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title: "A solid cylinder of mass \\(M\\) and cross-sectional area \\(A\\) floats vertically in a fluid whose density increases linearly with depth \\(z\\) according to \\(\\rho(z) = \\rho_0 (1 + b z)\\), where \\(\\rho_0\\) and \\(b\\) are positive constants and \\(z\\) is measured downward from the fluid surface. The total upward buoyant force exerted on the cylinder when its submerged depth is \\(z\\) is given by \\(F_B(z) = \\rho_0 g A \\left( z + \\dfrac{1}{2} b z^2 \\right)\\). The cylinder rests in static equilibrium at depth \\(z = z_0\\). If the cylinder is displaced slightly from its equilibrium depth and released, what is the angular frequency \\(\\omega\\) of its small vertical oscillations?"
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url: "https://nerd-notes.com/ubq/117663/"
date_modified: "2026-08-04T07:49:38+00:00"
---

# A solid cylinder of mass \(M\) and cross-sectional area \(A\) floats vertically in a fluid whose density increases linearly with depth \(z\) according to \(\rho(z) = \rho_0 (1 + b z)\), where \(\rho_0\) and \(b\) are positive constants and \(z\) is measured downward from the fluid surface. The total upward buoyant force exerted on the cylinder when its submerged depth is \(z\) is given by \(F_B(z) = \rho_0 g A \left( z + \dfrac{1}{2} b z^2 \right)\). The cylinder rests in static equilibrium at depth \(z = z_0\). If the cylinder is displaced slightly from its equilibrium depth and released, what is the angular frequency \(\omega\) of its small vertical oscillations?

A solid cylinder of mass \(M\) and cross-sectional area \(A\) floats vertically in a fluid whose density increases linearly with depth \(z\) according to \(\rho(z) = \rho_0 (1 + b z)\), where \(\rho_0\) and \(b\) are positive constants and \(z\) is measured downward from the fluid surface. The total upward buoyant force exerted on the cylinder when its submerged depth is \(z\) is given by \(F_B(z) = \rho_0 g A \left( z + \dfrac{1}{2} b z^2 \right)\). The cylinder rests in static equilibrium at depth \(z = z_0\). If the cylinder is displaced slightly from its equilibrium depth and released, what is the angular frequency \(\omega\) of its small vertical oscillations?

![A schematic diagram showing a vertical cylindrical container filled with fluid. A solid cylinder of mass M and cross-sectional area A floats vertically in the fluid. A vertical coordinate axis labeled z points downward, with origin z = 0 at the top surface of the fluid. The submerged base of the cylinder is located at depth z = z_0, which is marked by a horizontal dashed line labeled z_0. A small vertical double-headed arrow next to the cylinder is labeled y to represent vertical displacement from equilibrium. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785829778-BK2r3S.jpg)

- **A.** \(\omega = \sqrt{\dfrac{\rho_0 g A}{M}}\)
- **B.** \(\omega = \sqrt{\dfrac{\rho_0 g A \left(1 + \frac{1}{2} b z_0\right)}{M}}\)
- **C.** \(\omega = \sqrt{\dfrac{\rho_0 g A (1 + b z_0)}{M}}\)
- **D.** \(\omega = \sqrt{\dfrac{\rho_0 g A (1 + 2 b z_0)}{M}}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117663/*