A thin rod of length 'L' is lying along the x-axis with its ends at x=0 and x=L its linear (mass/length) varies with `x as k((x)/(L))^n`, where n can be zero of any positive number. If to position `x_(CM)` of the centre of mass of the rod is plotted against 'n', which of the following graphs best apporximates the dependence of `x_(CM)` on n?

Options

A.
B.
C.
D.

Answer

D

Explanation

`x_(c.m.) = (int x dm)/(int dm) = (int x rho dx)/(int rho dx)`
`= (int_(0)^(L) x k((x)/(L))^(n) dx)/(int_(0)^(L) k((x)/(L))^(n) dx) = (int_(0) .^(L) x^(n + 1) dx)/(int_(0)^(L) x^(n) dx)`
`= (|(x^(n +2))/(n + 2)|_(0)^(L))/(|(x^(n + 1))/( n + 1)|_(0)^(L)) = ((n + 1)/(n + 2)) L`
`= ((n + 2 - 1)/(n + 2))L = L - (L)/(n + 2)`
`n = 0 , x_(c.m.) = (L)/(2)`
For any value of `n ,x_(c.m.) lt L`