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Question

A rope of uniform mass m and length L is hanging from a support. A block of mass M is attached to the rope at the lower end. If a transverse pulse is produced at the lower end, find the time taken by the pulse to reach the upper end.


A
2Lg[(1+Mm)Mm]
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B
32Lg[(2+Mm)Mm]
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C
Lg[(1+Mm)Mm]
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D
32Lg[(1+Mm)Mm]
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Solution

The correct option is A 2Lg[(1+Mm)Mm]

Linear mass density of rope (μ)=mL
Tension at point A, TA=μxg+Mg
Velocity of transverse pulse at point A
v=TAμ=(μx+M)gμ
dxdt=(x+Mμ)g[v=dxdt]
To find time taken for wave pulse to travel to the upper end, integrate both sides setting the limit of x from 0 to L
L0dx(x+Mμ)=gtt=0dt
2[(x+Mμ)]L0=g[t]t0
2[(L+Mμ)Mμ]=gt
t=2[(L+Mμ)Mμ]g=2L+MLm2MLmg[μ=mL]
t=2Lg[(1+Mm)Mm]

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