Briefing Document: Pendulum Toy and Real Time ODE 10 Pendulum JavaScript Model Simulation Applet HTML5

This briefing document summarizes the main themes, important ideas, and facts presented in the provided excerpts from "Pendulum Toy" and the "Real Time ODE 10 Pendulum JavaScript Model Simulation Applet HTML5" resource page.

Source 1: Pendulum Toy

• Title and Authors: Pendulum Toy, by weelookang@gmail.com; Fu-Kwun Hwang; Francisco Esquembre; Wolfgang Christian; Félix J. García.
• Licensing and Creation: Released under a Creative Commons Attribution license. Compiled with EJS 6.1 BETA (201115) in 2021.

Main Themes and Important Ideas:

• This source appears to be a software tool or simulation related to pendulum motion, created using Easy JavaScript Simulations (EJS).
• The mention of "Compiled with EJS 6.1 BETA" and the licensing information suggests a focus on providing an open and modifiable tool for exploring pendulum physics.
• The list of authors indicates a collaborative effort in developing this educational resource.

Source 2: Real Time ODE 10 Pendulum JavaScript Model Simulation Applet HTML5 - Open Educational Resources / Open Source Physics @ Singapore

• Platform and Purpose: This resource is part of the Open Educational Resources / Open Source Physics @ Singapore initiative. It provides an interactive JavaScript HTML5 applet simulating a pendulum using Real Time Ordinary Differential Equation (ODE) solving.
• Accessibility: The applet is designed to be embedded in webpages via an <iframe> tag and is compatible with various devices and operating systems, including desktops, laptops, tablets, and smartphones.
• Educational Focus: The page explicitly outlines guiding questions, learning outcomes, and suggested learning experiences related to simple harmonic motion (SHM), of which the simple pendulum, under certain conditions, is an example.

Main Themes and Important Ideas:

• Simple Harmonic Motion (SHM): A central theme, with guiding questions like "What are the characteristics of periodic motion? How can we study and describe such motion?" and "How do we analyse simple harmonic motion?".
• Pendulum as an Oscillator: The resource uses the pendulum as a primary example of a free oscillation, stating under "1.1.1 Example 1: Simple pendulum": "Static picture of a pendulum bob given an initial horizontal displacement and released to swing freely to produce to and fro motion Dynamic picture of a pendulum bob given an initial horizontal displacement and released to swing freely to produce to and fro motion".
• Learning Outcomes: The resource clearly defines what students should be able to do after engaging with it, including:
• Describing free oscillations.
• Investigating oscillator motion experimentally and graphically.
• Understanding and using terms like amplitude, period, frequency, and angular frequency.
• Recognizing and using the defining equation of SHM: "a = - ω2 x".
• Recalling and using the displacement equation: "x= x0 ω sin( ωt )".
• Understanding and using velocity equations: "v = v0 cos ( ω t ) , v = ± ω ( x 0 2 - x 2 )".
• Describing changes in displacement, velocity, and acceleration graphically during SHM.
• Describing energy interchange (kinetic and potential) in SHM.
• Understanding damped and forced oscillations, including resonance, with practical examples like car suspension systems.
• Graphically describing the amplitude of forced oscillations near the natural frequency and factors affecting resonance.
• Appreciating the uses and dangers of resonance.
• Learning Experiences: The resource suggests various activities for students, emphasizing both experimental and simulation-based learning:
• Examining different graphs representing SHM and understanding their interpretations.
• Designing and conducting experiments (or using simulations) to investigate factors affecting the period of oscillations, like a pendulum or spring-mass system. It notes that "it is necessary for oscillations to be small for the motion to be considered as SHM".
• Investigating phase using turntables or simulations, linking SHM to uniform circular motion.
• Building models for SHM using dataloggers or simulations, focusing on observing motion and interpreting graphs before mathematical details.
• Exploring the modeling of molecular oscillations as SHM (for small amplitudes).
• Exploring the benefits and dangers of resonance.
• Simulation and Interactivity: The inclusion of an embed code (<iframe width="100%" height="100%" src="https://iwant2study.org/lookangejss/02_newtonianmechanics_8oscillations/ejss_model_SHM01realtimeODE/SHM01_Simulation.xhtml " frameborder="0"></iframe>) highlights the interactive nature of the resource. Links to various simulations are also provided.
• Small Angle Approximation: The resource explicitly addresses the limitations of the simple harmonic motion model for a simple pendulum, stating in "1.1.2.A1": ": 10 degrees for error of , depending on what is the error acceptable, small angle is typically about less than 10 degree of swing from the vertical."
• Other Resources and Credits: The page includes links to external resources like GeoGebra and PhET simulations, along with credits to the developers, including those mentioned in the "Pendulum Toy" source, suggesting a shared development community or related projects.

Connections Between the Sources:

• Both sources are related to the study of pendulum motion.
• The "Pendulum Toy" likely represents one of the simulation tools that could be used as part of the "Learning Experiences" suggested in the "Real Time ODE 10 Pendulum JavaScript Model Simulation Applet HTML5" resource.
• The shared authors (Fu-Kwun Hwang, Francisco Esquembre, Wolfgang Christian, Félix J. García) strongly suggest a connected development effort within the Open Source Physics community. Both resources utilize Easy JavaScript Simulations (EJS).

Conclusion:

These resources provide valuable tools and pedagogical frameworks for understanding pendulum motion and its connection to simple harmonic motion. The "Pendulum Toy" appears to be a specific simulation tool, while the "Real Time ODE 10 Pendulum JavaScript Model Simulation Applet HTML5" page offers a broader educational context, including learning objectives, suggested activities, and links to various simulations (potentially including "Pendulum Toy" or similar tools). The emphasis on interactive simulations and open educational resources makes these valuable for both students and educators. The explicit mention of the small angle approximation for SHM in a simple pendulum is a key concept highlighted.

Frequently Asked Questions about Pendulum Motion and Simple Harmonic Motion

1. What is periodic motion, and how does a pendulum exemplify it? Periodic motion is any motion that repeats itself at regular time intervals. A pendulum exhibits periodic motion as its bob swings back and forth, returning to its original position and direction of motion after a consistent period of time. This repetitive oscillation is a fundamental characteristic of periodic motion.

2. How is the motion of a simple pendulum related to Simple Harmonic Motion (SHM)? For small angles of displacement (typically less than 10 degrees from the vertical), the motion of a simple pendulum closely approximates Simple Harmonic Motion. SHM is a specific type of periodic motion where the restoring force is directly proportional to the displacement and acts in the opposite direction. In a simple pendulum with small oscillations, the component of gravity acting to restore the bob to its equilibrium position is approximately proportional to the displacement from that equilibrium.

3. What are the key properties used to describe Simple Harmonic Motion, such as that of a pendulum? Several key properties describe SHM:

• Amplitude (x₀): The maximum displacement from the equilibrium position. For a pendulum, this is the maximum angle or horizontal distance the bob reaches from its resting point.
• Period (T): The time taken for one complete oscillation (one full swing back and forth).
• Frequency (f): The number of oscillations per unit time, usually measured in Hertz (Hz). It is the inverse of the period (f = 1/T).
• Angular Frequency (ω): Related to the frequency by the equation ω = 2πf. It represents the rate of change of the angular displacement.

4. What factors affect the period of oscillation of a simple pendulum? The period of a simple pendulum depends primarily on two factors:

• Length of the pendulum (L): The period is directly proportional to the square root of the length. A longer pendulum will have a longer period.
• Acceleration due to gravity (g): The period is inversely proportional to the square root of the acceleration due to gravity. A pendulum will swing faster (shorter period) in a stronger gravitational field. Interestingly, for small oscillations, the mass of the pendulum bob and the initial angle of release (as long as it's within the small angle approximation) do not significantly affect the period. Damping forces, however, will eventually cause the oscillations to decrease in amplitude over time but ideally do not change the period.

5. How can we mathematically describe the displacement, velocity, and acceleration of a mass undergoing SHM, like a pendulum with small oscillations? The displacement (x) of a mass in SHM can be described as a sinusoidal function of time (t), such as x = x₀ cos(ωt + φ) or x = x₀ sin(ωt + φ), where φ is the phase constant. From this, we can derive expressions for velocity (v) and acceleration (a) by differentiation:

• Velocity (v): v = -ωx₀ sin(ωt + φ) or v = ω√(x₀² - x²)
• Acceleration (a): a = -ω²x This shows that the acceleration is always directed opposite to the displacement and is proportional to it, which is the defining characteristic of SHM.

6. How does energy transform during the Simple Harmonic Motion of a pendulum? During the oscillation of a simple pendulum (ignoring damping), there is a continuous interchange between potential energy (PE) and kinetic energy (KE). At the maximum displacement (amplitude), the pendulum momentarily stops, and all its energy is gravitational potential energy. As it swings towards the equilibrium position, the PE is converted into KE, reaching its maximum at the equilibrium point where the speed is highest and PE is minimum (ideally zero). As it swings towards the other extreme, KE is converted back into PE. The total mechanical energy (PE + KE) remains constant.

7. What is damping, and how does it affect the oscillations of a real pendulum? Damping refers to forces (such as air resistance and friction at the pivot point) that oppose the motion of an oscillator. In a real pendulum, damping causes the amplitude of the oscillations to gradually decrease over time until the pendulum eventually comes to rest at its equilibrium position. While ideally damping doesn't change the period of oscillation for small damping forces, significant damping can slightly affect it. Different degrees of damping exist, including underdamping (oscillations decrease gradually), critical damping (returns to equilibrium in the shortest time without oscillating), and overdamping (returns to equilibrium slowly without oscillating).

8. What are forced oscillations and resonance, and can they occur with a pendulum? A forced oscillation occurs when an external periodic force is applied to an oscillating system like a pendulum. The system will oscillate at the driving frequency of the external force. Resonance is a phenomenon that occurs when the driving frequency of the external force is close to the natural frequency of the oscillating system (the frequency at which it would oscillate freely). At resonance, the amplitude of the forced oscillations becomes very large, even if the driving force is relatively small. Resonance can occur with a pendulum if it is subjected to an external periodic force whose frequency is near its natural frequency, leading to a significant increase in its swing amplitude. Resonance can be both beneficial (e.g., in musical instruments) and detrimental (e.g., causing structural failure).