A non-uniform thin rod of length `L` is palced along X-axis so that one of its ends is at the origin. The linear mass density of rod is `lambda = lambda_(0)x`. The centre of mass of rod divides the length of the rod in the ratio:

Answer

`(2//1)`

Explanation

As shown in Fig. mass of small element of length `dx` is
`dm = lambda dx = lambda_(0) x dx`
`:. x_(cm) = (int_(0)^(L) x dm)/(int dm)= (int_(0)^(L)x (lambda_(0) x dx))/(int_(0)^(L) lambda_(0) x dx)`
`x_(cm) = ([x^(3)//3]_(0)^(L))/([x^(2)//2]_(0)^(L)) = (L^(3)//3)/(L^(2)//2) = (2L)/(3)`
If `x_(1) = (2L)/(3), x_(2) = L - (2L)/(3) = (L)/(3)`
`(x_(1))/(x_(2)) = (2L//3)/(L//3) = (2)/(1)`