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title: "A hypothetical spherical planet of radius \\(R\\) and total mass \\(M\\) has a volume mass density that varies with distance \\(r\\) from its center according to \\(\\rho(r) = C r\\) for \\(r \\le R\\), where \\(C\\) is a positive constant, and \\(\\rho(r) = 0\\) for \\(r > R\\). Which of the following pairs of expressions correctly gives the magnitude of the gravitational field \\(g(r)\\) inside the planet (\\(r \\le R\\)) and outside the planet (\\(r > R\\)) in terms of \\(G\\), \\(M\\), \\(R\\), and \\(r\\)?"
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url: "https://nerd-notes.com/ubq/123995/"
date_modified: "2026-09-28T13:30:04+00:00"
---

# A hypothetical spherical planet of radius \(R\) and total mass \(M\) has a volume mass density that varies with distance \(r\) from its center according to \(\rho(r) = C r\) for \(r \le R\), where \(C\) is a positive constant, and \(\rho(r) = 0\) for \(r > R\). Which of the following pairs of expressions correctly gives the magnitude of the gravitational field \(g(r)\) inside the planet (\(r \le R\)) and outside the planet (\(r > R\)) in terms of \(G\), \(M\), \(R\), and \(r\)?

A hypothetical spherical planet of radius \(R\) and total mass \(M\) has a volume mass density that varies with distance \(r\) from its center according to \(\rho(r) = C r\) for \(r \le R\), where \(C\) is a positive constant, and \(\rho(r) = 0\) for \(r > R\). Which of the following pairs of expressions correctly gives the magnitude of the gravitational field \(g(r)\) inside the planet (\(r \le R\)) and outside the planet (\(r > R\)) in terms of \(G\), \(M\), \(R\), and \(r\)?

- **A.** For \(r \le R\), \(g(r) = \dfrac{G M r}{R^3}\); for \(r > R\), \(g(r) = \dfrac{G M}{r^2}\)
- **B.** For \(r \le R\), \(g(r) = \dfrac{G M r}{R^3}\); for \(r > R\), \(g(r) = \dfrac{G M R^2}{r^4}\)
- **C.** For \(r \le R\), \(g(r) = \dfrac{G M r^2}{R^4}\); for \(r > R\), \(g(r) = \dfrac{G M R^2}{r^4}\)
- **D.** For \(r \le R\), \(g(r) = \dfrac{G M r^2}{R^4}\); for \(r > R\), \(g(r) = \dfrac{G M}{r^2}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/123995/*