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Question

# A rod of length L is placed along the x− axis between x=0 and x=L. The linear density (mass/length) λ of the rod varies with the distance x from the origin as λ=kx2. Here k is a positive constant. Find the position of the centre of mass of this rod.

A
[L2,0,0]
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B
[3L4,0,0]
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C
[L4,0,0]
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D
[L,0,0]
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Solution

## The correct option is B [3L4,0,0] Take the element dx of massdm situated at x from origin. Then, dm=λdx=kx2dx COM of the element has co-ordinates (x,0,0) Therefore, x− co-ordinate of COM of the rod will be xCOM=L∫0xdm∫dm=L∫0x.kx2dxL∫0kx2dx =L∫0kx3dxL∫0kx2dx=kL∫0x3dxkL∫0x2dx =[x44]L0[x33]L0=34L4L3 ∴xCOM=34L ∵ Rod is symmetrical about y and z axes, the y− co-ordinate of COM of the rod is yCOM=∫ydm∫dm=0 The z− co-ordinate of COM of the rod is zCOM=∫zdm∫dm=0 So, the center of mass of the rod lies at [3L4,0,0]

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