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Question

A wave pulse starts propagating in the x-direction along a non-uniform wire of length 20 m with mass per unit length given by μ=μ0+ax and under a tension of 400 N. Find the time taken by the pulse to travel from the lighter end (x=0) to the heavier end. μ0=102 kg/m and a=16×103 kg/m2

A
0.8 sec
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B
0.4 sec
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C
0.6 sec
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D
1 sec
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Solution

The correct option is B 0.4 sec
The speed of the wave pulse in a medium is given by
V=Tμ=Tμ0+ax=dxdt
To find the time taken for wave pulse to go from end to end, integrate above equation with limits of x set from 0 to L=20 m.
t0Tdt=L0μ0+ax dx
T[t]t0=23a(μ0+aL)32(μ0)32
t=23Ta(μ0+aL)32(μ0)32
=23×400×16×103(102+16×103×20)32(102)32
=2.08×0.1880.4 s

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