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

1.	If the linear density of a rod of length L varies as lambda=(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» <html><body><p></p>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. <br/> <img src="https://doubtnut-static.s.llnwi.net/static/physics_images/AKS_NEO_CAO_PHY_XI_V01_B_C07_SLV_023_S01.png" width="80%"/> <br/> 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 `<a href="https://interviewquestions.tuteehub.com/tag/dm-432223" style="font-weight:bold;" target="_blank" title="Click to know more about DM">DM</a>=lambda` dx`=(<a href="https://interviewquestions.tuteehub.com/tag/kx-1064991" style="font-weight:bold;" target="_blank" title="Click to know more about KX">KX</a>^(2))/(L)dx` <br/> 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^(<a href="https://interviewquestions.tuteehub.com/tag/4-311707" style="font-weight:bold;" target="_blank" title="Click to know more about 4">4</a>))/(4))/((L^(3))/(3))=(<a href="https://interviewquestions.tuteehub.com/tag/3l-310759" style="font-weight:bold;" target="_blank" title="Click to know more about 3L">3L</a>)/(4)`</body></html>