About

1.1.3 Introduction to terms used LO (c)

1.1.3.1 Q1: Consider an object P oscillating between point A and B about the origin (0,0), assuming the usual Cartesian Coordinate System apply.  Observe the Model and suggests possible meaning of the following points with the most suitable descriptions.

Central equilibrium position _ B O P A  

Instantaneous position _ B O P A  

Maximum amplitude _ B O P A m

Minimum amplitude _ B O P A m

Given the equation x = x₀ sin ( ω t ) can describe SHM, suggests the usual symbols associated to the physical quantity

Central equilibrium position  _ 0 x x0 -x0 w t T f  

Instantaneous position or displacement given by vector OP  _ 0 x x0 -x0 w t T f m

Maximum amplitude  _ 0 x x0 -x0 w t T f m

Minimum amplitude  _ 0 x x0 -x0 w t T f m

Time taken for one complete oscillation, for example Path from O→A→O→B→O   _ 0 x x0 -x0 w t T f s

Number of oscillations performed per unit time _ 0 x x0 -x0 w t T f 1/s. Hence, f and T are related by the equation ${\displaystyle f=\frac{1}{T}}$

Angular frequency _ 0 x x0 -x0 w t T f rad/s. Since one complete oscillation is 2π radians, ω and f are related by ω = 2π f
 
 

1.1.3.2 Example

1.1.3.2.1 The displacement of a spring mass system from a fixed point is as shown. From the graph, determine the
(a)     amplitude,  
(b)     period,     
(c)     frequency,      
(d)     angular frequency, of the oscillations.
[2.00 m, 6.28 s, 0.159 Hz, 1.00 rad s^–1]

1.1.3.2.2  Model:

1. Run Sim
2. http://iwant2study.org/ospsg/index.php/69

Suggested Activity Q1: run model with different starting x to explore the meaning of amplitude
Suggested Activity Q2: run model with different mass m and spring constant k to explore different period T, frequency f and angular frequency ω
 

 

Translations

Code	Language		Translator	Run

Credits

This email address is being protected from spambots. You need JavaScript enabled to view it.; Francisco Esquembre; Flix Jess Garcia Clemente

http://iwant2study.org/lookangejss/02_newtonianmechanics_8oscillations/ejss_model_SHM04/SHM04_Simulation.xhtml

Briefing Document: 📈Horizontal Spring Mass Model Simulation

1. Overview

This document reviews the "Horizontal Spring Mass Model X vs t JavaScript HTML5 Applet Simulation Model," a resource hosted by Open Educational Resources / Open Source Physics @ Singapore. The core of this resource is an interactive simulation designed to illustrate the principles of Simple Harmonic Motion (SHM) using a spring-mass system. The document will cover the key concepts introduced, the applet's functionality and suggested uses, and related resources.

2. Key Themes & Concepts

• Simple Harmonic Motion (SHM): The primary focus is to demonstrate SHM through the oscillation of a mass attached to a spring. The simulation visualizes the relationship between displacement (x) and time (t).
• Terminology of Oscillations: The document explicitly defines core concepts related to oscillatory motion:
• Central Equilibrium Position: The resting position of the mass when no external force is acting.
• Instantaneous Position/Displacement: "Instantaneous position or displacement given by vector OP m." This emphasizes the vector nature of displacement from the equilibrium position.
• Maximum Amplitude: The furthest distance the mass travels from the equilibrium position.
• Minimum Amplitude: This appears to be an error, as a minimum amplitude should be 0 for typical SHM. This is likely an oversight in the documentation.
• Period (T): "Time taken for one complete oscillation, for example Path from O→A→O→B→O s". The time it takes for one complete cycle of motion.
• Frequency (f): "Number of oscillations performed per unit time 1/s." The number of oscillations per second. Related to the period by f = 1/T.
• Angular Frequency (ω): "Since one complete oscillation is 2π radians, ω and f are related by ω = 2π f". Measured in radians per second, it's a measure of the rate of oscillation in terms of angle.
• Mathematical Representation: The simulation is connected to the SHM equation: x = x0 sin (ωt), where x is displacement, x0 is the amplitude, ω is the angular frequency, and t is time.
• Relationship Between Parameters: The resource highlights the interdependencies between period, frequency, and angular frequency with the equations f = 1/T and ω = 2πf.

3. Simulation Functionality & Suggested Activities

• Interactive Applet: The simulation is embedded in the page via an iframe, enabling users to interact with a spring-mass model directly within their web browser. The link to the actual simulation is provided: http://iwant2study.org/lookangejss/02_newtonianmechanics_8oscillations/ejss_model_SHM04/SHM04_Simulation.xhtml
• Exploration Activities: The document suggests two primary activities:
• Q1: "run model with different starting x to explore the meaning of amplitude." This encourages users to vary the initial displacement to observe its effect on the system's oscillation.
• Q2: "run model with different mass m and spring constant k to explore different period T, frequency f and angular frequency ω." This exploration is designed to allow users to investigate how system parameters (mass and spring constant) affect oscillatory behavior.

4. Example Application

• Graphical Analysis: The document provides a sample problem demonstrating how to analyze a displacement-time graph to determine the amplitude, period, frequency, and angular frequency. It gives the sample answers as [2.00 m, 6.28 s, 0.159 Hz, 1.00 rad s–1https://youtu.be/fHBLMbKi4vA" target="_blank" rel="noopener noreferrer" style="box-sizing: border-box; color: rgb(0, 102, 204); text-decoration: underline;">https://youtu.be/fHBLMbKi4vA, titled "Progressive mathematical modeling with OSP@SG by lookang lawrence wee," which presumably provides further guidance on using the models.
• Android App: There is a related app available on the Google Play Store: https://play.google.com/store/apps/details?id=com.ionicframework.shm08app156228&hl=en, indicating the platform is supported on multiple media.
• Extensive List of Related Simulations: The provided document also includes an extremely long list of different simulations that this website contains. This suggests that the simulation about the Spring Mass Model is part of a larger effort to provide a diverse range of interactive tools for teaching STEM related topics. The simulations cover topics including:
• Mechanics (e.g. Collisions, Projectile Motion, Pendulums)
• Waves and Oscillations
• Optics (e.g. Diffraction, Interference)
• Electromagnetism
• Thermodynamics
• Mathematics (e.g. Vectors, Geometry)
• Interactive Games

7. Credits and Licensing

• Authorship: The simulation is credited to "Francisco Esquembre; Flix Jess Garcia Clemente".
• License: The content is licensed under a "Creative Commons Attribution-Share Alike 4.0 Singapore License," while the "EasyJavaScriptSimulations Library" has a separate commercial use license from the University of Murcia. This highlights the importance of acknowledging sources and respecting licensing when using open education resources.

8. Conclusion

The "Horizontal Spring Mass Model" simulation is a valuable resource for teaching and learning about simple harmonic motion. It combines an interactive applet, definitions of key concepts, practical examples, and related resources to provide a comprehensive learning experience. The emphasis on exploring the effect of different parameters on the simulation through experimentation is in line with best practices in pedagogical approaches to science education.

Simple Harmonic Motion Study Guide

Quiz

Instructions: Answer the following questions in 2-3 sentences each.

1. What is the central equilibrium position in the context of Simple Harmonic Motion (SHM)?
2. What is meant by the instantaneous position of an oscillating object?
3. Define the maximum amplitude in SHM, and how does it relate to the object’s motion?
4. What does the period of oscillation represent, and what is its unit of measurement?
5. How is the frequency of oscillation defined, and what are its units?
6. Explain the relationship between the frequency (f) and the period (T) of oscillation.
7. What is angular frequency (ω), and how is it related to frequency (f)?
8. Describe how the simulation model can be used to explore the meaning of amplitude.
9. What is one way to use the simulation model to investigate the period, frequency and angular frequency?
10. If the displacement of a spring mass system is described by x = x0 sin ( ωt ), what do the symbols x, x0, ω and t represent?

Answer Key

1. The central equilibrium position is the point where the oscillating object would rest when not disturbed; it is the zero-displacement point around which oscillations occur. In the Cartesian coordinate system, this is often the origin (0,0).
2. The instantaneous position is the location of the oscillating object at any given moment. It is represented by a vector from the origin to the object's current location, indicating both its distance and direction from the equilibrium point.
3. Maximum amplitude is the greatest displacement of the object from its equilibrium position during oscillation. It represents the furthest the object moves from its central resting position in either direction.
4. The period of oscillation is the time it takes for an oscillating object to complete one full cycle of motion. It is measured in seconds (s).
5. The frequency of oscillation is the number of complete cycles an object makes per unit of time. It is measured in Hertz (Hz) or oscillations per second (1/s).
6. The frequency (f) and period (T) are inversely related; frequency is the reciprocal of the period (f = 1/T), and the period is the reciprocal of the frequency (T=1/f).
7. Angular frequency (ω) represents the rate of change of the phase of an oscillating object's motion. It is related to the regular frequency by the equation ω = 2πf and is measured in radians per second (rad/s).
8. By running the model with different starting positions for the oscillating mass, we can observe how the mass travels from different starting points, which changes the maximum displacement of the mass from its central equilibrium position.
9. The model can be run with varying masses and spring constants. Observing the resulting changes in the speed of oscillations and the time it takes for the object to complete one full cycle is how to explore the period, frequency and angular frequency.
10. In the equation x = x0 sin ( ωt ), x is the instantaneous position of the mass at a time t; x0 is the maximum amplitude of the oscillation; ω is the angular frequency and t is time.

Essay Questions

Instructions: Answer the following essay questions in a well-organized manner, using evidence from the provided materials to support your claims.

1. Explain the relationships between amplitude, period, frequency, and angular frequency in the context of Simple Harmonic Motion (SHM). How do these concepts manifest in the horizontal spring-mass model, and what physical variables influence them?
2. Using the provided simulation model as a reference, describe how changing the spring constant and mass affects the period and frequency of oscillation in a horizontal spring-mass system. Provide examples of how you might adjust the simulation to explore these effects.
3. The concept of SHM is fundamental in physics. Discuss how the equation x = x0 sin ( ω t ) mathematically represents this motion, paying particular attention to what each term describes and how they relate to the overall system.
4. How can educators use the simulation tools detailed in this document to enhance the understanding of SHM among students? Describe the educational advantages offered by using simulation models as learning tools.
5. Considering the connections between the simulation and the real world, how might an understanding of SHM be applied to other areas of physics, science, and engineering?

Glossary of Key Terms

• Simple Harmonic Motion (SHM): A type of periodic motion where the restoring force is directly proportional to the displacement and acts in the opposite direction to that displacement, resulting in oscillations around an equilibrium point.
• Central Equilibrium Position: The point at which an oscillating object would rest if undisturbed; it's the position of zero displacement.
• Instantaneous Position (Displacement): The location of an oscillating object at a specific moment, described as a vector from the equilibrium position.
• Maximum Amplitude: The greatest displacement of the oscillating object from its equilibrium position during its motion.
• Period (T): The time taken for an oscillating object to complete one full cycle of motion, measured in seconds (s).
• Frequency (f): The number of complete oscillations made by an object per unit of time, measured in Hertz (Hz) or oscillations per second (1/s).
• Angular Frequency (ω): The rate of change of the phase of an oscillating object's motion; related to frequency by ω = 2πf, measured in radians per second (rad/s).
• Displacement (x): The distance an object is from its equilibrium point in Simple Harmonic Motion (SHM).
• Spring Constant (k): A measure of the stiffness of a spring; it is the ratio of the force applied to the spring and how much it extends or compresses.

Apps

https://play.google.com/store/apps/details?id=com.ionicframework.shm08app156228&hl=en

Video

https://youtu.be/fHBLMbKi4vA   Progressive mathematical modeling with OSP@SG by lookang lawrence wee

 https://notebooklm.google.com/notebook/4f0a2080-cc5b-42d3-b27b-a1566121f8b3/audio

Cast of Characters

• Francisco Esquembre: A developer or contributor involved in creating or working on the Easy JavaScript Simulations (EJS) framework. He is mentioned as part of the credits for the project and as a participant in the WebEJS beta workshop. He is likely a key figure in the technical development of the simulation framework.
• Flix Jess Garcia Clemente: Another individual involved in the development or support of the EJS framework, credited alongside Francisco Esquembre.
• lookang lawrence wee: Identified as the person behind the YouTube video "Progressive mathematical modeling with OSP@SG." He also seems to be the user for the website. Likely a leading figure in OER and Open Source Physics @ Singapore.

 

Frequently Asked Questions about Simple Harmonic Motion (SHM) Simulations

1. What is the central equilibrium position in the context of Simple Harmonic Motion (SHM)? The central equilibrium position is the point where an oscillating object, like a mass on a spring, would rest if it were not disturbed. It's typically the zero point in our coordinate system (0,0), where there's no net force acting on the object in the case of SHM, and the position where the restoring force is zero. The object oscillates around this position.
2. What is meant by instantaneous position or displacement in SHM, and how is it represented? Instantaneous position or displacement refers to the object's location at any given moment during its oscillation. It’s represented as a vector from the central equilibrium position to the object's current location. This displacement is a changing value throughout the motion. For the applet described, it would be the vector OP where O is the origin and P is the current position of the mass.
3. How do we define amplitude in the context of SHM, and what are maximum and minimum amplitude? Amplitude in SHM is the maximum displacement of the object from its central equilibrium position. It’s the farthest the object moves away from the center. The maximum amplitude is the greatest distance reached in either direction from equilibrium. The minimum amplitude, for SHM, is always zero, because in SHM it oscillates between two maxima, passing through zero.
4. What is the period of oscillation and how is it measured? The period (T) is the time it takes for an object to complete one full cycle of oscillation. For example, in the provided description, from O→A→O→B→O. It is measured in seconds (s). It can be measured by finding the time between two successive identical points in the oscillation cycle (such as peak to peak or trough to trough).
5. What is frequency and how does it relate to the period of an oscillation? Frequency (f) is the number of oscillations performed per unit time, typically measured in cycles per second or Hertz (Hz), where 1 Hz = 1/s. It is inversely related to the period (T) by the equation f = 1/T, meaning a higher frequency corresponds to a shorter period, and vice versa.
6. What is angular frequency and what is its significance in describing SHM? Angular frequency (ω) is a measure of the rate of rotation in radians per second (rad/s). It is related to the frequency (f) by the equation ω = 2π * f. Angular frequency is used to simplify calculations, and it’s useful for describing oscillatory motion in a mathematical formula, such as the equation describing SHM, x = x₀ sin(ωt).
7. How can the provided simulation applet be used to explore the concepts of SHM? The applet can be used in several ways: First, by varying the starting position x of the oscillating object to explore how the amplitude is related to this initial displacement. Secondly, you can experiment with different masses (m) and spring constants (k) to observe changes in the period (T), frequency (f), and angular frequency (ω) of the oscillations. The simulation allows for a hands-on way of understanding these core SHM parameters.
8. What are some practical applications or educational tools provided alongside the simulation model? Besides the simulation itself, the resource offers several learning tools: There are suggested activities to guide experimentation. There are also links to app versions of the simulation for mobile platforms, and a video that further discusses mathematical modeling with similar resources. The site provides examples of various projects in the learning space using EJS simulations, demonstrating interactive educational tools for many subjects and settings. It also provides a web address to the simulation, where students can interact with it.