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