About

<h2>Horizontal Circular Motion of Mass on a Table</h2>

<p>A particle with mass m is moving with constant speed v along a circular orbit (radius r ). The centripetal force \( F=\frac{mv^2}{r} \) is provided by gravitation force from another mass \(M=\frac{F}{g} \). A string is connected from mass m to the origin then connected to mass M . Because the force is always in the r direction, so the angular momentum \( \widehat{L} = m\widehat{r} \widehat{v} \)  is conserved. i.e. \(L=mr^2\omega \)  is a constant. For particle with mass m:</p>

<p> \(m \frac{d^2r}{dt^2}=m\frac{dv}{dt}=mv^2r−Mg=\frac{L^2}{mr^3}−Mg \)</p>

<p> \( \omega = Lmr^2 \) </p>

<h2>Controls</h2>

<p>You can change the hang mass M or the on the table mass m or the radius r with sliders. The mass M also changed to keep the mass m in circular motion when you change r. However, if you change mass M , the equilibrium condition will be broken. </p>

 

Translations

Code	Language		Translator	Run
id	id	id

Credits

Fu-Kwun Hwang - Dept. of Physics, National Taiwan Normal Univ. and lookang; lookang; tinatan

http://iwant2study.org/lookangejss/02_newtonianmechanics_6circle/ejss_model_circular3dfr02wee/circular3dfr02wee_Simulation.xhtml

Briefing Document: Horizontal 3D Circular Motion of Mass on a Table

1. Overview

This document reviews an interactive JavaScript WebGL model simulating horizontal circular motion of a mass on a table. The model, developed as an open educational resource, allows users to explore the physics of circular motion by manipulating variables and observing their effects. It's designed to be accessible across various platforms (desktops, laptops, mobile devices). The model emphasizes the relationship between centripetal force, angular momentum, and the parameters of the system.

2. Key Themes and Concepts

• Uniform Circular Motion: The core concept is a mass (m) moving at a constant speed (v) in a circular orbit of radius (r). The model focuses on the forces required to sustain this motion.
• Centripetal Force: The model clearly illustrates that a centripetal force is necessary to maintain circular motion. This force is provided by the gravitational force from a hanging mass (M), connected to the mass on the table via a string through a frictionless hole. The formula ( F=\frac{mv^2}{r} ) is explicitly stated.
• Angular Momentum: The model emphasizes the conservation of angular momentum in this system. Because the force is always radial, the angular momentum ( \widehat{L} = m\widehat{r} \widehat{v} ) is conserved. The specific equation (L=mr^2\omega ) is used to highlight the relationship between angular momentum, mass, radius, and angular velocity.
• Equilibrium Condition: The model demonstrates how an equilibrium condition is maintained by balancing the centripetal force with the gravitational force of the hanging mass.
• Impact of Variable Change: The model allows the user to manipulate mass (m) and radius (r) and observes how the hanging mass (M) is adjusted to maintain the equilibrium. If the user directly changes M, the equilibrium will be broken and oscillations will occur.
• Mathematical Representation: The model uses mathematical equations to describe the physics, as indicated:
• (m \frac{d^2r}{dt^2}=m\frac{dv}{dt}=mv^2r−Mg=\frac{L^2}{mr^3}−Mg ) This is described as the equation of motion for mass m.
• ( \omega = \frac{L}{mr^2} ) This is described as an expression for angular velocity.

3. Important Ideas and Facts

• Interactive Simulation: The model is designed to be highly interactive, with sliders that allow users to adjust variables in real-time. "You can change the hang mass M or the on the table mass m or the radius r with sliders."
• Relationship between m, r and M: The hanging mass M is automatically adjusted to maintain circular motion when the parameters of the moving mass (m) or radius (r) is changed. The documentation states “The mass M also changed to keep the mass m in circular motion when you change r.”
• Disturbance of Equilibrium: The model shows that if you change mass M, then “the equilibrium condition will be broken.” When this happens, "the system will oscilliate up and down."
• Open Educational Resource: The simulation is freely available for use, indicating a commitment to open education, and is licensed under a Creative Commons Attribution license.
• Educational Alignment: The model's learning content is aligned with the Singaporean A-Level physics syllabus, specifically covering kinematics of uniform circular motion, centripetal acceleration, and centripetal force. The website links directly to the relevant PDF content.

4. Target Audience & Learning Outcomes

• Target Audience: The model is intended for Junior College (high school) level physics students.
• Learning Outcomes: The expected learning outcomes listed are:
• Express angular displacement in radians.
• Understand and use the concept of angular velocity.
• Apply the formula v = rω to solve problems.
• Describe motion in a curved path due to a perpendicular force.
• Understand centripetal acceleration in uniform circular motion.
• Use formulas (a = rω^2) and (a = \frac{v^2}{r}) to solve problems.
• Use formulas (F = mrω^2) and (F = \frac{mv^2}{r}) to solve problems.

5. Technical Details

• Technology: The model is built using JavaScript and WebGL, making it accessible on modern web browsers. It is described as an "HTML5 JavaScript WebGL version".
• Accessibility: The site emphasizes its accessibility: "Android/iOS including handphones/Tablets/iPads," "Windows/MacOSX/Linux including Laptops/Desktops", "ChromeBook Laptops."
• Embeddable: The model can be embedded in other webpages using an iframe.
• Credits: The model is credited to Fu-Kwun Hwang, lookang, and tinatan.

6. Connections to other Resources

• Other Simulations: The site lists a wide range of other related physics simulations from the same developers. This includes models on Kepler Orbits, Pendulums, Gravity, and other topics. This suggests that this simulation is part of a larger suite of educational resources.
• Related Resources: Several links to external resources are provided including a GeoGebra model, and links to blog posts discussing the various models and their history of development.
• Authoring Tool: The site mentions "Easy JavaScript/Java Simulation Authoring and Modeling Tool," which suggests an interest in teachers being able to create their own simulations, in addition to using existing resources.

7. Key Quotes

• "A particle with mass m is moving with constant speed v along a circular orbit (radius r ). The centripetal force ( F=\frac{mv^2}{r} ) is provided by gravitation force from another mass (M=\frac{F}{g} )."
• "Because the force is always in the r direction, so the angular momentum ( \widehat{L} = m\widehat{r} \widehat{v} ) is conserved. i.e. (L=mr^2\omega ) is a constant."
• "You can change the hang mass M or the on the table mass m or the radius r with sliders."
• "However, if you change mass M , the equilibrium condition will be broken."

8. Conclusion

The "Horizontal 3D Circular Motion of Mass on a Table" simulation is a valuable resource for teaching and learning about circular motion. Its interactive nature, clear explanations, and alignment with educational standards make it a useful tool for both students and teachers. The availability of the simulation as an open educational resource encourages wider adoption and exploration of physics concepts.

 

Horizontal Circular Motion Study Guide

Quiz

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

1. In the context of the simulation, what provides the centripetal force required for the mass 'm' to move in a circle?
2. What is the relationship between the hanging mass (M) and the circular motion of mass (m)?
3. What is angular momentum, and why is it conserved in this horizontal circular motion model?
4. Explain the meaning of the equation (F=\frac{mv^2}{r}) in the context of this simulation.
5. If the hanging mass M is changed, what happens to the equilibrium condition and the motion of mass m?
6. How does the simulation demonstrate the relationship between angular velocity (ω) and linear velocity (v)
7. What is the significance of the ability to change parameters such as mass 'm' and radius 'r' in this simulation?
8. According to the learning outcomes, what is one practical application of understanding ( v = rω )?
9. How does the simulation relate to the concepts of centripetal acceleration as described by (a=rω^2) and ( a = \frac{v^2}{r} )?
10. What is the role of the string in the simulation in terms of force and motion?

Quiz Answer Key

1. The centripetal force is provided by the gravitational force acting on the hanging mass M. This force is transmitted through the string and pulls the mass m toward the center of the circular path.
2. The hanging mass M provides the force necessary for the centripetal acceleration of mass m. Specifically, the weight of mass M creates the centripetal force.
3. Angular momentum is a measure of an object's rotation, and it is conserved because the force on mass m acts along the radial direction only. This means there is no torque which would change the angular momentum.
4. The equation (F=\frac{mv^2}{r}) shows that the centripetal force (F) required for an object of mass (m) to move in a circle of radius (r) at velocity (v) is directly proportional to the mass and the square of the velocity, and inversely proportional to the radius.
5. Changing the hanging mass M breaks the equilibrium condition. The system will no longer have the correct force to maintain a constant radius, and mass m will then oscillate up and down.
6. The simulation allows manipulation of radius (r), which directly changes the linear velocity (v) at a given angular velocity (ω). The relationship is shown with the formula (v=rω).
7. By changing 'm' and 'r', users can explore the relationships between centripetal force, angular momentum, and the dynamics of circular motion, allowing users to manipulate these parameters to observe how they influence each other.
8. Understanding v = rω allows one to calculate linear speed from angular speed, allowing the calculation of the speed of an object moving in a circle given the radius and angular speed.
9. The simulation demonstrates centripetal acceleration in two ways: (a=rω^2) and ( a = \frac{v^2}{r}). These two expressions show that acceleration is related to radius, angular velocity, and linear velocity and that both describe the acceleration towards the center of a circular path.
10. The string transmits the gravitational force from mass M to mass m, causing mass m to move in a circular path as the centripetal force always acts towards the center of the circle.

Essay Questions

Instructions: Choose one of the following prompts and construct a detailed essay, drawing upon the provided source material.

1. Discuss the significance of angular momentum conservation in the context of the horizontal circular motion model. How does the model demonstrate this principle, and what are the implications for the motion of mass 'm'?
2. Analyze the relationship between centripetal force, mass, velocity, and radius in the simulation, using the equations provided. Explain how modifying different parameters of the simulation affect the motion and the equilibrium.
3. Explain how the simulation relates to the learning outcomes provided, demonstrating an understanding of all six listed learning outcomes.
4. Compare and contrast the theoretical model presented with the actual, interactive simulation. How does the simulation enhance understanding of the abstract concepts related to circular motion?
5. Evaluate the effectiveness of the simulation as a learning tool for understanding circular motion, including its interactive elements, visual representations, and the ability to modify parameters.

Glossary of Key Terms

• Centripetal Force: A force that acts on a body moving in a circular path and is directed toward the center around which the body is moving. It is essential for maintaining circular motion.
• Angular Momentum: A measure of the amount of rotation of a body in motion, also describing an object's resistance to changes in its rotation. In this scenario, it's the product of mass, radial position, and angular velocity.
• Angular Velocity (ω): The rate at which an object rotates or revolves, measured in radians per second.
• Linear Velocity (v): The speed of an object in a specific direction, which, in circular motion, is tangential to the circle's path.
• Radius (r): The distance from the center of a circle to any point on its circumference.
• Equilibrium: A state where all forces acting upon an object are balanced, resulting in no net force and no acceleration. In this context, it refers to the specific balance of forces required for mass m to move in uniform circular motion at a specific radius.
• Radians: A unit of angular measure, defined as the angle subtended at the center of a circle by an arc equal in length to the radius of the circle.
• Centripetal Acceleration (a): The acceleration of a body moving along a circular path, which is always directed toward the center of the circle. It is not an acceleration that changes the speed of the object, but the direction of its velocity.
• Gravitational Force: The force of attraction between two masses. In the simulation, the weight of the hanging mass (M) provides the centripetal force needed for the mass (m) to move in a circle.

Learning Content

Motion in a Circle Content taken from http://www.seab.gov.sg/content/syllabus/alevel/2017Syllabus/9749_2017.pdf

1. Kinematics of uniform circular motion
2. Centripetal acceleration
3. Centripetal force

 

Learning Outcomes

Candidates should be able to:

1. express angular displacement in radians
2. show an understanding of and use the concept of angular velocity to solve problems
3. recall and use v = rω to solve problems
4. describe qualitatively motion in a curved path due to a perpendicular force, and understand the centripetal acceleration in the case of uniform motion in a circle
5. recall and use centripetal acceleration a = rω ² , and \( a = \frac{v^2 }{r} \) to solve problems
6. recall and use centripetal force F = mrω ² , and \( F = \frac{mv^2}{r} \) to solve problems.

 

For Teachers

A particle with mass \(m\) is moving with constant speed \(v\) along a circular orbit (radius \(r\)). The centripetal force \(F=m\frac{v^2}{r}\) is provided by gravitation force from another mass \(M=F/g\).
A string is connected from mass m to the origin then connected to mass \(M\).
Because the force is always in the \(\hat{r}\) direction, so the angular momentum \(\vec{L}=m\,\vec{r}\times \vec{v}\) is conserved. i.e. \(L=mr^2\omega\) is a constant.

For particle with mass m:

\( m \frac{d^2r}{dt^2}=m\frac{dv}{dt}= m \frac{v^2}{r}-Mg=\frac{L^2}{mr^3}- Mg \)
\( \omega=\frac{L}{mr^2}\)

The following is a simulation of the above model.

When mass m or radius r is changed with sliders, equilibrium condition is recalulated for constant circular motion.
However, if mass M is changed, the equilibrium condition will be broken, and the system will oscilliate up and down. 

Research

[text]

Video

 Ejs Open Source Horizontal Circular Motion java applet by lookang lawrence wee

 Version:

1. http://weelookang.blogspot.sg/2016/03/horizontal-3d-webgl-circular-motion-of.html HTML5 JavaScript WebGL version by Loo Kang Wee and Tina Tan
2. http://weelookang.blogspot.sg/2010/07/lesson-on-circular-motion-with-acjc.html 09 July 2010 Computer Lab hands on learning session on  Ejs Open Source Vertical Circular Motion of mass m attached to a rod java applet side view of the same 3D view with teacher explaining the physical setup of the mass m and mass M attached by a string through a table with a fricitionless hole in the middle of table for string to go through and student working on their own desktop
3. http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=1883.0 remixed Java applet by Loo Kang Wee
4. http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=1454.0 original Java applet by Fu-Kwun Hwang

Other Resources

https://www.geogebra.org/m/rnkrmcx8 by Tan Seng Kwang

Frequently Asked Questions About Horizontal Circular Motion

1. What is the main concept being explored in the simulation of horizontal circular motion?
2. The simulation demonstrates the principles of uniform circular motion, where a mass 'm' moves at a constant speed 'v' along a circular path of radius 'r'. The centripetal force, required to maintain this circular motion, is provided by the gravitational force from another mass 'M' connected by a string. It also shows how the angular momentum is conserved in this system.
3. How is the centripetal force generated in this horizontal circular motion model?
4. The centripetal force in this model is generated by the gravitational force of a hanging mass M, connected to mass m (on the table) via a string that runs through the center of the circular path. This gravitational force acts as the inward-pulling force necessary for the circular motion of the table mass m.
5. What happens to the angular momentum of the mass moving in a circle and why?
6. The angular momentum (L) of the mass (m) moving in a circle is conserved (i.e., remains constant). This is because the centripetal force acts only along the radial direction, and there is no torque (rotational force) acting on the mass about the center of rotation. The formula for this conserved angular momentum is L = m * r^2 * ω, where ω is the angular velocity.
7. What are the key equations related to the motion of the mass 'm' in this setup?
8. The primary equations governing the motion are:

• Centripetal force: F = mv²/r, which is also equal to Mg (gravitational force).
• Relationship between linear velocity and angular velocity: v = rω.
• Centripetal acceleration: a = rω² = v²/r.
• Conservation of angular momentum: L = mr²ω = constant.
• The radial force equation as: ( m \frac{d^2r}{dt^2}=m\frac{dv}{dt}=mv^2r−Mg=\frac{L^2}{mr^3}−Mg ) and ( \omega=\frac{L}{mr^2})

1. How can users interact with the simulation, and what parameters can be adjusted?
2. Users can interact with the simulation by using sliders to change the values of the mass 'm' on the table, the hanging mass 'M', and the radius 'r' of the circular path. Changing the radius or mass m will recalculate the equilibrium condition in the simulation to maintain circular motion. However, changing the mass M will break the equilibrium, causing the system to oscillate vertically.
3. What learning outcomes are targeted by this simulation?
4. The simulation aims to help students understand concepts such as angular displacement in radians, angular velocity, the relationship between linear and angular velocity (v = rω), centripetal acceleration (a = rω², a = v²/r), and centripetal force (F = mrω², F = mv²/r). It reinforces that circular motion is caused by a perpendicular force.
5. What happens if the equilibrium condition is broken in the simulation?
6. If the equilibrium condition is broken, specifically by changing the hanging mass 'M', the system will no longer be in a stable circular motion. Instead, the mass 'm' will oscillate up and down. This happens because the gravitational force provided by 'M' no longer perfectly balances the centripetal force required for the original radius, r.
7. Besides the simulation, are there any other resources or references associated with this model?
8. Yes, there are various resources and references, including the original Java applets, an HTML5/JavaScript/WebGL version, links to other related interactive models, and videos on how to use the applets and the theory behind them. These resources, such as those from GeoGebra, provide further context for the model.