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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(10x). Here k is a positive constant. Find the position of the centre of mass of this rod.


A
(1.32 m,0 m,0 m)
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B
(3 m,0 m,0 m)
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C
(2.57 m,0 m,0 m)
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D
(3.9 m,0 m,0 m)
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Solution

The correct option is C (2.57 m,0 m,0 m)
Here, given length of rod L=6 m


Take the element dx of mass dm length situated at x distance from origin.
Then, dm=λdx
dm=k(10x)dx

The COM of the element has the co-ordinates (x,0,0)
Therefore, x co-ordinate of COM of the rod will be
xCOM=xdmdm=6 m0x.k(10x)dx6 m0k(10x)dx
=k60(10xx2)dxk60(10x)dx=[10x22x33]60[10xx22]60
=10×362216310×6362=1807210×618
xCOM=10842=2.57 m

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 (2.57,0,0) m

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