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AP Physics C: Mechanics
2.1 Systems and Center of Mass
IntermediateMCQMathematical15.8k
A rectangular plate in the xy-plane with its lower-left corner at the origin. The plate extends horizontally along the x-axis from 0 to L and vertically along the y-axis from 0 to W. The density of shading increases smoothly along the x-axis from left to right, being white at x = 0 and dark gray at x = L. A horizontal arrow along the bottom edge is labeled x, reaching x = L, and a vertical arrow along the left edge is labeled y, reaching y = W. No other labels, lines, text, or axes appear.
A rectangular plate with quadratic mass density variation along the x-axis.
A thin rectangular plate of length \(L\) and width \(W\) lies in the \(xy\)-plane bounded by \(x = 0\) to \(x = L\) and \(y = 0\) to \(y = W\). The surface mass density of the plate is non-uniform and given by \(\sigma(x) = \sigma_0 \left(\dfrac{x}{L}\right)^2\), where \(\sigma_0\) is a constant. What is the \(x\)-coordinate of the center of mass of the plate?

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