---
title: "A block of mass \\(m\\) is attached to one end of an ideal horizontal spring of spring constant \\(k\\). The other end of the spring is fixed to a rigid wall. The block rests on a frictionless horizontal table. The block is pulled to a displacement \\(x = A\\) from its equilibrium position (\\(x = 0\\)) and released from rest. The block then undergoes simple harmonic motion."
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url: "https://nerd-notes.com/ubq/110891/"
date_modified: "2026-04-09T01:54:56+00:00"
---

# A block of mass \(m\) is attached to one end of an ideal horizontal spring of spring constant \(k\). The other end of the spring is fixed to a rigid wall. The block rests on a frictionless horizontal table. The block is pulled to a displacement \(x = A\) from its equilibrium position (\(x = 0\)) and released from rest. The block then undergoes simple harmonic motion.

A block of mass \(m\) is attached to one end of an ideal horizontal spring of spring constant \(k\). The other end of the spring is fixed to a rigid wall. The block rests on a frictionless horizontal table. The block is pulled to a displacement \(x = A\) from its equilibrium position (\(x = 0\)) and released from rest. The block then undergoes simple harmonic motion.

![A horizontal table with a vertical wall on the left. A coiled spring connects the wall to a rectangular block labeled 'm'. A dashed vertical line passes through the center of the block, pointing down to an axis label 'x = A'. To the left of the block, another dashed vertical line indicates the unstretched length of the spring, pointing down to an axis label 'x = 0'. A horizontal axis below the table indicates the positive x direction to the right.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1775699695-oodLa8.jpg)

**Part a)** **Derive** an expression for the square of the block's speed, \(v^2\), when it is at an arbitrary position \(x\) between \(-A\) and \(A\). Express your answer in terms of \(m\), \(k\), \(A\), \(x\), and fundamental constants as appropriate. *(3 points)*

**Part b)** A student uses a motion sensor to record the block's speed and position. The student plots a graph of \(v^2\) on the vertical axis as a function of \(x^2\) on the horizontal axis. The resulting graph is a straight line. *(2 points)*

**Part c)** The student repeats the experiment, this time replacing the original block with a new block of mass \(M\), where \(M > m\). The new block is pulled to the same amplitude \(A\) and released. **Indicate** whether the magnitude of the slope of the new \(v^2\) versus \(x^2\) graph will be greater than, less than, or equal to the magnitude of the slope of the original graph. - [ ] Greater than - [ ] Less than - [ ] Equal to **Justify** your answer. *(3 points)*

**Part d)** **Derive** an expression for the magnitude of the maximum acceleration \(a_{\text{max}}\) of the block during its motion. Express your answer in terms of \(m\), \(k\), \(A\), and fundamental constants as appropriate. *(2 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/110891/*