written by
Gary Gladding
published by
the University of llinois Physics Education Research Group

This interactive homework problem presents a block attached to a massless spring on a frictionless surface. Given an initial velocity and distance from the equilibrium point, the problem takes learners step-by-step through the components of simple harmonic motion. It provides a conceptual analysis and explicit help to set up the appropriate solution. The problem is accompanied by a sequence of questions designed to encourage critical thinking and conceptual analysis.

This tutorial is part of a larger collection of interactive problems developed by the Illinois Physics Education Research Group.

Editor's Note:This problem can help students recognize the connection between the oscillation of a mass on a spring and the sinusoidal nature of simple harmonic motion. It provides help with the related free-body diagram, graphs depicting SHM, and support in using the Work-Kinetic Energy Theorem to solve. See Related Materials for an interactive simulation of spring motion, recommended by the editors.

9-12: 2A/H1. Mathematics is the study of quantities and shapes, the patterns and relationships between quantities or shapes, and operations on either quantities or shapes. Some of these relationships involve natural phenomena, while others deal with abstractions not tied to the physical world.

2C. Mathematical Inquiry

9-12: 2C/H3. To be able to use and interpret mathematics well, it is necessary to be concerned with more than the mathematical validity of abstract operations and to take into account how well they correspond to the properties of the things represented.

4. The Physical Setting

4F. Motion

9-12: 4F/H1. The change in motion (direction or speed) of an object is proportional to the applied force and inversely proportional to the mass.

9-12: 4F/H7. In most familiar situations, frictional forces complicate the description of motion, although the basic principles still apply.

9-12: 4F/H8. Any object maintains a constant speed and direction of motion unless an unbalanced outside force acts on it.

9. The Mathematical World

9B. Symbolic Relationships

6-8: 9B/M3. Graphs can show a variety of possible relationships between two variables. As one variable increases uniformly, the other may do one of the following: increase or decrease steadily, increase or decrease faster and faster, get closer and closer to some limiting value, reach some intermediate maximum or minimum, alternately increase and decrease, increase or decrease in steps, or do something different from any of these.

9-12: 9B/H2a. Symbolic statements can be manipulated by rules of mathematical logic to produce other statements of the same relationship, which may show some interesting aspect more clearly.

9-12: 9B/H4. Tables, graphs, and symbols are alternative ways of representing data and relationships that can be translated from one to another.

9-12: 9B/H5. When a relationship is represented in symbols, numbers can be substituted for all but one of the symbols and the possible value of the remaining symbol computed. Sometimes the relationship may be satisfied by one value, sometimes by more than one, and sometimes not at all.

11. Common Themes

11B. Models

9-12: 11B/H1a. A mathematical model uses rules and relationships to describe and predict objects and events in the real world.

Common Core State Standards for Mathematics Alignments

High School — Algebra (9-12)

Seeing Structure in Expressions (9-12)

A-SSE.1.a Interpret parts of an expression, such as terms, factors, and coefficients.

A-SSE.2 Use the structure of an expression to identify ways to rewrite it.

Creating Equations^{?} (9-12)

A-CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

Reasoning with Equations and Inequalities (9-12)

A-REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

High School — Functions (9-12)

Interpreting Functions (9-12)

F-IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.^{?}

F-IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.^{?}

Building Functions (9-12)

F-BF.1.a Determine an explicit expression, a recursive process, or steps for calculation from a context.

Trigonometric Functions (9-12)

F-TF.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.^{?}

Common Core State Reading Standards for Literacy in Science and Technical Subjects 6—12

Key Ideas and Details (6-12)

RST.11-12.3 Follow precisely a complex multistep procedure when carrying out experiments, taking measurements, or performing technical tasks; analyze the specific results based on explanations in the text.

Craft and Structure (6-12)

RST.11-12.4 Determine the meaning of symbols, key terms, and other domain-specific words and phrases as they are used in a specific scientific or technical context relevant to grades 11—12 texts and topics.

RST.11-12.5 Analyze how the text structures information or ideas into categories or hierarchies, demonstrating understanding of the information or ideas.

Range of Reading and Level of Text Complexity (6-12)

RST.11-12.10 By the end of grade 12, read and comprehend science/technical texts in the grades 11—CCR text complexity band independently and proficiently.

This resource is part of a Physics Front Topical Unit.

Topic: Periodic and Simple Harmonic Motion Unit Title: Simple Harmonic Motion

This interactive problem takes learners step-by-step through the components of simple harmonic motion. It will help students recognize the connection between the oscillation of a mass on a spring and the sinusoidal nature of SHM. It provides help with the related free-body diagram, graphs depicting SHM, and support in using the Work-Kinetic Energy Theorem to do the calculations.

<a href="https://www.compadre.org/precollege/items/detail.cfm?ID=6505">Gladding, Gary. Illinois PER Interactive Examples: Block and Spring SHM. Urbana: University of llinois Physics Education Research Group, June 16, 2006.</a>

G. Gladding, (University of llinois Physics Education Research Group, Urbana, 2006), WWW Document, (https://per.physics.illinois.edu/per/IE/ie.pl?phys111/ie/12/IE_block_and_spring).

G. Gladding, Illinois PER Interactive Examples: Block and Spring SHM, (University of llinois Physics Education Research Group, Urbana, 2006), <https://per.physics.illinois.edu/per/IE/ie.pl?phys111/ie/12/IE_block_and_spring>.

Gladding, G. (2006, June 16). Illinois PER Interactive Examples: Block and Spring SHM. Retrieved September 20, 2020, from University of llinois Physics Education Research Group: https://per.physics.illinois.edu/per/IE/ie.pl?phys111/ie/12/IE_block_and_spring

Gladding, Gary. Illinois PER Interactive Examples: Block and Spring SHM. Urbana: University of llinois Physics Education Research Group, June 16, 2006. https://per.physics.illinois.edu/per/IE/ie.pl?phys111/ie/12/IE_block_and_spring (accessed 20 September 2020).

Gladding, Gary. Illinois PER Interactive Examples: Block and Spring SHM. Urbana: University of llinois Physics Education Research Group, 2006. 16 June 2006. 20 Sep. 2020 <https://per.physics.illinois.edu/per/IE/ie.pl?phys111/ie/12/IE_block_and_spring>.

@misc{
Author = "Gary Gladding",
Title = {Illinois PER Interactive Examples: Block and Spring SHM},
Publisher = {University of llinois Physics Education Research Group},
Volume = {2020},
Number = {20 September 2020},
Month = {June 16, 2006},
Year = {2006}
}

%A Gary Gladding %T Illinois PER Interactive Examples: Block and Spring SHM %D June 16, 2006 %I University of llinois Physics Education Research Group %C Urbana %U https://per.physics.illinois.edu/per/IE/ie.pl?phys111/ie/12/IE_block_and_spring %O text/html

%0 Electronic Source %A Gladding, Gary %D June 16, 2006 %T Illinois PER Interactive Examples: Block and Spring SHM %I University of llinois Physics Education Research Group %V 2020 %N 20 September 2020 %8 June 16, 2006 %9 text/html %U https://per.physics.illinois.edu/per/IE/ie.pl?phys111/ie/12/IE_block_and_spring

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