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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

1. m₁/√m₂
2. (m₁ + m₂)/m₂
3. m₂/√m₁
4. (m₁ + m₂)/m₁

Correct answer: (m₁ + m₂)/m₂

Solution

The speed of a transverse wave in a rope depends on the tension and the linear mass density. As the wave travels upward, the tension increases due to the weight of the rope and the block. The wavelength is proportional to the square root of the tension, which is proportional to the total mass below the point of the wave. At the top, the tension is due to the entire mass (m₁ + m₂), while at the bottom, it is due to m₂. Thus, the ratio λ₂/λ₁ = √((m₁ + m₂)/m₂).

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