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((x)/(L))^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?

Options

A.
B.
C.
D.

Answer

A

Explanation

`X_(cm)=(intxdm)/(intdm)=(intxrhoAdx)/(intrhoAdx)=(int_(0)^(L)xk(x^(n))/(L^(n))dx)/(int_(0)^(L)k(x^(n))/(L^(n))dx)`
`=(k[(x^(n+2))/((n+2)L^(n))]_(0)^(L))/(k[(x^(n+1))/((n+1)L^(n))]_(0)^(L))=L((n+1)/(n+2))`

If n=0,1,2,3,4,5,…
then `X_(cm)=(L)/(2),(2L)/(3),(3L)/(4),(4L)/(5),(5L)/(6)`
As n increases, the centre of mass shifts towards the end x = L