A rod of length L is placed along the x-axis between x = 0 and x = L. The linear density (mass/length) `lamda` of the rod varies with the distance x from the origin as `lamda` = 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 = `lamda` dx = R x dx
The COM of the element has coordinates (x, 0, 0).
Therefore, x-coordinates 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-coordinates of COM of the rod is
`y_("COM")=(intydm)/(intdm)=0` (as y = 0)
Similarly, `Z_("COM")=0`
Hence, the centre of mass of the rod lies at
`[(2L)/3,0,0]`