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Question

# The mass per unit length of a rod of length 3 m is directly proportional to distance x from its one end. The centre of mass of the rod from that end will be at

A
1.5 m
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B
2 m
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C
2.5 m
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D
3 m
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Solution

## The correct option is B 2 mIn a case where mass density is unifoum, COM= L2 In the given situation, mass density is changing, as low at left side and incresing while moving towards right. So, in this condition COM will be shifts towards right. xcom=m1x1+m2x2+....m1+m2+.... xcom=∫x dmdm where, dmdx=λ=kxdx=(kx)=dx then, ∫x(kx)dxkx dx=∫x2dx∫xdx using the limit from 0 to L, =∫L0x2dx∫L0xdx=[x33]L0[x22]L0=2L3 So ,the total shift of COM = 2L3×3=2L Hence, now the new COM is in 2L position of the rod.

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