A uniform thin rope of length $12\, m$ and mass $6\, kg$ hangs vertically from a rigid support and a block of mass $2\, kg$ is attached to its free end. A transverse short wavetrain of wavelength $6\, cm$ is produced at the lower end of the rope. What is the wavelength of the wavetrain (in $cm$ ) when it reaches the top of the rope $?$
JEE MAIN 2020, Diffcult
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$V \propto \lambda \quad T _{2}=8 g$
$T _{1}=2 g$
$\frac{ V _{1}}{ V _{2}}=\frac{\lambda_{1}}{\lambda_{2}}$
$\lambda_{2}=\frac{ V _{2}}{ V _{1}} \lambda_{1}=\sqrt{\frac{ T _{2}}{ T _{1}}} \times \lambda_{1}$
$=\sqrt{\frac{8 g }{2 g }} \lambda_{1}=2 \times 6=12 cm$
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