Question

source of frequency 10 kHz when vibrated over the mouth of a closed organ pipe, the air column is in unison at 300 K. The beats produced when temperature rises by 1 K is​

Answer 1

Answer: The beats produced when temperature rises by 1 K is​ 16.67 Hz

Explanation:

Initial frequency = f =  10 kHz

formula for frequency = f = $\frac{v}{4l}$

We know that , velocity of sound is given by the formula,

v =  $\sqrt{\frac{vRT}{M} }$    where,

R = the universal gas constant = 8.314 J/mol K,

T = the absolute temperature

M = the molecular weight of the gas in kg/mol

$v$ = the adiabatic constant, characteristic of the specific gas

so, frequency f can also be written as

f = $\frac{\sqrt{\frac{vRT}{M} }}{4l}$

from the above expression it is evident that

frequency is directly proportional to the square root of the temperature.

f α $\sqrt{T}$   ......................equation 1

Let $f_1$ be final frequency and $T_1$ be the final temperature

∴ $\frac{f_1}{f}$ = $\sqrt{\frac{T_1}{T} }$ = $\sqrt{1 + \frac{1}{300} }$ = 1 + $\frac{1}{600}$

∴ $f_1$ = f ( 1 + $\frac{1}{600}$  )

We need to find beats produced,

Beats produced is the difference in frequency Δf

Δf = $f_1$ - $f$ ..........................equation 1

$f$ = 10 kHz given in the question

substituting this value of f in

Δf = $\frac{10}{600}$ X 1000 = 16.67 Hz

Δf = 16.67 Hz

Answer 2

The number of beats produced are 16.67 Hz when temperature rises by 1 K

Explanation:

• Initial frequency = f =  10 kHz
• Formula for frequency = f = v / 4l
• Velocity of sound formula is given below.

V =  √vRT/M   

so, frequency f can also be written as

f = √ vRT / M / 4l

We can see that frequency is ∝ to √T.

Now

f ∝ √T  --- (1)

• f1 = Final frequency
• T1 = Final temperature

Beats produced = Δf = f1 - f ----- (2)

we are given that f = 10 kHz

Δf  = 10 / 600 x 1000 = 16.67 Hz

Thus the number of beats produced are 16.67 Hz when temperature rises by 1 K.