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Question

Linear mass density of a bar is ρ=ρoLoL+ρo where ρo is constant and Lo is length of bar. Its centre of mass from one of the end is
(Assume the other dimensions of the bar to be unity)

A
3Lo5
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B
4Lo7
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C
5Lo9
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D
2Lo5
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Solution

The correct option is C 5Lo9
Let us take a small mass dm at distance L from one of the end of width dl


We know, C.O.M of continuous body,
rC.O.M=Ldmdm
=Lo0L(ρoLoL+ρo)dLLo0(ρoLoL+ρo)dL [ρ=dmdL=ρoLoL+ρo]
rC.O.M=[ρoLo×L33+ρoL22]Lo0[ρoLo×L22+ρoL]Lo0
=ρoL2o3+ρoL2o2ρoLo2+ρoLo
rC.O.M=59Lo
Centre of mass from one of the end is 5Lo9.

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