While measuring the speed of sound by performing a resonance column experiment, a student gets the first resonance condition at a column length of `18 cm` during winter. Repeating the same experiment during summer, she measures the column length to be `x cm` for the second resonance. Then

Options

A. `54gtxgt36`
B. `36gtxgt18`
C. `18gtx`
D. `xgt54`

Answer

D

Explanation

The speed of the sound in air can be written as
`v_(1)=sqrt((gammaRT)/(M))`
Here we are assuming `M` is the average molar mass of the air and `gamma` is the adiabatic constant of air hence we can write and summer temperature
for `1st` resonance `L_(1)=(lambda)/(4)=18cm`
`f=(v_(1))/(lambda)=(v_(1))/(4L_(1))`
At temperature `T_(1)`
`L_(1)=(v_(1))/(4f)=18 cm implies v_(1)=72f cm//s`
For `2nd` resonance `L_(2)=(3lambda)/(4)`
`f=(v_(2))/(lambda)=(3v_(2))/(4L_(2))`
At temperature `T_(2)`
`V_(2)=(4fL_(2))/(3)`
At `T_(2)` is summers `v_(2)gtv_(1)`
i.e., `72fgt(4fL_(2))/(3)impliesL_(2)54cm`