A 3.00 kHz tone is being produced by a speaker with a diameter of 0.150 m. The air temperature changes from 0 to 29°C. Assuming air to be an ideal gas, find the change in the diffraction angle .
Please tell me where I am going wrong.
first I converted the temp to kelvin
0 = 273k
29°C = 302k
then I converted the kHz to Hz
3.0 kHz = 3000 Hz
I found the wavelenght for the gas at 0
273/3000 = 0.091
then
sin (theta)= 122(0.091/0.150 m)
Sin (theta) = 74.0133
when I put 74.0133(Sin^-1) into the calculator it comes up as error.
Whoa, dudette: Wavelength is not temperature/frequency. YOu need to use temp to get speed of sound, then wavelength is speedsound/freq.
It appears that you made an error in the calculation of the wavelength of the gas at 0°C. The wavelength of a sound wave in a gas can be determined using the formula:
wavelength = speed of sound / frequency
To calculate the speed of sound at a given temperature, you can use the formula:
speed of sound = √(γ * R * temperature)
where γ is the adiabatic index for air (approximately 1.4), R is the specific gas constant for air (approximately 287 J/(kg·K)), and temperature is in Kelvin.
Let's go step by step to fix your calculation:
1. Convert the temperature to Kelvin:
0°C = 273 K
29°C = 302 K
2. Calculate the speed of sound at 0°C:
speed of sound = √(1.4 * 287 * 273) ≈ 331.4 m/s
3. Calculate the wavelength at 0°C:
wavelength = speed of sound / frequency = 331.4 / 3000 ≈ 0.1105 m
Now, to find the change in the diffraction angle, we need to consider the change in the wavelength with the change in temperature.
1. Calculate the speed of sound at 29°C:
speed of sound = √(1.4 * 287 * 302) ≈ 346.6 m/s
2. Calculate the wavelength at 29°C:
wavelength = speed of sound / frequency = 346.6 / 3000 ≈ 0.1155 m
3. Find the change in wavelength:
Δwavelength = wavelength at 29°C - wavelength at 0°C = 0.1155 - 0.1105 ≈ 0.005 m
Finally, to find the change in the diffraction angle, you can use the formula:
Δθ = arcsin(Δwavelength / diameter)
Δθ = arcsin(0.005 / 0.150) ≈ arcsin(0.0333) ≈ 1.91 degrees
Therefore, the change in the diffraction angle is approximately 1.91 degrees.