question_answer

If \[{{\lambda }_{1}},{{\lambda }_{2}}\] and \[{{\lambda }_{3}}\]are the wavelengths of the waves giving resonance with the fundamental, first and second overtones respectively of a closed organ pipe. Then the ratio of wavelengths \[{{\lambda }_{1}}:{{\lambda }_{2}}:{{\lambda }_{3}}\]is

A)  \[1:3:5\]                             

B)  \[1:2:3\]

C)  \[5:3:1\]                             

D)  \[1:\frac{1}{3}:\frac{1}{5}\]

Correct Answer: D

Solution :

                 According to the question \[l=\frac{{{\lambda }_{1}}}{2}\] \[\Rightarrow \]                               \[{{\lambda }_{1}}=2l\]                 \[l=\frac{3{{\lambda }_{2}}}{2}\Rightarrow {{\lambda }_{2}}=\frac{2l}{2}\]                 \[l=\frac{5{{\lambda }_{3}}}{2}\Rightarrow {{\lambda }_{3}}=\frac{2l}{5}\] \[\therefore \]  \[{{\lambda }_{1}}:{{\lambda }_{2}}:{{\lambda }_{3}}=2l:\frac{2l}{3}:\frac{2l}{5}\]                                    \[=1:\frac{1}{3}:\frac{1}{5}\]