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.

Options

A. (i), (iv)
B. (i), (iii)
C. (iii), (iv)
D. (ii),(iii),(iv)

Answer

A

Explanation

As temperature decreases, frequency of air column decreases `(f prop v prop sqrt(T))`.
Beat frequency decreases, i.e., frequency of air column is greater than tuning fork.
Let `n`: frequency of tunninh fork
`51^(@)C, f = n +4, 16^(@)C, f' = n+1`
`(f)/(f') = sqrt((273+51)/(273+16)) = sqrt((324)/(289)) = (18)/(17)`
`(n+4)/(n+1) = (18)/(17) rArr n=50 Hz`
It may be that at `16^(@)C`, frequency of air column goes below `n`.
`51^(@)C = n+4, 16^(@)C,f' = n-1`
`(n+4)/(n-1) = sqrt((273+51)/(273+16)) = (18)/(17)`
`n=86 Hz`