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title: "A block of mass \\(m\\) rests on a horizontal, frictionless surface and is attached to a non-linear spring. The spring exerts a restoring force given by \\(F(x) = -kx – \\beta x^3\\), where \\(k\\) and \\(\\beta\\) are positive constants and \\(x\\) is the displacement of the block from its equilibrium position at \\(x = 0\\). The block is pulled to a displacement \\(x = d\\) and released from rest. Which of the following expressions gives the maximum speed \\(v_{\\text{max}}\\) of the block, and what is the limiting value of \\(v_{\\text{max}}\\) as \\(\\beta \\to 0\\)?"
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url: "https://nerd-notes.com/ubq/120662/"
date_modified: "2026-08-23T04:42:31+00:00"
---

# A block of mass \(m\) rests on a horizontal, frictionless surface and is attached to a non-linear spring. The spring exerts a restoring force given by \(F(x) = -kx – \beta x^3\), where \(k\) and \(\beta\) are positive constants and \(x\) is the displacement of the block from its equilibrium position at \(x = 0\). The block is pulled to a displacement \(x = d\) and released from rest. Which of the following expressions gives the maximum speed \(v_{\text{max}}\) of the block, and what is the limiting value of \(v_{\text{max}}\) as \(\beta \to 0\)?

A block of mass \(m\) rests on a horizontal, frictionless surface and is attached to a non-linear spring. The spring exerts a restoring force given by \(F(x) = -kx - \beta x^3\), where \(k\) and \(\beta\) are positive constants and \(x\) is the displacement of the block from its equilibrium position at \(x = 0\). The block is pulled to a displacement \(x = d\) and released from rest. Which of the following expressions gives the maximum speed \(v_{\text{max}}\) of the block, and what is the limiting value of \(v_{\text{max}}\) as \(\beta \to 0\)?

![A horizontal flat surface is bounded on the left by a vertical wall with hatching. A horizontal coiled spring extends from the vertical wall to the left side of a rectangular block labeled m. A horizontal dashed line below the surface shows two vertical tick marks: the left tick mark is aligned with the center of the block at equilibrium and is labeled x = 0; the right tick mark is labeled x = d with a small dashed rectangle representing the displaced block. A horizontal right-pointing arrow above the displacement interval is labeled d. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787460151-byooRx.jpg)

- **A.** \(v_{\text{max}} = d\sqrt{\dfrac{2k}{m} + \dfrac{2\beta d^2}{m}}\), and as \(\beta \to 0\), \(v_{\text{max}} \to d\sqrt{\dfrac{2k}{m}}\)
- **B.** \(v_{\text{max}} = d\sqrt{\dfrac{k}{m} + \dfrac{\beta d^2}{4m}}\), and as \(\beta \to 0\), \(v_{\text{max}} \to 0\)
- **C.** \(v_{\text{max}} = d\sqrt{\dfrac{k}{m} + \dfrac{\beta d^2}{m}}\), and as \(\beta \to 0\), \(v_{\text{max}} \to d\sqrt{\dfrac{k}{m}}\)
- **D.** \(v_{\text{max}} = d\sqrt{\dfrac{k}{m} + \dfrac{\beta d^2}{2m}}\), and as \(\beta \to 0\), \(v_{\text{max}} \to d\sqrt{\dfrac{k}{m}}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/120662/*