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Question

The centre of mass of a non uniform rod of length L whose mass per unit length varies as ρ= kx2/L (where k is a constant and x is the distance measured from one end) is at the following distance from the same end

A
3L/4
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B
L/4
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C
2L/3
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D
L/3
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Solution

The correct option is A 3L/4
Given,
density of rod is ρ=kx2L
Consider small distance ds centre of mass is dm
dm=ρdxdm=kx2Ldx
Now centre of mass od non uniform rod is ,
dm=ρdxdm=L0xdmL0dm
Substitute value of dm in above equation therefore,
M=L0xkx2LdxL0kx2LdxM=L0x3dxL0x2dxM=[x44]L0[x33]L0M=3L4
Distance from the centre of mass is 3L/4
Hence, correct option is (A)

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