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title: "A flat plate of cross-sectional area \\(A\\) moves in the \\(+x\\)-direction at a constant speed \\(v_0\\) through a cloud of stationary dust particles. At time \\(t = 0\\), the front face of the plate is located at \\(x = 0\\). The volume mass density of the dust cloud varies with position according to \\(\\rho(x) = \\rho_0 \\left(\\dfrac{x}{L}\\right)^2\\), where \\(\\rho_0\\) and \\(L\\) are positive constants. As the dust particles collide with the plate, they stick to its surface. Which of the following expressions gives the magnitude of the external force \\(F(t)\\) required to keep the plate moving at constant speed \\(v_0\\) as a function of time \\(t\\)?"
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url: "https://nerd-notes.com/ubq/117641/"
date_modified: "2026-08-04T07:49:26+00:00"
---

# A flat plate of cross-sectional area \(A\) moves in the \(+x\)-direction at a constant speed \(v_0\) through a cloud of stationary dust particles. At time \(t = 0\), the front face of the plate is located at \(x = 0\). The volume mass density of the dust cloud varies with position according to \(\rho(x) = \rho_0 \left(\dfrac{x}{L}\right)^2\), where \(\rho_0\) and \(L\) are positive constants. As the dust particles collide with the plate, they stick to its surface. Which of the following expressions gives the magnitude of the external force \(F(t)\) required to keep the plate moving at constant speed \(v_0\) as a function of time \(t\)?

A flat plate of cross-sectional area \(A\) moves in the \(+x\)-direction at a constant speed \(v_0\) through a cloud of stationary dust particles. At time \(t = 0\), the front face of the plate is located at \(x = 0\). The volume mass density of the dust cloud varies with position according to \(\rho(x) = \rho_0 \left(\dfrac{x}{L}\right)^2\), where \(\rho_0\) and \(L\) are positive constants. As the dust particles collide with the plate, they stick to its surface. Which of the following expressions gives the magnitude of the external force \(F(t)\) required to keep the plate moving at constant speed \(v_0\) as a function of time \(t\)?

![A horizontal x-axis points to the right with origin x = 0. A flat vertical plate of cross-sectional area A moves to the right at constant velocity v_0. To the right of x = 0, a shaded region represents a cloud of stationary dust particles, with the shading density increasing quadratically with distance x. An arrow labeled v_0 indicates the plate's direction of motion. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785829766-XX616A.jpg)

- **A.** \(\dfrac{\rho_0 A v_0^4 t^2}{3 L^2}\)
- **B.** \(\dfrac{\rho_0 A v_0^4 t^2}{2 L^2}\)
- **C.** \(\dfrac{\rho_0 A v_0^4 t^2}{L^2}\)
- **D.** \(\dfrac{2 \rho_0 A v_0^4 t^2}{L^2}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117641/*