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Problem 1 - Olympiad

A 1.2-meter-long string is tightened such that the left half has tension T and the right half has tension 4T. A standing wave pattern is formed with a node at the junction. What is the ratio of the frequencies of the harmonics formed in the left and right halves?

Correct: A

The frequency of a harmonic on a string is proportional to sqrt(T/μ), where μ is the linear mass density. Since the string is cut at the junction, μ is constant for both halves. The left segment has f₁ ∝ sqrt(T), and the right segment has f₂ ∝ sqrt(4T) = 2f₁. Hence, the ratio is 1:2.