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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. Then, the x coordinate of the centre of mass of this rod will be Ln, where n is (Answer upto two decimal places)

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Solution


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=L0xdmdm=L0x.kx2dxL0kx2dx
=L0kx3dxL0kx2dx=kL0x3dxkL0x2dx
=[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=ydmdm=0
The z co-ordinate of COM of the rod is zCOM=zdmdm=0
So, the center of mass of the rod lies at [3L4,0,0]

x coordinate of COM=3L4=L4/3
So, n=43=1.33

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