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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, then constant λ0 is

A
πM2L
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B
πM3L
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C
πM4L
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D
2πML
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Solution

The correct option is A πM2L
Consider the rod as

We consider length of the rod from bottom to top.
We have been given that
λ=λ0sin(πzL)
dmdz=λ0sin(πzL)
M0dm=L0λ0sin(πzL)dz
M=λ0Lπ[cos(πzL)]L0
M=λ0Lπ[1+1]
λ0=πM2L

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