A) \[1:n\]
B) \[{{n}^{2}}:1\]
C) \[\sqrt{n}:1\]
D) \[n:1\]
Correct Answer: C
Solution :
[c] Velocity of longitudinal waves\[{{v}_{1}}=\sqrt{\frac{Y}{\rho }}\] and velocity of transverse waves \[{{v}_{2}}=\sqrt{\frac{T}{m}}=\sqrt{\frac{T}{\rho s}}\] \[\therefore \frac{{{v}_{1}}}{{{v}_{2}}}=\sqrt{\frac{Y}{T/s}}=\sqrt{\frac{Y}{Y\left( \frac{\Delta l}{l} \right)}}=\sqrt{n}\] \[\left[ \therefore \Delta l=\frac{l}{n} \right]\] Now \[f\propto v\,\,\,\therefore \,\,\,\frac{{{f}_{1}}}{{{f}_{2}}}=\frac{{{v}_{1}}}{{{v}_{2}}}=\sqrt{n}\] In the above expression, \[\rho \] = density of string, s = area of cross-section of string, Y= Young's modulus.
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