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A thin rod of length 'L' is lying along ...

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?

A

B

C

D

Text Solution

Verified by Experts

The correct Answer is:
D
`x_(cm)=(int"xdm")/(int"dm")=(int"x k"(x^(n))/(L^(n))"dx")/(intk(x^(n))/(L^(n))"dx")`
`=(int_(0)^(L)x^(n+1)"dx")/(int_(0)^(L)x^(n)"dx")=((L^(n+2))/(n+2))/((L^(n+1))/(n+1))`
`x=((n+1)/(n+2))L`
`at =0" "x_(cm)=(L)/(2)` at `nrarroo" "x_(cm)=L`
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