Question

A student has a variable length air tube that is open at both ends and a $345Hz$ tuning fork. The student finds that the tube resonates with the tuning fork when the tube is a specific length. As the student slowly lengthens the air tube, the student notices no more resonance until the tube is $0.487$ meters longer than the previous resonant length.
From this information, the student calculates the speed of sound. What value does he calculate?

• A. $84m/s$
• B. $342m/s$
• C. $168m/s$
• D. $351m/s$
• E. $336m/s$

Answer 1

The correct option is E $336m/s$

Let the length of the air column initially be $L$ and speed of sound be $v$.

Case 1: $v_1=\frac{1\times v}{2L}$

$\therefore$ $345=\frac{v}{2L}$ ..............(1)

Case 2 : Length of the air column $L^{\prime }=L+0.487$

$\therefore$ Next resonating frequency $v^{\prime }=v_1=\frac{2v}{2L^{\prime }}=\frac{v}{L^{\prime }}$

$\therefore$ $345=\frac{v}{L+0.487}$ ..............(2)

From (1) and (2) we get $\frac{v}{2L}=\frac{v}{L+0.487}$ $⟹L=0.487$ m

Using (1), $345=\frac{v}{2\times 0.487}$ $⟹v=336$ $m/s$