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title: "A block of mass \\(m\\) on a horizontal surface is attached to an ideal spring with spring constant \\(k\\), where \\(x = 0\\) represents the spring’s unstretched equilibrium position. The coefficient of kinetic friction between the block and the surface increases with position according to \\(\\mu_k(x) = bx\\), where \\(b\\) is a positive constant and \\(x > 0\\). The block is pulled to position \\(x = x_0\\) and released from rest. Which of the following expressions represents the speed of the block as it first passes through the equilibrium position \\(x = 0\\)?"
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url: "https://nerd-notes.com/ubq/117665/"
date_modified: "2026-08-04T07:49:40+00:00"
---

# A block of mass \(m\) on a horizontal surface is attached to an ideal spring with spring constant \(k\), where \(x = 0\) represents the spring’s unstretched equilibrium position. The coefficient of kinetic friction between the block and the surface increases with position according to \(\mu_k(x) = bx\), where \(b\) is a positive constant and \(x > 0\). The block is pulled to position \(x = x_0\) and released from rest. Which of the following expressions represents the speed of the block as it first passes through the equilibrium position \(x = 0\)?

A block of mass \(m\) on a horizontal surface is attached to an ideal spring with spring constant \(k\), where \(x = 0\) represents the spring's unstretched equilibrium position. The coefficient of kinetic friction between the block and the surface increases with position according to \(\mu_k(x) = bx\), where \(b\) is a positive constant and \(x > 0\). The block is pulled to position \(x = x_0\) and released from rest. Which of the following expressions represents the speed of the block as it first passes through the equilibrium position \(x = 0\)?

![A horizontal surface with a vertical wall on the far left. An ideal spring with spring constant k extends horizontally from the wall to a rectangular block of mass m. The origin x = 0 is marked with a dashed vertical line at the spring's relaxed position. The block is positioned at x = x_0 to the right of x = 0. An arrow labeled v = 0 points at the block. Below the surface, an arrow points to the right labeled x, with text below reading mu_k(x) = bx. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-diagram-1-1785829780-PdZZT4.jpg)

- **A.** \(x_0 \sqrt{\dfrac{k + bmg}{m}}\)
- **B.** \(x_0 \sqrt{\dfrac{2k - bmg}{m}}\)
- **C.** \(x_0 \sqrt{\dfrac{k - bmg}{m}}\)
- **D.** \(x_0 \sqrt{\dfrac{k - 2bmg}{m}}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117665/*