A thin rod of length L is lying along the x -axis from x=0 to x=L . Its linear density λ varies as λ = k(x/L)^

Question

A thin rod of length L is lying along the x -axis from x=0 to x=L . Its linear density λ varies as λ = k(x/L)^n . Which graph best approximates the dependence of the position x_CM of the centre of mass on n ( n ≥ 0 )? ![image](https://pub-7faa453bd1ec4bf0a04cc031c6f058de.r2.dev/images/dc9b84a2-0b8f-

Accepted Answer

x_CM = x dm/dm = _0^L x λ dx/_0^L λ dx = _0^L x × k(x/L)^n dx/_0^L k(x/L)^n dx x_CM = _0^L x^n+1 dx_0^L x^n dx = [x^n+2/(n+2)]_0^L[x^n+1/(n+1)]_0^L = ^n+2/(n+2)L^n+1/(n+1) = n+1/n+2 L At n=0 (uniform rod), x_CM = L/2 .As n → ∞ , x_CM → L .Thus, x_CM starts at L/2 and increases towards L as n increases.

A thin rod of length $L$ is lying along the $x$ -axis from $x = 0$ to $x = L$ . Its linear density $λ$ varies as $λ = k (x / L)^{n}$ . Which graph best approximates the dependence of the position $x_{CM}$ of the centre of mass on $n$ ( $n \geq 0$ )?

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A thin rod of length L is lying along the x -axis from x=0 to x=L. Its linear density λ varies as λ=k(x/L)n. Which graph best approximates the dependence of the position xCM​ of the centre of mass on n (n≥0)?