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 integer. If position `X_(cm)` of 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

Linear density , `mu = ("Mass")/("Length") = k ((x)/(L))^(n)`
where n = 0 , `mu` = constant . The centre of mass will be at the centre `(x = (L)/(2))` of the rod . As n increases to 1, 2, 3…., linear density goes on increasing . The centre of mass shifts beyond x = L/2 towards x = L . For sufficiently large value of n the centre of mass tends to be the other end , x = L of the rod .
`therefore` Graph (a) is the best approximately of dependence of `x_(CM)` on n .