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AP Physics C: Mechanics
5.4 Rotational Inertia
AdvancedMCQMathematicalConceptual25.1k
A quantitative graph showing total rotational inertia on the vertical axis and distance squared on the horizontal axis. The vertical axis is labeled \(I\ (\text{kg}\cdot\text{m}^2)\) with major tick marks and numerical labels at 0.00, 0.20, 0.40, 0.60, 0.80, and 1.00. The horizontal axis is labeled \(d^2\ (\text{m}^2)\) with major tick marks and numerical labels at 0.00, 0.05, 0.10, 0.15, 0.20, and 0.25. A light gray rectangular grid aligns with these tick marks. Exactly five data points are plotted as small filled circular markers at coordinates (0.04, 0.32), (0.08, 0.44), (0.12, 0.56), (0.16, 0.68), and (0.20, 0.80). A single straight solid black line passes through all five points, starting at the vertical intercept at (0.00, 0.20) and extending linearly with a constant positive slope through (0.25, 0.95). No other lines, curves, background shading, or markings appear on the graph.
Graph of total rotational inertia \(I\) versus distance squared \(d^2\).
A student investigates the rotational inertia of an irregularly shaped object of unknown mass \(M\). The object is clamped to a horizontal mounting disk that rotates about a fixed vertical axis through its center. The disk has a known rotational inertia \(I_{\text{disk}} = 0.080\text{ kg}\cdot\text{m}^2\) about this axis. The object is positioned such that its center of mass is at various radial distances \(d\) from the rotation axis. For each distance \(d\), the total rotational inertia \(I\) of the disk-object system is measured, and a graph of \(I\) versus \(d^2\) is plotted. The resulting best-fit line passes through the points \((0.04\text{ m}^2, 0.32\text{ kg}\cdot\text{m}^2)\) and \((0.20\text{ m}^2, 0.80\text{ kg}\cdot\text{m}^2)\). Based on the graph, what are the total mass \(M\) of the object and its rotational inertia \(I_{\text{cm}}\) about an axis through its center of mass parallel to the rotation axis?

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