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Question

A rod of length 6 m is placed along the x axis between x=0 and x=6 m. The linear density (mass/length) λ of the rod varies with the distance x from the origin as λ=k(100x2). Here k is a positive constant. Find the position of the centre of mass of this rod.


A
(2.79 m,0 m,0 m)
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B
(2 m,0 m,0 m)
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C
(3 m,0 m,0 m)
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D
(6 m,0 m,0 m)
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Solution

The correct option is A (2.79 m,0 m,0 m)
Given, length of rod L=6 m


Taking element of length dx situated at distance x
Mass of this element is dm=λdx
=k(100x2)dx
COM of the element has co-ordinates (x,0,0)

Therefore, x co-ordinate of COM of this rod will be
xCOM=60xdm60dm=60x.k(100x2)dx60k(100x2)dx
=k60(100xx3)dxk60(100x2)dx=[100x22x44]60[100xx33]60
=180032460072=1476528 m
xCOM=2.79 m

y co-ordinate of COM of the rod = yCOM=ydmdm=0
z co-ordinate of COM of the rod = zCOM=zdmdm=0
So, center of mass of the rod lies at (2.79 m,0 m,0 m)

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