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A stretched wire vibrates in its fundamental mode at a frequency of 420 Hz. What would be the fundamental frequency if the wire were one fourth as long, its diameter were quadrupled, and its tension were increased two-fold?

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    With 4 times the diameter (and assuming the same material density), the mass per unit length would increase by a factor of 16. The ratio (tension)/(linear density) would be 2/16 = 1/8 of its previous value. The wave speed is proportional to the square root of that, or 0.3536 as large as before.

    The fundamental frequency is proportional to (wave speed)/(length). This changes by a factor 0.3536/0.250 = 1.414

    The new fundamential frequency is 594 Hz.

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