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) \(41.2^{\circ}\).

Key concepts

Wavelength Calculation

Understanding how to calculate the wavelength of sound is crucial in physics. Specifically, it involves using the formula that connects the speed of sound, the frequency of the sound wave, and its wavelength.

• We denote wavelength by \( \lambda \).
• The speed of sound is represented by \( v \).
• Frequency is denoted by \( f \).

The formula used is:\[ \lambda = \frac{v}{f} \]This equation essentially tells us that the wavelength is the speed of sound divided by its frequency.
For example, if the speed of sound is 300 m/s and the frequency is 600 Hz, plugging these values into the equation gives us \( \lambda = 0.5 \) meters. This means that each sound wave measures half a meter from peak to peak.

Sound Frequency

Sound frequency refers to the number of sound wave cycles that pass a given point per second. It is measured in Hertz (Hz).

• Higher frequencies lead to higher-pitched sounds.
• Lower frequencies result in lower-pitched sounds.

In the context of the exercise, the sound frequency given is 600 Hz. This tells us that in every second, 600 complete sound wave cycles are emitted by the source.
Frequency is vital in determining the characteristics of the sound we hear. It directly influences the wavelength when the speed of sound is held constant, as showcased in the previous section. So, a precise understanding of frequency is essential for any further calculations and predictions related to sound and its behavior.

Angle of Diffraction

The angle of diffraction refers to the angle at which a wave bends around an obstacle or passes through a slit. When considering single-slit diffraction, the formula to determine the angle of the first minimum is:\[ \sin(\theta) = \frac{m \lambda}{a} \]Here:

• \( m \) represents the order of the minimum, and it is 1 for the first minimum.
• \( \lambda \) is the wavelength.
• \( a \) is the width of the slit, or door in this example.

Using this formula, if the wavelength is 0.5 meters and the slit width is 0.75 meters:
The result for the sine of the angle becomes:\[ \sin(\theta) = \frac{1 \times 0.5}{0.75} = \frac{2}{3} \]To find the angle \( \theta \), use the inverse sine function:\[ \theta = \sin^{-1}\left(\frac{2}{3}\right) \approx 41.81^{\circ} \]This angle indicates where the first minimum of the wavefront is located on either side of the central maximum.

Speed of Sound

The speed of sound is a fundamental concept in acoustics, representing how fast sound travels through a medium. It is determined by the medium's properties – mainly its temperature, elasticity, and density.

• The typical speed of sound in air at room temperature is approximately 343 m/s.
• In this specific exercise, the speed is given as 300 m/s, possibly indicating a different environmental condition.

This speed is integral to calculating wavelengths when the frequency is known (as shown previously).
Additionally, different mediums will carry sound at different speeds; for example, sound travels faster in water and solids than in air.
Understanding the speed of sound helps not just in calculating wavelengths, but also in practical applications such as sound localization and Doppler effect analysis.