---
title: "A block of mass \\(m\\) is attached to an ideal spring with spring constant \\(k\\) on a horizontal, frictionless surface. The block is displaced to a position \\(x = +A\\) relative to its equilibrium position at \\(x = 0\\). At time \\(t = 0\\), the block is released from rest. The position of the block as a function of time \\(t\\) is given by the equation \\(x(t) = A \\cos(\\omega t)\\), where \\(\\omega\\) is the angular frequency of the oscillation."
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url: "https://nerd-notes.com/ubq/117773/"
date_modified: "2026-08-04T07:56:21+00:00"
---

# A block of mass \(m\) is attached to an ideal spring with spring constant \(k\) on a horizontal, frictionless surface. The block is displaced to a position \(x = +A\) relative to its equilibrium position at \(x = 0\). At time \(t = 0\), the block is released from rest. The position of the block as a function of time \(t\) is given by the equation \(x(t) = A \cos(\omega t)\), where \(\omega\) is the angular frequency of the oscillation.

A block of mass \(m\) is attached to an ideal spring with spring constant \(k\) on a horizontal, frictionless surface. The block is displaced to a position \(x = +A\) relative to its equilibrium position at \(x = 0\). At time \(t = 0\), the block is released from rest. The position of the block as a function of time \(t\) is given by the equation \(x(t) = A \cos(\omega t)\), where \(\omega\) is the angular frequency of the oscillation.

![A horizontal line representing a surface. On the left is a vertical hatched region representing a wall. A horizontal spring connects the wall to a rectangular block labeled 'm'. The surface is labeled as frictionless. A vertical dashed line passes through the center of the block, labeled 'x = +A'. Another vertical dashed line to the left of the block indicates the equilibrium position, labeled 'x = 0'. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830180-ntnmyb.jpg)

**Part a)** Using calculus, **derive** an expression for the velocity \(v(t)\) and acceleration \(a(t)\) of the block as functions of time. Express your answers in terms of \(A\), \(\omega\), and \(t\). *(2 points)*

**Part b)** On the axes provided, **sketch** graphs of the block's position \(x\), velocity \(v\), and acceleration \(a\) as functions of time \(t\) for one complete period \(T\). *(3 points)*

**Part c)** Using your expressions from part (a), **show that** the relationship between the block's acceleration and position satisfies the defining condition for simple harmonic motion. *(2 points)*

**Part d)** The initial conditions are changed such that at \(t = 0\), the block is at the equilibrium position \(x = 0\) moving in the positive \(x\)-direction. *(3 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117773/*