---
title: "A spring of ideal spring constant \\(k\\) hangs vertically from a ceiling. When the spring is unextended, its bottom end is at position \\(y = 0\\). The positive \\(y\\)-direction is defined as downward. A block of mass \\(m\\) is attached to the spring and gently lowered until it hangs at rest at its equilibrium position \\(y_{eq}\\). The block is then pulled down an additional distance \\(A\\) to a maximum position \\(y_{max} = y_{eq} + A\\) and released from rest."
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url: "https://nerd-notes.com/ubq/111711/"
date_modified: "2026-04-20T09:12:16+00:00"
---

# A spring of ideal spring constant \(k\) hangs vertically from a ceiling. When the spring is unextended, its bottom end is at position \(y = 0\). The positive \(y\)-direction is defined as downward. A block of mass \(m\) is attached to the spring and gently lowered until it hangs at rest at its equilibrium position \(y_{eq}\). The block is then pulled down an additional distance \(A\) to a maximum position \(y_{max} = y_{eq} + A\) and released from rest.

A spring of ideal spring constant \(k\) hangs vertically from a ceiling. When the spring is unextended, its bottom end is at position \(y = 0\). The positive \(y\)-direction is defined as downward. A block of mass \(m\) is attached to the spring and gently lowered until it hangs at rest at its equilibrium position \(y_{eq}\). The block is then pulled down an additional distance \(A\) to a maximum position \(y_{max} = y_{eq} + A\) and released from rest.

![Three vertical configurations side-by-side. Left: A bare vertical spring attached to a ceiling, its bottom end aligns with a dashed horizontal line labeled 'y = 0'. Middle: The same spring with a rectangular block of mass m attached, stretched so the block aligns with a dashed horizontal line labeled 'y_eq'. Right: The spring and block stretched further down, with the block aligning with a dashed horizontal line labeled 'y_max = y_eq + A'. To the far left of the setups, a vertical downward-pointing arrow indicates the y-axis, labeled 'y (positive downward)'.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1776211290-Lsgf6x.jpg)

**Part a)** **Derive** an expression for the equilibrium position \(y_{eq}\) of the block. Express your answer in terms of \(m\), \(k\), and fundamental constants as appropriate. *(2 points)*

**Part b)** Using conservation of energy, **derive** an expression for the speed \(v\) of the block as a function of its position \(y\) during its oscillation. Express your answer in terms of \(m\), \(k\), \(A\), \(y\), \(y_{eq}\), and fundamental constants as appropriate. *(3 points)*

**Part c)** Using your expression from part (b), **derive** an expression for the maximum speed \(v_{max}\) of the block. Express your answer in terms of \(m\), \(k\), and \(A\) only. Show your algebraic steps. *(2 points)*

**Part d)** A student claims that if the same experiment (pulling the block down a distance \(A\) from equilibrium and releasing it) were performed on a planet with a greater acceleration due to gravity, the maximum speed of the block during its oscillation would be greater. **Indicate** whether the student's claim is correct or incorrect. - [ ] Correct - [ ] Incorrect **Justify** your answer using physical principles and your previous derivations. *(2 points)*

**Part e)** On the axes below, **sketch** a graph of the block's kinetic energy \(K\) as a function of position \(y\) from \(y = y_{eq} - A\) to \(y = y_{eq} + A\). **Label** the maximum value of \(K\) on the vertical axis. *(3 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/111711/*