If the linear density of a rod of length L varies as lamda=(kx^(2))/(L) where k is a constant and x is the distance of any point from one end, then find the distance of centre of mass from the end at x=0.

If the linear density of a rod of length L varies as lamda=(kx^(2))/(L

1.	If the linear density of a rod of length L varies as lamda=(kx^(2))/(L) where k is a constant and x is the distance of any point from one end, then find the distance of centre of mass from the end at x=0.
Answer» Solution :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=lamdadx=(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)`