Wave Mechanics Vibration Of String Question 31
Question: Two uniform strings A and B made of steel are made to vibrate under the same tension. if the first overtone of A is equal to the second overtone of B and if the radius of A is twice that of B, the ratio of the lengths of the strings is
[EAMCET 2003]
Options:
A) 1: 2
B) 1 : 3
C) 1 : 4
D) 1 : 6
Show Answer
Answer:
Correct Answer: B
Solution:
First overtone of string $ A $ = Second overtone of string B.
$ \Rightarrow $ Second harmonic of $ A $ = Third harmonic of B
$ \Rightarrow $ $ n _{2}=n _{3} $
$ \Rightarrow $ $ {{[ 2(n _{1}) ]} _{A}}={{[ 3(n _{1}) ]} _{B}} $ ( $ \because $
$ n _{1}=\frac{1}{2l}\sqrt{\frac{T}{\pi r^{2}\rho }} $ )
Therefore $ 2[ \frac{1}{2l _{A}r _{A}}\sqrt{\frac{T}{\pi \rho }} ]=3[ \frac{1}{2l _{B}r _{B}}\sqrt{\frac{T}{\pi \rho }} ] $
$ \frac{l _{A}}{l _{B}}=\frac{2}{3}\frac{r _{B}}{r _{A}}\Rightarrow \frac{l _{A}}{l _{B}}=\frac{2}{3}\times \frac{r _{B}}{(2r _{B})}=\frac{1}{3} $
BETA
