---
title: "Two blocks of masses \\(m\\) and \\(2m\\) are placed on a frictionless horizontal surface and connected by an ideal spring of force constant \\(k\\) and relaxed length \\(L_0\\).  The blocks are pulled apart such that the total length of the spring becomes \\(L_0 + \\Delta x\\), and the system is released from rest at time \\(t = 0\\). Let \\(x(t)\\) denote the displacement of the spring from its relaxed length at time \\(t\\). Which of the following differential equations correctly describes the motion of the system in terms of \\(x(t)\\), and what is the angular frequency \\(\\omega\\) of the resulting oscillation?"
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url: "https://nerd-notes.com/ubq/124697/"
date_modified: "2026-09-28T14:09:09+00:00"
---

# Two blocks of masses \(m\) and \(2m\) are placed on a frictionless horizontal surface and connected by an ideal spring of force constant \(k\) and relaxed length \(L_0\).

The blocks are pulled apart such that the total length of the spring becomes \(L_0 + \Delta x\), and the system is released from rest at time \(t = 0\). Let \(x(t)\) denote the displacement of the spring from its relaxed length at time \(t\). Which of the following differential equations correctly describes the motion of the system in terms of \(x(t)\), and what is the angular frequency \(\omega\) of the resulting oscillation?

Two blocks of masses \(m\) and \(2m\) are placed on a frictionless horizontal surface and connected by an ideal spring of force constant \(k\) and relaxed length \(L_0\).

The blocks are pulled apart such that the total length of the spring becomes \(L_0 + \Delta x\), and the system is released from rest at time \(t = 0\). Let \(x(t)\) denote the displacement of the spring from its relaxed length at time \(t\). Which of the following differential equations correctly describes the motion of the system in terms of \(x(t)\), and what is the angular frequency \(\omega\) of the resulting oscillation?

![A side-view diagram of two rectangular blocks on a horizontal surface. A horizontal solid line at the bottom represents the frictionless surface. Resting on the surface on the left is a square block labeled m. Resting on the surface on the right is a wider rectangular block labeled 2m, having the same height as the left block but twice the width. Connecting the right vertical face of block m to the left vertical face of block 2m is a horizontal coiled line representing an ideal spring labeled k with exactly six coils. Below the surface, a horizontal dashed line is bounded by two vertical tick marks aligned with the inner faces of the blocks, with a double-headed horizontal arrow between the tick marks labeled L_0 + \Delta x. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604549-MGrE0t.jpg)

- **A.** \(\dfrac{d^2x}{dt^2} + \dfrac{k}{3m}x = 0\), with \(\omega = \sqrt{\dfrac{k}{3m}}\)
- **B.** \(\dfrac{d^2x}{dt^2} + \dfrac{3k}{2m}x = 0\), with \(\omega = \sqrt{\dfrac{3k}{2m}}\)
- **C.** \(\dfrac{d^2x}{dt^2} + \dfrac{2k}{m}x = 0\), with \(\omega = \sqrt{\dfrac{2k}{m}}\)
- **D.** \(\dfrac{d^2x}{dt^2} + \dfrac{3k}{m}x = 0\), with \(\omega = \sqrt{\dfrac{3k}{m}}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124697/*