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Question

# A uniform string of length l is fixed at both ends such that tension T is produced in it. The string is excited to vibrate with maximum displacement amplitude a0. The maximum kinetic energy of the string for its first overtone is

A

a20π2T4l

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B

a20π2Tl

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C

a20π2T2l

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D

a20π2T3l

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Solution

## The correct option is B a20π2Tl First overtone : Frequency, n1=vλ1=1l√Tm Considering an elemental length dx of string at a distance from left end, its mass = m dx. Its oscillating energy =12(mdx)a20.sin2(2πxl)(2πn1)2 =2π2a20Tl2sin2(2πxl)dx ∴ Total oscillation energy of the string =2π2a20Tl2∫l0sin2(2πxl)dx or maximum kinetic energy of string in its overtone=a20π2Tl

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