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title: "A block of mass \\(M\\) is on a horizontal, frictionless surface and is attached to a fixed vertical wall by a nonlinear spring. The spring exerts a restoring force on the block given by \\(F_s(x) = -kx – \\beta x^3\\), where \\(x\\) is the displacement of the block from the spring’s equilibrium position (\\(x = 0\\)), and \\(k\\) and \\(\\beta\\) are positive constants. The block is initially held at rest at \\(x = 0\\). At time \\(t = 0\\), a constant external horizontal force of magnitude \\(F_0\\) is applied to the block in the \\(+x\\)-direction. Which of the following expressions represents the speed \\(v(x)\\) of the block as a function of its displacement \\(x\\)?"
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url: "https://nerd-notes.com/ubq/120583/"
date_modified: "2026-08-23T04:41:42+00:00"
---

# A block of mass \(M\) is on a horizontal, frictionless surface and is attached to a fixed vertical wall by a nonlinear spring. The spring exerts a restoring force on the block given by \(F_s(x) = -kx – \beta x^3\), where \(x\) is the displacement of the block from the spring’s equilibrium position (\(x = 0\)), and \(k\) and \(\beta\) are positive constants. The block is initially held at rest at \(x = 0\). At time \(t = 0\), a constant external horizontal force of magnitude \(F_0\) is applied to the block in the \(+x\)-direction. Which of the following expressions represents the speed \(v(x)\) of the block as a function of its displacement \(x\)?

A block of mass \(M\) is on a horizontal, frictionless surface and is attached to a fixed vertical wall by a nonlinear spring. The spring exerts a restoring force on the block given by \(F_s(x) = -kx - \beta x^3\), where \(x\) is the displacement of the block from the spring's equilibrium position (\(x = 0\)), and \(k\) and \(\beta\) are positive constants. The block is initially held at rest at \(x = 0\). At time \(t = 0\), a constant external horizontal force of magnitude \(F_0\) is applied to the block in the \(+x\)-direction. Which of the following expressions represents the speed \(v(x)\) of the block as a function of its displacement \(x\)?

![A horizontal frictionless surface is shown with a vertical wall at the far left. A block of mass \(M\) rests on the horizontal surface, connected to the wall by a horizontal coiled spring. The spring extends horizontally from the vertical wall to the left face of the block. A horizontal arrow labeled \(F_0\) originates at the right face of the block and points horizontally to the right. Below the horizontal surface, a horizontal axis is oriented with an arrowhead pointing to the right, labeled \(x\). A vertical dashed line extends downward from the center of the block to a tick mark labeled \(0\) on the \(x\)-axis, indicating the equilibrium position \(x = 0\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787460102-vsHr1l.jpg)

- **A.** \(v(x) = \sqrt{\dfrac{2}{M}\left(F_0 x - k x^2 - \beta x^4\right)}\)
- **B.** \(v(x) = \sqrt{\dfrac{1}{M}\left(F_0 x - \dfrac{1}{2} k x^2 - \dfrac{1}{4} \beta x^4\right)}\)
- **C.** \(v(x) = \sqrt{\dfrac{2}{M}\left(F_0 x - \dfrac{1}{2} k x^2 - \dfrac{1}{4} \beta x^4\right)}\)
- **D.** \(v(x) = \sqrt{\dfrac{2}{M}\left(F_0 x - \dfrac{1}{2} k x^2 - \dfrac{1}{2} \beta x^4\right)}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/120583/*