A straight rod of length L has one of its end at the origin and the other at `X=L`. If the mass per unit length of the rod is given by `Ax` where A is constant, where is its centre of mass?

Explanation

Let the mass of an element of length dx of the rod located at a distance x away from left end is Axdx.

The x-coordinate of the center of mass is given by
`x_(CM) = 1/Mintxdm = 1/M int_(0)^(L)x (Axdx)`
`=(int_(0)^(L)x.Axdx)/(int_(0)^(L)Axdx)=[(Ax^(3)//3)]_(0)^(L)/[Ax^(2)//2]_(L)^(0) =2/3L`
The y-coordinate is`y_(cm) = (1/m)intydm=0`
and similarly, `z_(cm) =0`
Hence, the center of mass is at (2/3L, 0,0) or at 2/3L in the middle of the rod, i.e, at `(2/3)L` from one end.