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AP Physics C: Mechanics
2.1 Systems and Center of Mass
IntermediateMCQMathematical23.2k
A horizontal thin rod lies along an x-axis. The left end of the rod is aligned with a vertical tick mark labeled 0 at the origin, and the right end is aligned with a tick mark labeled L. Below the rod, a horizontal double-headed dimension line spans from 0 to L and is labeled L. Shading inside the rod is light gray at the left end and becomes progressively darker toward the right end to indicate increasing density. A horizontal coordinate axis with an arrow pointing right labeled x runs directly beneath the rod. No other labels, lines, text, or axes appear.
Non-uniform rod of length L oriented along the x-axis.
A thin non-uniform rod of length \(L\) is positioned along the \(x\)-axis from \(x = 0\) to \(x = L\). The linear mass density of the rod varies quadratically according to the function \(\lambda(x) = \lambda_0\left(1 + \dfrac{x^2}{L^2}\right)\), where \(\lambda_0\) is a positive constant. Which of the following expressions represents the correct integral setup to determine the center of mass coordinate \(x_{\text{cm}}\) of the rod?

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