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title: "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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url: "https://nerd-notes.com/ubq/117655/"
date_modified: "2026-08-04T07:49:31+00:00"
---

# 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, \(z_{\text{cm}}\)?

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, \(z_{\text{cm}}\)?

![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.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785829771-IZKzNl.jpg)

- **A.** \(\dfrac{3}{8} R\)
- **B.** \(\dfrac{2}{5} R\)
- **C.** \(\dfrac{1}{2} R\)
- **D.** \(\dfrac{8}{15} R\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117655/*