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

`50 Hz`

Explanation

`(v_(51))/(lambda) - n = 4 `
`(v_(51))/( lambda) = n + 4` (i)
and `(v_(16))/(lambda) - n = 1` (ii)
`(v_(16))/(lambda) = n + 1`
Dividing Eq. (ii) by Eq.(i)
`(v_(51))/(v_(16)) = ( n + 4)/( n + 1)`
As ` v prop sqrt(T)`
`rArr sqrt((273 + 51)/(273 + 16)) = ( n + 4)/( n + 1)`
or
` sqrt(( 324)/(289)) = ( n + 4)/( n + 1) or (18)/( 17) = ( n + 4)/( n + 1)`
`n = 50 Hz`.