Progressive Wave in a String: Practice Problem & Solution
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 $\lambda_2/\lambda_1$ is: (2016 - I)
Solution Explained:
To solve this problem, we apply the core principles of Progressive Wave in a String. Understanding the underlying formula is key to arriving at the correct answer below:
$v = f\lambda \Rightarrow \lambda \propto v \propto \sqrt{T}$. At the bottom, tension $T_1 = m_2 g$. At the top, $T_2 = (m_1+m_2)g$. Thus $\lambda_2/\lambda_1 = \sqrt{T_2/T_1} = \sqrt{\frac{m_1+m_2}{m_2}}$.
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