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AP Physics C: Mechanics
2.1 Systems and Center of Mass
AdvancedMCQMathematical18.1k
A three-dimensional diagram showing a solid hemisphere positioned in a Cartesian coordinate system. The flat circular base of radius \(R\) lies in the horizontal \(xy\)-plane, centered at the origin \((0,0,0)\). The \(z\)-axis points vertically upward through the center of the base to the top apex of the hemisphere at \(z = R\). A circular cross-sectional slice of thin thickness \(dz\) is shaded light gray at a height \(z\) above the base, with its radius labeled \(r = \sqrt{R^2 - z^2}\). An arrow indicating height \(z\) extends vertically from the origin along the \(z\)-axis to the center of the slice. A dimension line of length \(R\) marks the total height along the \(z\)-axis from \(z = 0\) to \(z = R\). No other labels, lines, text, or axes appear.
Solid hemisphere with height-dependent mass density.
A solid hemisphere of radius \(R\) has its flat circular base centered at the origin in the \(xy\)-plane, with its curved surface extending along the positive \(z\)-axis from \(z = 0\) to \(z = R\). The volume mass density of the hemisphere varies with height \(z\) above the base according to \(\rho(z) = \rho_0 \left(\dfrac{z}{R}\right)\), where \(\rho_0\) is a positive constant. Which of the following expressions represents the \(z\)-coordinate of the hemisphere's center of mass, \(z_{\text{cm}}\)?

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