On vibrating air column at 627 centigrade and a tuning fork simultaneously 6 Beats per. Second are heard the frequency of fork is less than that of air column no Beats are heard at _48 centigrade then frequency of tuning

To approach this problem, we first need to understand how beats are produced and how temperature affects the frequency of sound in a vibrating air column. The scenario involves a vibrating air column at a high temperature of 627 degrees Celsius and a tuning fork that produces beats when played simultaneously. We're also told that no beats are heard at 48 degrees Celsius. Let's break this down step by step to determine the frequency of the tuning fork.

Understanding Beats and Frequency

When two sound sources are played together, beats occur when the two frequencies are close but not identical. The number of beats per second is determined by the difference in frequencies of the two sources. In this case, we have:

• Frequency of air column: f_air
• Frequency of tuning fork: f_fork
• Beats per second: 6

The relationship can be expressed as:

|f_air - f_fork| = 6

Since the frequency of the tuning fork is less than that of the air column, we can state:

f_air - f_fork = 6

Frequency and Temperature Relationship

The speed of sound in air increases with temperature. The formula for the speed of sound (v) in air is given by:

v = 331.4 + 0.6 * T

Where T is the temperature in degrees Celsius. Therefore, at 627 degrees Celsius, we can calculate the speed of sound:

• v_air = 331.4 + 0.6 * 627
• v_air = 331.4 + 376.2 = 707.6 m/s

Next, we need to find out the frequency of the air column at this temperature. Frequency (f) is related to speed (v) and wavelength (λ) by the formula:

f = v / λ

Assuming we have a certain wavelength λ for the air column, we can express the frequency of the air column as:

f_air = 707.6 / λ

Finding the Tuning Fork Frequency

Now, at a lower temperature of 48 degrees Celsius, we calculate the new speed of sound:

• v_air_48 = 331.4 + 0.6 * 48
• v_air_48 = 331.4 + 28.8 = 360.2 m/s

At this temperature, since no beats are heard, this implies that the tuning fork frequency equals the frequency of the air column:

f_fork = f_air_48 = 360.2 / λ

Calculating the Frequencies

We can set up our equations using the information we have. We know:

• From the first scenario: f_air - f_fork = 6
• From the second scenario: f_fork = 360.2 / λ

Substituting the expression for f_air into the first equation gives us:

(707.6 / λ) - (360.2 / λ) = 6

This simplifies to:

(707.6 - 360.2) / λ = 6

347.4 / λ = 6

Thus, we can find λ:

λ = 347.4 / 6 ≈ 57.9 m

Final Calculation of Tuning Fork Frequency

Now, substituting λ back to find the tuning fork frequency:

f_fork = 360.2 / 57.9 ≈ 6.22 Hz

So, we can conclude that the frequency of the tuning fork is approximately 6.22 Hz. This calculation illustrates the relationship between temperature, frequency, and the phenomenon of beats in sound waves. By understanding these concepts, you can analyze different scenarios involving sound waves and their interactions with each other.