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AP Physics C: Mechanics
5.6 Newton’s Second Law in Rotational Form
5.4 Rotational Inertia
5.3 Torque
AdvancedMCQMathematical23.5k
A horizontal rod of length \(L\) lies along the horizontal axis, supported at its left end by a circular pivot pin attached to a fixed vertical mount. The left end of the rod is labeled \(x = 0\) and the right free end is labeled \(x = L\). The rod is shaded with a smooth grayscale gradient that is light gray near \(x = 0\) and transitions to dark gray near \(x = L\) to indicate non-uniform mass density. A downward vertical arrow near the pivot indicates the direction of gravitational acceleration, labeled \(g\). A horizontal double-headed dimension arrow spans below the rod from \(x = 0\) to \(x = L\) and is labeled \(L\). No other labels, lines, text, or axes appear.
A non-uniform rod pivoted at one end and held horizontally.
A thin, non-uniform rod of length \(L\) is pivoted about a frictionless horizontal axle through one end. The rod has a linear mass density given by \(\lambda(x) = k x^3\), where \(x\) is the distance from the pivot and \(k\) is a positive constant. The rod is held in a horizontal orientation and released from rest. Immediately after release, what is the magnitude of the angular acceleration of the rod about the pivot?

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