The centre of mass of a non uniform rod of length L whose mass per unit length `lambda = (K x^(2))/(L)`, where K is a constant and x is the distance from one end is

Explanation

Let the x-axis be along the length of the rod and origin at one of its ends. As rod is along x-axis, for all points on it y and z co - ordinates are zero.

Centre of mass will be on the rod. Now consider an element of rod of length dx at a distance x from the origin, then `dm = lambda dx = (Kx^(2))/(L)dx`, so,
`X_(CM)=(int_(0)^(L)xdm)/(int_(0)^(L)dm)=(int_(0)^(L)x(kx^(2))/(L)dx)/(int_(0)^(L)(kx^(2))/(L)dx)=(int_(0)^(L)x^(3)dx)/(int_(0)^(L)x^(2)dx)=((l^(4))/(4))/((L^(3))/(3))=(3L)/(4)`