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The density of a non-uniform rod of leng...

The density of a non-uniform rod of length `1m` is given by `rho (x) = a (1 + bx^(2))`
where a and b are constants and `0 le x le 1`.
The centre of mass of the rod will be at

A

`(3(2 + b))/4(3 + b)`

B

`(4(2 + b))/3(3 + b)`

C

`(3(3 + b))/4(2 + b)`

D

`(4(3 + b))/3(2 + b)`

Text Solution

Verified by Experts

The correct Answer is:
a

Mass of a small element of length dx of the rod at a distance x from the one end of the rod is
`dm=pdx = a(1+bx^(2))dx`
The center of mass of the rod is
`X_(CM) = (int_(0)^(l)xdm)/(int_0)^(l)dm = (int_(0)^(l)xa(1+bx^(2))dx)/(int_(0)^(l)a(1+bx^(2))dx)`
`=(int_(0)^(l)(x+bx^(3))dx)/(int_(0)^(l)(1+bx^(2))dx)=([x^(2)/2+(bx^(4))/(4)]_(0)^(1))/([x+(bx^(3))/(3)]_(0)^(1))`
`=([1/2+b/4])/([1+b/3])=(3(2+b))/(4(3+b))`
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