A rod of length L is placed along the x-axis between x = 0 and x = L. The linear density (mass/ length) `lambda` of the rod varies with the distance x from the origin as `lambda = Rx`. Here, R is a positive constant. Find the position of centre of mass of this rod.

Explanation

Mass of element dx situated at x = x is `dm = lambda dx = Rx` dx
The COM of the element has coordinates (x, 0, 0).
Therefore, x-coordinate of COM of the rod will be

`x_(COM)=(int_(0)^(L)xdm)/(intdm)`
`=(int_(0)^(L)(x)(Rx)dx)/(int_(0)^(L)(Rx)dx)=(Rint_(0)^(L)x^(2)dx)/(Rint_(0)^(L)xdx)`
`=([(x^(3))/(3)]_(0)^(L))/([(x^(2))/(2)]_(0)^(L))=(2L)/(3)`
The y-coordinate of COM of the rod is `y_(COM)`
`=(int y dm)/(int dm)=0" "`(as y = 0)
Similarly, `z_(COM)=0`
Hence, the centre of mass of the rod lies at `[(2L)/(3),0,0]`