A uniform rope of length L and mass m₁ hangs vertically from a rigid support. A block of mass m₂ is attached to the free end of the rope. A transverse pulse of wavelength λ₁ is produced at the lower end of the rope. The wavelength of the pulse when it reaches the top of the rope is λ₂. The ratio (λ…

Question

A uniform rope of length L and mass m₁ hangs vertically from a rigid support. A block of mass m₂ is attached to the free end of the rope. A transverse pulse of wavelength λ₁ is produced at the lower end of the rope. The wavelength of the pulse when it reaches the top of the rope is λ₂. The ratio (λ₂)/(λ₁) is

Accepted Answer

(4) √(m₁+m₂)/(m₂). Given: a rope of mass m₁ hanging from a support, block m₂ at its lower end; a pulse starts at the bottom and travels to the top. Idea: the pulse keeps its frequency as it travels, so λ∝ v, and v=√T/μ changes only because the tension does. At the lower end the rope supports only the block: T₁=m₂g. At the top it supports the block and the whole rope: T₂=(m₁+m₂)g. (λ₂)/(λ₁)=(v₂)/(v₁)=√(T₂)/(T₁). (λ₂)/(λ₁)=√(m₁+m₂)/(m₂).

Step-by-step solution

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