146 Hz bass tone? (b) At the speed calculated in part (a), how long in years (365.25 days) would it take the person to move a distance of one meter?

Question

As Section 27.5 discusses, sound waves diffract or bend around the edges of a doorway. Larger wavelengths diffract more than smaller wavelengths. (a) The speed of sound is 343 m/s. With what speed would a 57.0 kg person have to move through a doorway to diffract to the same extent as a 146 Hz bass tone? (b) At the speed calculated in part (a), how long in years (365.25 days) would it take the person to move a distance of one meter?

Answer 1

To answer part (a) of the question, we need to find the speed at which the person would have to move through a doorway to diffract to the same extent as a 146 Hz bass tone.

We know that larger wavelengths diffract more than smaller wavelengths, so we need to compare the wavelength of the bass tone to the width of the doorway.

To find the wavelength, we can use the formula:

wavelength = speed of sound / frequency

Given:
speed of sound (v) = 343 m/s
frequency (f) = 146 Hz

Plugging in these values, we can find the wavelength:

wavelength = 343 m/s / 146 Hz

Using a calculator, we get:

wavelength = 2.354 m

Now, to find the speed at which the person would have to move through the doorway, we need to consider the diffraction of sound waves.

Diffraction occurs when the size of an obstacle (in this case, the doorway) is comparable to the wavelength of the sound wave. In this scenario, the width of the doorway becomes an obstacle for diffraction.

We can use the formula for diffraction:

sin(theta) = (wavelength / width)

where theta is the angle of diffraction, wavelength is the length of the sound wave, and width is the size of the obstacle (doorway width).

Solving for theta, we get:

theta = sin^(-1)((wavelength / width))

For diffraction to occur to the same extent, the person would need to move at an angle equal to theta.

The velocity of the person moving through the doorway can be calculated using the formula for velocity:

velocity = speed / cos(theta)

Substituting the values, we get:

velocity = (343 m/s) / cos(theta)

Now, using a calculator, we can find the value of theta, and then calculate the velocity at that angle.

For part (b) of the question, we need to calculate the time it would take for the person to move a distance of one meter at the velocity calculated in part (a).

We can use the formula:

time = distance / velocity

Given:
distance = 1 meter
velocity = calculated in part (a)

Substituting these values, we can calculate the time it would take for the person to move one meter.

Note: Remember to convert the time to years by dividing the calculated time by the number of seconds in a year (365.25 days).

Answer 2

To answer these questions, we can use the formula for the wavelength of a sound wave:

λ = v/f

where:
λ is the wavelength of the sound wave
v is the speed of sound
f is the frequency of the sound wave

(a) First, we can calculate the wavelength of the 146 Hz bass tone. Since larger wavelengths diffract more, we will assume this is the wavelength that the person needs to match when moving through the doorway.
Using the formula, we have:

λ = v/f
λ = 343 m/s / 146 Hz
λ ≈ 2.35 meters

To diffract to the same extent, the person needs to have a velocity such that their wavelength matches 2.35 meters. The person's mass is given as 57.0 kg.

We can calculate the velocity using the formula:

v = λf
v = 2.35 meters * 146 Hz
v ≈ 342.1 m/s

Therefore, the person needs to move through the doorway at a speed of approximately 342.1 m/s to diffract to the same extent as a 146 Hz bass tone.

(b) To find the time in years it would take for the person to move a distance of one meter at the calculated speed, we can use the formula:

time = distance / velocity
time = 1 meter / 342.1 m/s

Converting to years:

time = (1 meter / 342.1 m/s) * (1 year / 365.25 days)
time ≈ 2.88 × 10^-6 years

Therefore, at the calculated speed, it would take approximately 2.88 × 10^-6 years (or about 0.00000288 years) for the person to move a distance of one meter through the doorway.

Answer 3

(a)

λ(sound)= λ1
λ (person)= λ2
λ(sound)= λ (person)
λ1= λ2
λ1=v/f=343/146=2.35 m.
λ2=h/p=h/(m•v2)
v2=h/(m• λ2)= h/(m• λ1)=
=(6.63•10^-34)/57•2.35 =4.95•10^-36 m/s.

(b) t=s/v=1/4.95•10^-36=
=2.02•10^35 s=6.4•10^27 years