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Speed of transverse waves on a string
Concepts tested here
- hanging-rope-wavelength-ratio
All Questions
2016 1 question
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A uniform rope of length L and mass $m_1$ hangs vertically from a rigid support. A block of mass $m_2$ is attached to the free end of the rope. A transverse pulse of wavelength $\lambda_1$ is produced at the lower end of the rope. The wavelength of the pulse when it reaches the top of the rope is $\lambda_2$. The ratio $\frac{\lambda_2}{\lambda_1}$ is
Frequency is fixed and $\lambda \propto \sqrt{T}$.
Wave velocity $v = \sqrt{\frac{T}{\mu}}$ and $v = f\lambda$ with f constant, so $\lambda \propto \sqrt{T}$
Tension at the bottom $= m_2g$; tension at the top $= (m_1 + m_2)g$
$\frac{\lambda_1}{\lambda_2} = \sqrt{\frac{m_2g}{(m_1 + m_2)g}}$
$\frac{\lambda_2}{\lambda_1} = \sqrt{\frac{m_1 + m_2}{m_2}}$