About

This is a pendulum in 3D.

Let the angle between the pendulum and the vertical line is \(\theta\) and the angular velocity \(\omega=\frac{d\theta}{dt}\)

And the angle of the pendulum (projected to x-y plane) with the x-axis is  \(\phi\), and it's angular velocity \(\dot\phi=\frac{d\phi}{dt}\)

The [url=http://en.wikipedia.org/wiki/Lagrangian_mechanics]lagrange equation[/url] for the system is \(L=T-V = \tfrac{1}{2}m (L\dot\theta)^2+\tfrac{1}{2}m (L\sin\theta \dot{\phi})^2- (-mgL\cos\theta)\)

The equation of the motion is

\(\ddot\theta=\sin\theta\cos\theta\dot{\phi}^2-\frac{g}{L}\sin\theta\)  ...... from \(\frac{d}{dt}(\frac{\partial L}{\partial \dot{\theta}})-\frac{\partial L}{\partial \theta}=0\)

and

\( m L^2 \sin\theta^2 \dot{\phi}=const\) Angular momentum is conserved. ...... from \(\frac{d}{dt}(\frac{\partial L}{\partial \dot{\phi}})-\frac{\partial L}{\partial \phi}=0\)

And the following is the simulation of such a system:

When the checkbox ([b]circular loop[/b]) is checked, \(\omega=0\). and \(\dot{\phi}= \sqrt{\frac{g}{L\cos\theta}}\) It is a circular motion.

The vertical component tangential of the string balanced with the mass m, and the horizontal component tangential provide the centripetal force for circular motion.

You can uncheck it and change the period \(T=\frac{2\pi}{\dot{\phi}}\) ,

and you will find out the z-coordinate of the pendulum will change with time when

\(\omega\neq 0\) or \(\dot{\phi}\neq \sqrt{\frac{g}{L\cos\theta}}\)

You can also drag the blue dot to change the length of the pendulum.

 

Translations

Code	Language		Translator	Run

Credits

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

http://iwant2study.org/lookangejss/02_newtonianmechanics_3dynamics/ejss_model_pendulum3D/pendulum3D_Simulation.xhtml

 

Briefing Document: Pendulum 3D JavaScript HTML5 Applet Simulation Model

1. Introduction

This document provides a review of the "Pendulum 3D JavaScript HTML5 Applet Simulation Model" resource, available from Open Educational Resources / Open Source Physics @ Singapore. This resource is an interactive simulation designed for educational purposes, primarily for students studying dynamics and oscillations, likely at the junior college level and above. It allows users to explore the behavior of a pendulum in three dimensions, manipulating variables and observing the resulting motion. The simulation is built using JavaScript and HTML5, making it accessible on various devices, including desktops, laptops, tablets, and smartphones.

2. Main Themes and Key Concepts

• 3D Pendulum Dynamics: The core theme of this resource is the exploration of a pendulum's motion in three-dimensional space. It goes beyond the simplified two-dimensional pendulum usually taught in introductory physics.
• Lagrangian Mechanics: The simulation is grounded in the principles of Lagrangian mechanics. The description includes a calculation of the Lagrangian (L = T - V), where T is the kinetic energy and V is the potential energy of the system. The resource states: "The lagrange equation for the system is \((L=T-V = \tfrac{1}{2}m (L\dot\theta)^2+\tfrac{1}{2}m (L\sin\theta \dot{\phi})^2- (-mgL\cos\theta))\)".
• Equations of Motion: The resource provides the key equations of motion governing the 3D pendulum. These equations are derived from the Lagrangian formalism and describe how the angular position ((\theta)) and angular velocity \(((\omega))\), and the angle of the pendulum projected to the x-y plane\( ((\phi))\), change with time. "The equation of the motion is \((\ddot\theta=\sin\theta\cos\theta\dot{\phi}^2-\frac{g}{L}\sin\theta) ... and ( m L^2 \sin\theta^2 \dot{\phi}=const) \)Angular momentum is conserved."
• Angular Momentum Conservation: A crucial concept is the conservation of angular momentum, expressed in the resource by the equation \(( m L^2 \sin\theta^2 \dot{\phi}=const)\). This highlights a key principle in the system’s dynamics.
• Circular Motion (Special Case): The simulation includes a special "circular loop" mode where the pendulum's motion is confined to a horizontal circle. In this mode, the angular velocity in the vertical direction is zero ((\omega=0)). "When the checkbox ([bhttps://iwant2study.org/lookangejss/02_newtonianmechanics_3dynamics/ejss_model_pendulum3D/pendulum3D_Simulation.xhtml " frameborder="0"></iframe>
• Author and License: The simulation is attributed to Fu-Kwun Hwang (Dept. of Physics, National Taiwan Normal Univ.), lookang, and tina, and it is released under a Creative Commons Attribution-Share Alike 4.0 Singapore License.
• Educational Level: It is primarily suited for secondary, junior college level students and above, with specific application in the study of "Dynamics" and "Oscillations."
• Extensive Related Resources: The page also includes a very extensive list of links to a variety of other physics applets and simulations, covering a broad array of topics from mechanics to thermodynamics and wave physics. This demonstrates the rich collection of educational simulations produced by the team.
• Easy JavaScript Simulation Framework: The resource utilizes the Easy JavaScript Simulation framework (EJS), which allows the creation of the many interactive applets on the site, and provides a way for other people to create their own simulations.

4. Potential Uses in Education

• Visualization of Complex Motion: The simulation provides a powerful visual representation of 3D pendulum motion, which can be difficult to understand from equations alone.
• Interactive Learning: Students can actively explore the effects of changing parameters, enhancing their conceptual understanding of the underlying physics.
• Application of Theoretical Concepts: It helps bridge the gap between the theoretical concepts of Lagrangian mechanics and the real-world behavior of oscillating systems.
• Exploration of Special Cases: The circular motion mode allows students to connect the principles of pendulum motion to circular motion, a fundamental topic in introductory physics.
• Experimentation and Discovery: The tool encourages students to experiment with various initial conditions (such as different velocities and length of pendulum) and observe the resulting motion, thus allowing students to "discover" how different parameters impact the motion.

5. Conclusion

The "Pendulum 3D JavaScript HTML5 Applet Simulation Model" is a valuable resource for educators and students interested in exploring the complexities of pendulum motion and Lagrangian mechanics. Its interactive nature, accessibility, and grounding in fundamental physical principles make it a powerful tool for learning and teaching. The source also highlights a wide collection of other simulations that can also be used to enhance physics education.

Pendulum 3D Simulation Study Guide

Quiz

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

1. What does the simulation depict?
2. What are the variables theta and omega in relation to the pendulum's motion?
3. What is the significance of the "Lagrangian equation" presented in the text?
4. Explain the conditions under which the pendulum exhibits circular motion according to the simulation.
5. What does the equation \(( m L^2 \sin\theta^2 \dot{\phi}=const) \) represent in terms of the pendulum’s movement?
6. How does changing the period (T) of the pendulum impact the z-coordinate of the pendulum when the "circular loop" box is not checked?
7. What is the role of the blue dot in the simulation, and how can it be used to interact with the model?
8. Who are the credited authors and developers of the Pendulum 3D simulation?
9. Besides the Pendulum 3D simulation, name two other types of simulations or models available on the website.
10. What is the licensing for the educational resources provided by the website?

Quiz Answer Key

1. The simulation depicts a pendulum in a three-dimensional space, allowing for the observation and manipulation of its motion. It visualizes how the pendulum moves and the factors that influence its trajectory.
2. Theta \(((\theta))\) represents the angle between the pendulum and the vertical line, while omega \(((\omega))\) represents the angular velocity, or the rate of change of this angle with respect to time, \((\omega=\frac{d\theta}{dt})\).
3. The Lagrangian equation (L = T-V) is a mathematical expression that describes the system's energy, which is the difference between kinetic energy (T) and potential energy (V). This is a starting point for deriving equations of motion for the pendulum.
4. The pendulum will exhibit circular motion when the "circular loop" checkbox is checked. This sets the angular velocity along the z-axis to zero \(((\omega = 0)) and sets (\dot{\phi}= \sqrt{\frac{g}{L\cos\theta}})\), which is required for uniform circular motion.
5. The equation \(( m L^2 \sin\theta^2 \dot{\phi}=const)\) represents the conservation of angular momentum for the pendulum, meaning the total angular momentum of the pendulum remains constant if no external torques are applied.
6. When the circular loop box is unchecked and the period is changed \(((\omega\neq 0) or (\dot{\phi}\neq \sqrt{\frac{g}{L\cos\theta}}))\), the z-coordinate of the pendulum will vary over time, indicating the pendulum is not moving in a simple circle.
7. The blue dot represents a draggable control point that allows users to interact with the simulation by changing the length of the pendulum and observing the effect of the length on the pendulum’s behavior.
8. The Pendulum 3D simulation is credited to Fu-Kwun Hwang from the Department of Physics, National Taiwan Normal University, along with lookang and tina.
9. Two other simulation models available on the website are the "Magnetic Bars Field" and the "Gravity Model Example." Many additional simulations are listed in the linked document covering diverse topics, not limited to mechanics.
10. The educational resources on the website are licensed under a Creative Commons Attribution-Share Alike 4.0 Singapore License. This means that the materials can be shared and adapted, provided appropriate attribution is given.

Essay Questions

Instructions: Answer each of the following questions in a 4-5 paragraph essay.

1. Discuss the concept of Lagrangian mechanics and explain how it is applied to model the motion of the 3D pendulum simulation.
2. Analyze the relationship between the variables θ, ω, and φ in the context of the 3D pendulum simulation, explaining how changes in these variables impact the behavior of the pendulum.
3. Explain the significance of the two provided equations of motion in the simulation and discuss their importance to the analysis of the pendulum's motion.
4. Describe the "circular loop" checkbox functionality. How does this feature enhance the understanding of a pendulum’s behavior, and what physics concepts does it illustrate?
5. Critically evaluate the educational value of the Pendulum 3D simulation as a teaching tool for physics, contrasting its strengths and weaknesses compared to other traditional methods.

Glossary of Key Terms

• Pendulum: A body suspended from a fixed point that swings freely back and forth under the influence of gravity.
• 3D Simulation: A computational model that recreates an environment or system in three dimensions, allowing for a more immersive and realistic observation.
• Lagrangian Mechanics: A formulation of classical mechanics which emphasizes energy over force, using the Lagrangian (L=T-V) and the principle of least action.
• Lagrange Equation: A differential equation describing the motion of a system derived from Lagrangian mechanics, which uses the Lagrangian to express how the system’s energy changes.
• \(( \theta)\) (Theta): The angle between the pendulum and the vertical axis.
• \(( \omega\)) (Omega): Angular velocity of the pendulum's swing in the vertical plane (rate of change of theta), ( \omega=\frac{d\theta}{dt} ).
• \(( \phi)\) (Phi): The angle of the pendulum's projection onto the x-y plane relative to the x-axis.
• \(( \dot\phi)\): The angular velocity of the pendulum's rotation in the x-y plane.
• Angular Momentum: A measure of an object's rotational momentum, conserved in the absence of external torques.
• Circular Motion: The movement of an object along a circular path, at a constant or varying speed.
• Period (T): The time it takes for one complete cycle of the pendulum's motion.
• Centripetal Force: The force that keeps an object moving along a circular path, directed toward the center of the circle.
• Kinetic Energy (T): The energy possessed by an object due to its motion.
• Potential Energy (V): The energy stored within an object due to its position.

Other resources

1. http://weelookang.blogspot.sg/2015/11/pendulum-3d-model.html
2. http://iwant2study.org/lookangejss/02_newtonianmechanics_3dynamics/ejs/ejs_users-ntnu-fkh-pendulum3D.jar
3. http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=1191.new#new
4. http://iwant2study.org/lookangejss/02_newtonianmechanics_3dynamics/ejs/ejs_model_pendulum3D.jar

Frequently Asked Questions about the 3D Pendulum Simulation

1. What does the 3D Pendulum Simulation demonstrate?
2. The 3D Pendulum Simulation demonstrates the motion of a pendulum in three dimensions, allowing users to explore the relationship between the pendulum's angle with respect to the vertical \(((\theta))\), its angular velocity \(((\omega))\), the angle projected onto the x-y plane \(((\phi))\), and its angular velocity \(((\dot\phi))\). It showcases the application of Lagrangian mechanics to describe the system's behavior and the conservation of angular momentum. The simulation helps visualize how these factors influence the pendulum's movement, including both oscillatory and circular motion.
3. How are the physics of the pendulum modeled in this simulation?
4. The simulation uses Lagrangian mechanics to derive the equations of motion. The Lagrangian (L) is defined as the difference between the kinetic (T) and potential (V) energies of the pendulum: \( (L=T-V = \tfrac{1}{2}m (L\dot\theta)^2+\tfrac{1}{2}m (L\sin\theta \dot{\phi})^2- (-mgL\cos\theta))\). From this, the equations are derived: \((\ddot\theta=\sin\theta\cos\theta\dot{\phi}^2-\frac{g}{L}\sin\theta)\) which dictates the angular acceleration in the vertical plane and \( ( m L^2 \sin\theta^2 \dot{\phi}=const) \) which shows the conservation of angular momentum, related to changes in \((\phi)\). These equations govern how the pendulum's motion changes over time.
5. What is the significance of the "circular loop" checkbox in the simulation?
6. When the "circular loop" checkbox is selected, the simulation sets the angular velocity of the pendulum in the vertical plane \(((\omega))\) to zero. In this mode, the pendulum undergoes circular motion because \((\dot{\phi}= \sqrt{\frac{g}{L\cos\theta}})\), which means the horizontal component of the string's tension provides the centripetal force necessary for circular motion. It forces the pendulum to move in a horizontal circle.
7. What happens to the pendulum's motion when the "circular loop" checkbox is unchecked?
8. When the "circular loop" checkbox is unchecked, the user can manually change the period of the pendulum's horizontal motion \(((T = \frac{2\pi}{\dot{\phi}}))\). This adjustment causes the z-coordinate of the pendulum to vary over time, indicating that the pendulum's motion is no longer confined to a simple horizontal circle. It results in more complex, oscillatory motion involving vertical movement as well. It results in more complex motion when \((\omega\neq 0)\) or \((\dot{\phi}\neq \sqrt{\frac{g}{L\cos\theta}})\)
9. Can users interact with the simulation, and if so, how?

Yes, users can interact with the simulation in several ways. They can check or uncheck the "circular loop" checkbox to toggle between circular and more complex pendulum motion. They can also change the period of the horizontal motion when the "circular loop" mode is off. Additionally, the simulation allows users to drag a blue dot, which changes the length of the pendulum, influencing its period and behavior.

1. What is the target audience for this simulation?
2. The simulation is designed for secondary and junior college level students studying dynamics and oscillations. It is suitable for learning about advanced physics concepts like Lagrangian mechanics and the conservation of angular momentum. It is accessible on various devices including Android/iOS devices, Windows/MacOSX/Linux, and Chromebooks. The simulation is used as an interactive and visual aid in physics learning.
3. Is this simulation a standalone resource, or is it part of a larger collection?
4. The 3D pendulum simulation is part of a larger collection of interactive resources provided by Open Educational Resources / Open Source Physics @ Singapore. This collection includes various simulations, models, and tools for physics, mathematics, and other subjects. The listed resources show the vast collection including content created by various educators. It appears to be part of an extensive platform for interactive educational content.
5. What technologies are used to build the simulation and how is it accessible?
6. The 3D pendulum simulation is built using JavaScript and HTML5, making it accessible through web browsers on various devices. It uses the EasyJavaScriptSimulation framework. The use of these web-based technologies allows for wide accessibility without requiring any additional software installation. It is embedded in a webpage via an iframe which is a common way of embedding content from one website in another.