A thin rod of length ‘L’ is lying along the x-axis with its ends at x = 0 and x = L. Its linear density (mass/length) varies with x as $k{\left(\frac{x}{L}\right)}^n$,where n can be zero or any positive number. If the position X_CM of the centre of mass of the rod is plotted against ‘n’, which of the following graphs best approximates the dependence of X_CM on n?

• When n = 0, x = k where k is a constant. This means that the linear mass density is constant. In this case the centre of mass will be at the middle of the rod i.e., at L/2.
• n is positive and as its value increases, the rate of increase of linear mass density with increase in x increases. This shows that the centre of mass will shift towards that end of the rod where x = L as the value of n increases. 
• The linear mass density

              $$\begin{gathered} \lambda =k{\left(\frac{x}{L}\right)}^n \\ Here,\frac{x}{L}\le 1 \end{gathered}$$

With increase in the value of n, the centre of mass shift towards the end x = L such that first the shifting is at a higher rate with increase in
the value of n and then the rate decreases with the value of n.