---
title: "A block of known mass \\(m\\) is placed on a rough, horizontal surface. One end of a light spring of unknown spring constant \\(k\\) is attached to the block, and the other end is attached to a fixed wall. The coefficient of kinetic friction between the block and the surface is \\(\\mu_k\\). A student wants to determine the spring constant \\(k\\) by pulling the block to stretch the spring to various distances \\(\\Delta x\\) from its equilibrium length, releasing the block from rest, and measuring its initial acceleration \\(a\\).  Assume that the initial acceleration \\(a\\) is measured immediately after the block begins to move, such that the frictional force acting on the block is the kinetic friction force \\(f_k\\)."
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url: "https://nerd-notes.com/ubq/112736/"
date_modified: "2026-04-30T21:07:50+00:00"
---

# A block of known mass \(m\) is placed on a rough, horizontal surface. One end of a light spring of unknown spring constant \(k\) is attached to the block, and the other end is attached to a fixed wall. The coefficient of kinetic friction between the block and the surface is \(\mu_k\). A student wants to determine the spring constant \(k\) by pulling the block to stretch the spring to various distances \(\Delta x\) from its equilibrium length, releasing the block from rest, and measuring its initial acceleration \(a\).

Assume that the initial acceleration \(a\) is measured immediately after the block begins to move, such that the frictional force acting on the block is the kinetic friction force \(f_k\).

A block of known mass \(m\) is placed on a rough, horizontal surface. One end of a light spring of unknown spring constant \(k\) is attached to the block, and the other end is attached to a fixed wall. The coefficient of kinetic friction between the block and the surface is \(\mu_k\). A student wants to determine the spring constant \(k\) by pulling the block to stretch the spring to various distances \(\Delta x\) from its equilibrium length, releasing the block from rest, and measuring its initial acceleration \(a\).

Assume that the initial acceleration \(a\) is measured immediately after the block begins to move, such that the frictional force acting on the block is the kinetic friction force \(f_k\).

![A physical setup showing a horizontal surface with a block labeled 'm' resting on it. A spring is attached to the left side of the block, and the other end of the spring is attached to a vertical wall on the left. The surface is drawn with hatch marks indicating it is rough (friction). A dashed vertical line indicates the spring's equilibrium position, and the block is positioned to the right of this line. A horizontal bracket labeled '\Delta x' spans from the dashed line to the block. A vector arrow pointing to the left from the block is labeled 'a' to indicate the direction of initial acceleration.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1777583270-XOfQYQ.jpg)

**Part a)** **Derive** an expression for the magnitude of the initial acceleration \(a\) of the block immediately after it is released. Express your answer in terms of \(m\), \(k\), \(\Delta x\), \(\mu_k\), and fundamental constants, as appropriate. *(3 points)*

**Part b)** **Describe** an experimental procedure that the student could use to gather the data needed to determine the spring constant \(k\). In your description, identify the quantities to be measured and the equipment to be used to measure them. You may include a table like the one below to organize your equipment list. Include any steps necessary to reduce experimental uncertainty. | Quantity to be Measured | Symbol for Quantity | Equipment for Measurement | | :---: | :---: | :---: | | | | | | | | | | | | | *(4 points)*

**Part c)** The student collects the data and plots the initial acceleration \(a\) on the vertical axis and the stretch distance \(\Delta x\) on the horizontal axis to produce a straight-line graph. *(2 points)*

**Part d)** The student repeats the experiment using a new block with mass \(2m\) on the same rough surface. The student creates a new linear graph of \(a\) as a function of \(\Delta x\). *(4 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/112736/*