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122. Problem 17.32P (HRW) A uniform rope of mass m and length l hangs from a
ceiling. We have to show (a) that the speed of a transverse wave on the
rope is a function of y, the distance from the lower end, and is given by
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. (b)
We have to show that the time a transverse wave takes to travel the length of
the rope is given by
.
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Solution: Click For PDF Version (a) Speed of transverse waves on the sting will be a function of y, the distance on the string measured from its bottom end, because the tension in the string is determined by the weight of the string hanging below the level y. Therefore,
where m is the mass and l is the length of the string. As speed of a transverse wave in a string is given by the relation
where T is the tension and
(b) We will next calculate the time taken be a transverse wave for travelling the length of the string
By integrating this equation, we find
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is the
mass per unit length of the string. We thus have the result
