Question

A column of air at `51^(@) C` and a tuning fork produce `4` beats per second when sounded together. As the temperature of the air column is decreased, the number of beats per second tends to decrease and when the temperature is `16^(@) C` the two produce `1` beat per second. Find the frequency of the tuning fork.

Answer 1

Frequency of air column, `f prop v prop sqrt(T)`
`v`: speed of should in air,
`T`: absolute temperature
Let frequnecy of tuning fork is `n`.
`f - n = +- 4`
As temperature decreases, beat frequency decreases i.e
`f gt n`
As `51^(@)C`, frequency of air column:
`(n + 4) prop sqrt(273 + 51)` (i)
At `16^(@)C`, frequency of air column:
`(n + 1) prop sqrt(273 + 16)` (ii)
`(i)//(ii)`
`(n + 4)/(n + 1) = sqrt((324)/(289)) = (18)/(17)`
`17n + 17 xx 4 = 18n + 18 rArr n = 50 Hz`
At `16^(@)C`, frequency of air column may be smaller than tuning fork
`(n - 1) prop sqrt(273 + 16)` (iii)
`(i)//(iii)`
`(n + 4)/(n - 1) prop sqrt((324)/(289)) = (18)/(17)`
`17n + 17 xx 4 = 18n + 18 rArr n = 86 Hz`
The possible frequencies of tuning fork are `50 Hz` and `86 Hz`.