The width of the doorway is 83 cm, and the speed of sound is 343 m/s. Find the diffraction angle when the frequency is each of the following.
(a) 4.8 kHz
If theta is the diffraction angle.
Sine(theta)= lambda/slit width
where lambda = wave length
so you need to find the wavelength
(hint make sure it is the same units as the width)
and then substitute into the above.
I've tried this problem several time and got the wrong answer. Please explain where I am going wrong.
Sine (theata) = Lambda/ slit width
Lambda = wave speed/ wave frequency
Lambda = (34300cm/s)/ (4800Hz)
Lambda = 7.1458
therefore
sine (theata) = 7.1458 / 83cm
sine (theata) = 0.08609
theata = 0.0015
SineTheta= .08609, I agree
Theta= 4.94 degrees.
Can you please explain how you arrived at 4.94 degrees?
I put in my calculator .086, then pressed 2nd Sin, thus getting the angle whose sine is .086
Theta= arcSin .086= 4.94 deg
To find the diffraction angle in this problem, you correctly started by using the formula: Sine(theta) = lambda/slit width. However, it seems like there was a mistake when calculating the wavelength (lambda).
Wave speed = 343 m/s
Frequency = 4.8 kHz = 4800 Hz
To find the wavelength, you correctly used the formula: Lambda = wave speed / wave frequency. However, it looks like there was an error in the conversion of units.
Lambda = (343 m/s) / (4800 Hz) = 0.07146 m (not cm)
Since the width of the doorway is given in cm, we need to convert the wavelength to cm as well.
0.07146 m = 7.146 cm
Now, substitute the correct values into the formula: Sine(theta) = lambda / slit width.
Sine(theta) = 7.146 cm / 83 cm
Sine(theta) = 0.08609
To find the diffraction angle, we can use the inverse sine function. On your calculator, enter 0.08609 and then press the inverse sine button (usually labeled "sin^-1" or "2nd sin").
Theta = arcsin(0.08609) = 4.942 degrees
So, the diffraction angle when the frequency is 4.8 kHz is approximately 4.942 degrees.