Question

A sound source emits sound of \(600 \mathrm{~Hz}\) frequency, this sound enters by opened door of width \(0.75 \mathrm{~m} .\) Find the angle on one side at which first minimum is formed. The speed of sound \(=300 \mathrm{~ms}^{-1}\) (A) \(84.4^{\circ}\) (B) \(90^{\circ}\) (C) \(74.2^{\circ}\) (D) \(41.2^{\circ}\)

Short answer

The angle at which the first minimum is formed when sound enters through the opened door is approximately \(41.8^{\circ}\). The correct answer is (D) \(41.2^{\circ}\).

Step-by-step solution

Calculate the wavelength

To find the wavelength of the sound, use the formula: \(\lambda = \frac{v}{f}\) Where: - \(v\) is the speed of sound (given as 300 m/s), and - \(f\) is the frequency of the sound (given as 600 Hz). Plugging in the values: \(\lambda = \frac{300 \mathrm{ms^{-1}}}{600 \mathrm{Hz}} = 0.5 \mathrm{m}\)

Use single-slit diffraction formula

Now, we can use the formula for the angle of the first minimum in single-slit diffraction: \(sin(\theta) = \frac{m\lambda}{a}\) Using the given values and the calculated wavelength: - \(m\) is the order of the minimum (1, for the first minimum), - \(\lambda\) is 0.5 meters, - \(a\) is the width of the door (0.75 meters). Plugging in the values: \(sin(\theta) = \frac{1 \cdot 0.5 \mathrm{m}}{0.75 \mathrm{m}} = \frac{2}{3}\)

Calculate the angle

Now we have the value for the sine of the angle, we can calculate the angle itself: \(\theta = sin^{-1}(\frac{2}{3})\) After calculating, we find: \(\theta \approx 41.81^{\circ}\) If we look at the given options, we see that (D) is the closest option to our calculated angle.

Conclusion

Therefore, the angle at which the first minimum is formed when sound enters through the opened door is approximately \(41.8^{\circ}\) . The correct answer is (D) \(41.2^{\circ}\).