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Question

The mass per unit length of a non-uniform rod OP of length L varies as m=kxL, where k is a constant and x is the distance of point on the rod from end O. The distance of the centre of mass of the rod from end O is

A
L3
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B
2L3
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C
L2
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D
2L3
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Solution

The correct option is B 2L3
Consider a rod OP of length L lying along the x axis with O as the origin. Consider a small element AB of length dx at a distance x from O.

Mass of element AB(=dm)=mdx=kL(xdx)
The distance of the centre of mass from O is given by
xCM=(dM)x(dM)=kLL0x2dxkLL0xdx
=x33L0x22L0=L33L22=2L3
The correct choice is (b).

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