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Question

Consider a rope of total mass M and length L suspended at rest from a fixed mount. The rope has a linear mass density that varies with height as λ(z)=λ0sin(πzL), where λ0 is a constant. Constant gravitational acceleration g acts downward.

The tension along the rope as a function of distance z below the mount is

A
T=Mg2[1+cosπzL]
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B
T=Mg[1+cosπzL]
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C
T=Mg2[1+cosπLz]
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D
T=Mg2[1cosπzL]
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Solution

The correct option is A T=Mg2[1+cosπzL]

T=mg ....(i)
We have been given that
λ=λ0sin(πzL)
dmdz=λ0sin(πzL)
m0dm=L0λ0sin(πzL)dz
M=λ0Lπ[cos(πzL)]L0 ...(ii)
M=λ0Lπ[1+1]
λ0=πM2L
In eq(ii) we just need to alter the mass M to m and the limts from 0L to 0(Lz)
we get
m=λ0Lπ[cos(πzL)]Lz0
m=πM2LLπ[1cos(πLLπzL)]
m=M2[1+cosπzL]
Putting this in eq(i)
T=Mg2[1+cosπzL]

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