About

Projectile Motion with System of Masses and Spring

This is the simulation of the motion of two masses m and m₁ situated at the ends of a spring of length L₀ and negligible mass. The motion is restricted to two spatial dimensions, with the y-axis representing the vertical (if gravity is switched on).

We use Hooke's law for the spring force, and include a damping term that is proportional to the difference of the velocities of the masses on both ends of the spring.

Applying Newton's Second Law yields a second-order ordinary differential equation, which we solve numerically in the simulation and visualise the results.

Activities

1. Drag the red mass to impart an initial velocity, and see how the system evolves.
2. Observe what happens when you do the same, but with gravity switched on.
3. Try changing the relative mass of the blue ball, and notice how the centre of gravity shifts.
4. Try varying the spring constant and/or the damping coefficient while the simulation runs.

 

Translations

Code	Language		Translator	Run

Credits

Wolfgang Christian; Francisco Esquembre; Zhiming Darren TAN

Briefing Document: Projectile Motion with System of Masses and Spring Simulation

1. Overview

This document provides a summary of the "Projectile Motion with System of Masses and Spring" JavaScript HTML5 applet simulation model, developed by Darren Tan and hosted by Open Educational Resources / Open Source Physics @ Singapore. The simulation is designed to explore the dynamics of a two-mass system connected by a spring under the influence of various forces. It is an interactive tool intended for educational purposes, allowing users to manipulate variables and observe the resulting motion.

2. Key Features and Concepts

• System Description: The simulation models two masses, m and m1, connected by a spring of length L0 and negligible mass. The motion is constrained to two dimensions (x-y plane), with the y-axis representing the vertical direction, especially when gravity is enabled.
• Forces: The simulation incorporates two main forces:
• Spring Force: This force is calculated using Hooke's Law, which states that the force exerted by the spring is proportional to the extension or compression of the spring from its natural length.
• Damping Force: A damping term proportional to the velocity difference between the two masses is included. This force opposes the relative motion and leads to a loss of energy in the system.
• Newton's Second Law: The motion of the masses is determined by applying Newton's Second Law, which relates the net force on an object to its mass and acceleration. This results in a second-order ordinary differential equation, which is solved numerically by the simulation.
• Visualisation: The simulation visualizes the movement of the two masses and allows users to observe the effect of varying initial conditions and system parameters.

3. Interactive Activities

The simulation offers several interactive activities to explore the system's behavior:

• Initial Velocity: "Drag the red mass to impart an initial velocity, and see how the system evolves." This allows users to understand how initial conditions affect the trajectory of the two masses.
• Gravity: "Observe what happens when you do the same, but with gravity switched on." This activity allows observation of projectile motion of the coupled system under gravitational force.
• Mass Variation: "Try changing the relative mass of the blue ball, and notice how the centre of gravity shifts." This demonstrates how mass distribution affects the motion of the system and its center of mass.
• Parameter Variation: "Try varying the spring constant and/or the damping coefficient while the simulation runs." This activity allows users to investigate the impact of these parameters on the system's behavior and oscillations.

4. Educational Value

• Concept Reinforcement: The simulation offers a practical and visual way to understand the following concepts:
• Projectile Motion
• Hooke's Law
• Newton's Second Law
• Damping Forces
• Center of Gravity
• Numerical Solutions of Differential Equations.
• Active Learning: By interacting with the simulation and modifying the parameters, students can actively learn and explore the physics behind the motion.
• Visual Learning: The visualization of the motions makes the abstract mathematical models more understandable and concrete.

5. Key Quotes

• "This is the simulation of the motion of two masses m and m1 situated at the ends of a spring of length L0 and negligible mass." - This defines the primary system being modeled.
• "We use Hooke's law for the spring force, and include a damping term that is proportional to the difference of the velocities of the masses on both ends of the spring." - This describes the types of forces acting on the system.
• "Applying Newton's Second Law yields a second-order ordinary differential equation, which we solve numerically in the simulation and visualise the results." - This outlines the mathematical method used to model the physical system.
• "Drag the red mass to impart an initial velocity, and see how the system evolves." - Example of interactive activity.
• "Try changing the relative mass of the blue ball, and notice how the centre of gravity shifts." - Example of interactive activity.

6. Authors and Credits

• Authors: Wolfgang Christian, Francisco Esquembre, Zhiming Darren TAN
• Open Source Physics: The simulation is part of the Open Source Physics project.

7. Related Resources

The provided document also lists numerous related resources, including simulations on:

• Projectile motion with and without air resistance
• Simple harmonic motion
• Electromagnetism
• Other physics and math topics

This wide range of resources shows a commitment to open educational materials and tools for interactive learning of science concepts.

8. Conclusion

The "Projectile Motion with System of Masses and Spring" simulation is a valuable educational resource that enables interactive and visual learning of fundamental physics principles. By allowing users to manipulate parameters and observe the outcomes in real time, it promotes a deeper understanding of the underlying dynamics. The simulation, and the related resources on the site, are valuable for educators and students interested in physics, particularly mechanics and dynamics.

 

Projectile Motion with System of Masses and Spring Study Guide

Quiz

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

1. What physical system does the simulation model?
2. What forces are considered in the simulation?
3. How is the spring force calculated in the simulation?
4. What is the purpose of the damping term?
5. What numerical method is used to solve the equations of motion in the simulation?
6. What happens to the center of gravity when the mass of the blue ball is changed?
7. What are some of the parameters you can adjust during the simulation?
8. What does the simulation allow a user to visualize?
9. Describe one activity suggested for using the simulation?
10. What type of license is used for the materials, and what does it mean?

Quiz Answer Key

1. The simulation models the motion of two masses connected by a spring with negligible mass. The motion is restricted to two spatial dimensions, typically with the y-axis representing the vertical.
2. The forces considered are the spring force (calculated using Hooke's law) and a damping force. If gravity is switched on, it is also included.
3. The spring force is calculated using Hooke’s law which states that the force exerted by the spring is proportional to the displacement from its equilibrium length. In this case, the displacement is the difference between the current length and the original length, L0.
4. The damping term is included to simulate the energy loss in the system due to friction or other resistive forces. It is proportional to the difference in velocities of the two masses.
5. The simulation utilizes a numerical method to solve the second-order ordinary differential equation derived from Newton's Second Law. Numerical solutions are approximations generated by computers and are necessary because the equation is too complex to solve analytically.
6. Changing the mass of the blue ball will cause the center of gravity of the two-mass system to shift. If the blue ball's mass increases, the center of gravity shifts toward the blue ball.
7. During the simulation, a user can adjust parameters such as the initial velocity of the red mass, the presence of gravity, the relative mass of the blue ball, the spring constant, and the damping coefficient.
8. The simulation allows users to visualize the motion of the two masses in real-time, including their positions, velocities, and how the spring affects their trajectory over time.
9. One suggested activity is to drag the red mass to impart an initial velocity and observe how the system evolves. Another activity is to observe what happens with gravity switched on.
10. The materials are licensed under a Creative Commons Attribution-Share Alike 4.0 Singapore License. This license means that the material can be shared and modified, as long as appropriate credit is given to the original author, and the modified material is shared under the same license.

Essay Questions

Instructions: Answer the following essay questions using your understanding of the simulation model.

1. Discuss how the spring constant and the damping coefficient affect the overall motion of the system and the energy transfer within the simulation. Include specific examples of how changing these parameters might change the motion of the masses in different ways.
2. Explain how the simulation demonstrates the interplay between Hooke's Law and Newton's Second Law of Motion. Be sure to clearly outline how these laws apply in the context of the simulation model with two masses and a spring.
3. Analyze the role of gravity in the simulation and how its presence alters the behavior of the system. Compare and contrast scenarios with and without gravity and the effect on the trajectories of the two masses.
4. Evaluate the pedagogical value of using interactive simulations like the one described to teach physics concepts. Discuss specific examples of how the simulation could aid in student understanding of difficult concepts such as projectile motion and system dynamics.
5. Using the provided links as a jumping off point, compare and contrast the approaches to projectile motion seen in the other provided simulations. How might they be used in conjunction to improve comprehension of the topic.

Glossary of Key Terms

• Projectile Motion: The motion of an object thrown or projected into the air, subject to gravity and air resistance (in some cases).
• Hooke's Law: A law stating that the force needed to extend or compress a spring by some distance is proportional to that distance. Represented mathematically as F = -kx.
• Newton's Second Law: A law stating that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. Represented mathematically as F = ma.
• Damping: A force that acts to reduce the amplitude of oscillations or vibrations in a system, usually proportional to velocity.
• Spring Constant (k): A measure of the stiffness of a spring. A higher spring constant means the spring is stiffer and requires more force to stretch or compress.
• Damping Coefficient: A measure of how quickly a system loses energy due to damping forces.
• Ordinary Differential Equation: A mathematical equation involving derivatives of a function of one independent variable.
• Numerical Solution: An approximate solution to a mathematical problem that cannot be found by standard algebraic methods. It is an estimate calculated by computers.
• Center of Gravity: The point where the weight of an object is balanced, the point around which the object's weight is evenly distributed.
• Second Order Ordinary Differential Equation: A type of differential equation where the highest derivative in the equation is the second derivative.
• Creative Commons Attribution-Share Alike 4.0 Singapore License: A type of open copyright license that allows users to share and adapt the material, provided they give proper credit and release their adaptations under the same license.

Versions

1.  http://opensourcephysicssg.blogspot.sg/2017/10/mass-and-spring-projectile-by-darren-tan.html
2. https://zdtan.github.io/EjsS-2bodygrav/TwoBodyNewtonGravity.xhtml 

Other resources

1. http://www.compadre.org/Physlets/mechanics/illustration3_4.cfm Illustration 3.4: Projectile Motion by W. Christian and M. Belloni 
2. http://physics.weber.edu/amiri/director-dcrversion/newversion/airresi/AirResi_1.0.html Trajectory of a ball with air resistance by Farhang Amiri
3. http://www.walter-fendt.de/html5/phen/projectile_en.htm HTML5 version of Projectile Motion by Walter Fendt
4. http://www.compadre.org/OSP/items/detail.cfm?ID=7299&S=7 Ejs Intro 2DMotionLab Model by  Anne Cox, Wolfgang Christian, and Mario Belloni 
5. http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=623.0 Projectile motion with equations by Fu-Kwun Hwang
6. http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=1832.0 Airdrag by Fu-Kwun Hwang and ahmedelshfie
7. http://archive.geogebra.org/en/upload/files/english/lewws/basketballsimulation_counterspeed_simulationspeed_updated1r.html Simulation of BasketBall Throw by Lew W. S.
8. http://ophysics.com/k8.html by Tom Walsh
9. http://ophysics.com/k9.html by Tom Walsh

FAQ: Projectile Motion with System of Masses and Spring Simulation

1. What does the "Projectile Motion with System of Masses and Spring" simulation model? This simulation models the motion of two masses (m and m1) connected by a spring of negligible mass. It shows how these masses move in two dimensions, with a focus on vertical motion when gravity is applied. The simulation incorporates Hooke's Law for the spring's force and a damping term that depends on the difference in velocities between the two masses. It numerically solves the equations of motion derived from Newton’s Second Law and visualizes the results.
2. How can I interact with the simulation? The simulation provides a number of interactive activities. You can drag the red mass to give it an initial velocity and see how the system evolves. You can also observe what happens when gravity is turned on. Additionally, you can adjust the mass of the blue ball to see how it affects the system's center of gravity, and you can modify the spring constant and damping coefficient to see their effects in real-time.
3. What physical principles are demonstrated in this simulation? The simulation primarily demonstrates concepts related to projectile motion, Hooke's Law (spring forces), Newton's Second Law of Motion, and damped oscillations. It visually shows how forces and initial conditions affect the motion of connected masses. It also touches on center of gravity and damping effects.
4. What is Hooke's Law and how does it apply in this simulation? Hooke's Law describes the restoring force exerted by a spring when it is stretched or compressed. In this simulation, the spring force between the two masses is calculated using Hooke's Law, which states that the force is proportional to the displacement of the spring from its equilibrium length. This force plays a key role in the dynamics of the system.
5. What is the purpose of the damping term in the simulation? The damping term in the simulation represents forces like friction or air resistance, which dissipate energy from the system. This term is proportional to the difference in velocities between the two masses. It causes the oscillations and movement of the masses to gradually diminish over time, bringing them toward a state of rest. Without damping, the motion would continue indefinitely.
6. How does changing the masses of the two objects affect the simulation? Changing the relative masses of the blue and red balls has a direct impact on the system's center of gravity. By varying these masses, you can observe how the center of gravity shifts, influencing how the connected masses move and rotate, especially when gravity is enabled.
7. What do the spring constant and damping coefficient represent and how do they affect the motion? The spring constant, usually denoted as 'k', determines how much force the spring exerts for a given displacement from its equilibrium length. A higher spring constant results in a stiffer spring. The damping coefficient represents the amount of energy lost due to dissipative forces. A higher damping coefficient leads to a more rapidly diminishing oscillation. By varying these, one can observe how these parameters control the period, amplitude and overall duration of motion of the masses.
8. What tools were used to create this simulation, and where can I find other similar simulations? This simulation was created using JavaScript and HTML5, allowing it to be run directly in a web browser without the need for plugins. This project is part of Open Source Physics at Singapore. The source also provides links to many other physics simulations and related resources on the Open Source Physics website and beyond. These include models of projectile motion, air resistance, and other concepts in mechanics.