Find coordinates of mass center of a non-uniform rod of length L whose linear mass density lambda varies as lambda=a+bx, where x is the distance from the lighter end.

Explanation

Assume the rod lies along the x-axis with its lighter end on the origin to make mass distribution equation consistent with coordinate system. Making use of eq. , we have
`x_c=int(xdm)/M`rarr `x_c=(int_0^Lxlambdadx)/(int_0^Llambdadx)=(int_0^Lx(a+bx)dx)/(int_0^L(a+bx)dx)=((2bL+3a)L)/(3(bL+2a))`