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7.3.7 Gravitational Potential (symbol: ϕ and units: J kg^-1)

The gravitational potential, ϕ , at a point due to the gravitational field set up by a mass M is defined as the work done per unit mass in bringing a point mass from infinity to that point.

Mathematically, it can be shown that $ϕ = - \frac{G M}{r}$

7.3.7.1 Equation U = mϕ

1) This expression

ϕ = - \frac{G M}{r}

is similar to the expression for gravitational potential energy,

U = -m \frac{G M}{r}

. and they are related by U = mϕ.
ϕ = - G M/r

7.3.7.2 Scalar quantity

2) Gravitational potential is a scalar quantity. (i.e. it has no direction and a negative value simply means it is less than zero).

7.3.7.3 Always negative

3) This expression implies that ϕ is also always negative (less than zero) and by convention, the gravitational potential at infinity is also taken to be zero (maximum).
ϕ = - G M/r

7.3.7.4 Independent of test mass

4) Similar to gravitational field strength

g = \frac{G M}{r^{2}}

, gravitational potential

ϕ = - \frac{G M}{r}

is also independent of the mass of the point mass, m.

7.3.7.5 Vary according to inverse law with distance r

5) As distance r of the point mass from source mass increases, ϕ increases according to the equation

ϕ = - \frac{G M}{r}

7.3.8 Activity to do

ICT inquiry worksheet 1 (C) and (D), as well as the "G field and potential" EJS here. The HTML5 version is below, the Java worksheet customization to HTML5 is work in progress.

7.3.9 Summary

symbol	g	ϕ
name	Field strength	Potential
units	N kg^-1 or m s^-2	J kg^-1
meaning	Gravitational force per unit mass	Gravitational potential energy per unit mass
quantity	vector	scalar
equation	$\|g\| = \frac{G M}{r^{2}}$ towards the centre of the source mass	$ϕ = - \frac{G M}{r}$
relationship to mass	Force, $F = \frac{G M_{1} M_{2}}{r^{2}}$ = mg	Potential energy, $U = -m \frac{G M}{r}$ = mϕ
graph
computer model if M = 500.	-6.67500/(abs(r)r)	-6.67*500/abs(r)

7.3.10 Modeling

Try to input your own model for potential until you achieve a close fit to the data set graph from potential.
Hint:
What is the value of M is the model?
no need to key in x10^-11
abs in JavaScript is absolute | | that always make the value positive.
try something like -6.67*500/abs(r) in the equivalent for

ϕ = - \frac{G M}{r}

7.3.11 Model

Translations

Code	Language		Translator	Run

Credits

This email address is being protected from spambots. You need JavaScript enabled to view it.; Anne Cox; Wolfgang Christian; Francisco Esquembre

http://iwant2study.org/lookangejss/02_newtonianmechanics_7gravity/ejss_model_gravity06/gravity06_Simulation.xhtml

Briefing Document: 🍎Gravitational Potential Simulation Model

1. Introduction:

This document analyzes an online resource from Open Educational Resources / Open Source Physics @ Singapore, specifically focusing on a JavaScript HTML5 applet simulation model for gravitational potential. The resource, titled "7.3.7 Gravitational Potential JavaScript HTML5 Applet Simulation Model," provides a concise explanation of gravitational potential, its mathematical representation, and its relationship to gravitational potential energy. The resource also includes an interactive simulation (accessible via iframe) to explore these concepts.

2. Key Concepts and Definitions:

Gravitational Potential (ϕ): Defined as "the work done per unit mass in bringing a point mass from infinity to that point." This work is done against the gravitational force.
Mathematical Representation: The resource states that ϕ = -GM/r, where G is the gravitational constant, M is the mass of the source, and r is the distance from the center of the source mass.
Units: The units for Gravitational Potential are J/kg (Joules per Kilogram).
Gravitational Potential Energy (U): The resource highlights the close relationship between gravitational potential and gravitational potential energy.
Relationship: The potential energy (U) of a mass m at a certain point is given by U = mϕ, and the resource shows that U = -mGM/r.
The resource states "This expression ϕ = - G M r is similar to the expression for gravitational potential energy, U = -m G M r . and they are related by U = m ϕ ."

3. Key Characteristics of Gravitational Potential:

Scalar Quantity: "Gravitational potential is a scalar quantity. (i.e. it has no direction and a negative value simply means it is less than zero)." Unlike gravitational field strength, which is a vector, gravitational potential has magnitude but no direction.
Always Negative: "This expression implies that ϕ is also always negative (less than zero) and by convention, the gravitational potential at infinity is also taken to be zero (maximum)." The negative sign arises from the fact that the work is done by an external agent moving a mass against the gravitational field, meaning the work done is considered negative in this system's standard conventions.
Independent of Test Mass: "Similar to gravitational field strength g = G M r 2 , gravitational potential ϕ = - G M r is also independent of the mass of the point mass, m." The gravitational potential is a property of the source mass and its surrounding space, not the mass that is experiencing the potential.
Varies Inversely with Distance: "As distance r of the point mass from source mass increases, ϕ increases according to the equation ϕ = - G M r." The potential increases (becomes less negative, tending towards zero) as the distance from the source mass increases.

4. Simulation Model:

Interactive Iframe: The resource embeds an interactive simulation using an iframe. This simulation allows users to visualize the concept of gravitational potential, though this analysis does not contain direct information on the features or UI of the simulation. It links to https://iwant2study.org/lookangejss/02_newtonianmechanics_7gravity/ejss_model_gravity06/gravity06_Simulation.xhtml
Modeling Activities: The resource encourages users to explore the model by varying parameters, as detailed in sections 7.3.10 and 7.3.11. Specifically, users can "Try to input your own model for potential until you achieve a close fit to the data set graph from potential." The resource indicates a model with something like "-6.67*500/abs(r)" is an attempt to achieve this.
Underlying Equations: The resource presents the equations used for calculations and graphs in the model. "computer model if M = 500. -6.67*500/(abs(r)r) -6.67500/abs(r)". Here "-6.67" is the gravitational constant G and "500" is likely the value of the mass M in the simulation. The abs(r) likely represents the absolute value of r, to ensure that the radius is positive.

5. Key Differences Highlighted in the Summary (7.3.9):

FeatureGravitational Field Strength (g)Gravitational Potential (ϕ)SymbolgϕUnitsN kg⁻¹ or m s⁻²J kg⁻¹MeaningGravitational force per unit massGravitational potential energy per unit massQuantityVectorScalarEquationgRelationship to MassForce, F = GM₁M₂/r² = mgPotential energy, U = -mGM/r = mϕThis section clearly distinguishes between field strength and potential, emphasizing the vector vs scalar nature and the different relationship to force and energy respectively.

6. Additional Context:

Open Educational Resource: The resource is part of a larger collection of open educational materials from Open Source Physics @ Singapore, promoting accessible science education.
Interactive Learning: The website emphasizes the use of interactive simulations and applets as learning tools, consistent with modern pedagogical practices.
Use of JavaScript and HTML5: The simulation being developed with JavaScript and HTML5 suggests a commitment to accessibility across multiple platforms and devices.
Links to Related Resources: The resource includes links to Java-based models as well as other simulations covering a wide variety of physics and science topics, including: Newton's Cradle, Projectile Motion, Electromagnetism, Radioactive Decay, and much more. This is a very broad range of topics.

7. Conclusion:

The "Gravitational Potential JavaScript HTML5 Applet Simulation Model" provides an accessible and interactive explanation of the gravitational potential. Its key strength lies in its clear articulation of the concept, its mathematical basis, its relationship to potential energy and its interactive simulation. It stands as a resource consistent with the trends of open education and tech-driven learning. The resource distinguishes itself by presenting both mathematical understanding and model-based learning.

Gravitational Potential Study Guide

Quiz

Instructions: Answer each question in 2-3 complete sentences.

How is gravitational potential defined, and what are its units?
How does the mathematical expression for gravitational potential relate to gravitational potential energy?
Is gravitational potential a vector or a scalar quantity? Explain what this means.
Why is gravitational potential always a negative value?
What is the conventional value of gravitational potential at infinity? Why is this value chosen?
How does gravitational potential change with increasing distance from a source mass?
What is the key difference between gravitational field strength (g) and gravitational potential (ϕ) in terms of what they represent?
Is gravitational potential dependent on the mass of the point mass being considered?
How do the equations for gravitational force and gravitational potential energy relate to the concepts of field strength and potential?
Briefly describe one of the modeling activities suggested in the source for gravitational potential.

Quiz Answer Key

Gravitational potential is defined as the work done per unit mass in bringing a point mass from infinity to a specific point. Its units are Joules per kilogram (J/kg).
The gravitational potential (ϕ) is related to the gravitational potential energy (U) by the equation U = mϕ. This means that gravitational potential energy is the gravitational potential multiplied by the mass of the object.
Gravitational potential is a scalar quantity, which means it has magnitude but no direction. A negative value for potential simply indicates that it is less than zero, not a direction.
Gravitational potential is always negative because it is defined as the work done by the gravitational field to move a mass from infinity to a point. Since gravity is an attractive force, the work done by the field is negative.
The gravitational potential at infinity is conventionally taken to be zero. This is a reference point, and it simplifies calculations and comparisons of potential at other locations in the field.
Gravitational potential increases as the distance from the source mass increases. This increase follows an inverse relationship as described by the equation ϕ = -GM/r, approaching zero as the distance approaches infinity.
Gravitational field strength (g) represents the gravitational force per unit mass, while gravitational potential (ϕ) represents the gravitational potential energy per unit mass. The former is a vector and the latter is a scalar quantity.
No, gravitational potential is independent of the mass of the point mass. It is a property of the gravitational field created by the source mass and applies to any point mass placed within that field.
The equations for gravitational force (F = GMm/r^2) and potential energy (U = -GMm/r) relate to field strength (g) and potential (ϕ) respectively because force divided by mass gives g, and potential energy divided by mass gives ϕ. This establishes an important connection between force and energy in a gravitational context.
The source suggests modeling the potential by inputting values into an expression, such as -6.67*500/abs(r), to achieve a close fit to the data set graph. The activity is to find the value of M in the model through this process.

Essay Questions

Instructions: Write an essay that thoroughly addresses the question posed below.

Discuss the relationship between gravitational potential and gravitational potential energy. How are they related mathematically, and what are the key differences in their conceptual meanings?
Explain why gravitational potential is always a negative value. How does this negativity relate to the way we define gravitational potential and the work done by the gravitational field?
Compare and contrast gravitational potential (ϕ) and gravitational field strength (g). What are their units, and are they vector or scalar quantities? How do they relate to the fundamental concepts of force and energy?
Describe the role of simulation models in understanding gravitational potential. How can these models enhance learning and address the abstract nature of the concept?
Using the information provided, describe how to develop an accurate model of gravitational potential using JavaScript. How could this model be used to represent different situations, and what is its advantage over simply using equations?

Glossary of Key Terms

Gravitational Potential (ϕ): The work done per unit mass in bringing a point mass from infinity to a specific point in a gravitational field. Measured in Joules per kilogram (J/kg).
Gravitational Potential Energy (U): The energy an object possesses due to its position in a gravitational field. Measured in Joules (J).
Scalar Quantity: A quantity that has magnitude but no direction. Examples include gravitational potential and energy.
Vector Quantity: A quantity that has both magnitude and direction. Examples include gravitational force and field strength.
Gravitational Field Strength (g): The gravitational force experienced per unit mass. Measured in Newtons per kilogram (N/kg) or meters per second squared (m/s²).
Source Mass (M): The mass that is creating a gravitational field.
Point Mass (m): A mass that is treated as if all of its mass is concentrated at a single point.
Inverse Law: Describes a relationship in which a quantity varies in the inverse of another quantity. Gravitational potential varies inversely with distance as described by the formula ϕ = -GM/r.
Infinity: A theoretical point infinitely far away from a source mass. It is used as a reference point where gravitational potential is conventionally defined as zero.

Video

https://notebooklm.google.com/notebook/208794b1-9b97-4766-9640-bb3f996ff6b7/audio

Resources

http://iwant2study.org/lookangejss/02_newtonianmechanics_7gravity/ejs/ejs_model_GField_and_Potential_1D_v8wee.jar

Timeline of Events (as implied by the source):

Ongoing Development & Use of Open Educational Resources: The source describes a project focused on creating and sharing open educational resources, particularly interactive physics simulations using JavaScript and HTML5. This indicates a sustained effort over time.
Creation of the Gravitational Potential Simulation Model (gravity06): A specific JavaScript HTML5 applet simulation model, "gravity06," was developed to illustrate the concept of gravitational potential. This model allows users to explore the relationship between gravitational potential, distance, and mass.
Implementation of Gravitational Potential Formula: The model uses and demonstrates the equation ϕ = -GM/r, which represents gravitational potential, where G is the gravitational constant, M is the mass of the object creating the gravitational field, and r is the distance from that object. This includes the related energy equation U = mϕ
Focus on Scalar Nature of Potential: The model also reinforces the concept of gravitational potential as a scalar quantity.
Development of Related Resources: A variety of other interactive simulations and tools have been created and are linked to from the site. These cover a range of physics topics and also touch on other subjects like math and chemistry. These resources show the effort to create interactive educational tools using the Easy JavaScript Simulation (EJS) framework.
Worksheets and Activities: The project includes the development of inquiry-based worksheets (like ICT inquiry worksheet 1 (C) and (D)) and activities to be used with the simulations, which helps integrate the simulations into a lesson plan.
Ongoing Modeling and Customization: There is an emphasis on users engaging with the simulations, for instance by inputting different values and models in order to achieve a close fit to data sets
Collaboration and Contribution: The project involves multiple developers and contributors, suggesting a collaborative environment for the creation of these resources.
Use of EJS Framework: The mention of "Easy JavaScript Simulation (EJS)" and "WebEJS" throughout the document suggests that the tools use this framework for creating interactive simulations.
HTML5 & JavaScript focus: The use of these indicates a specific technological direction for accessibility to users across multiple platforms.
Licensing and Open-Source: The project emphasizes open-source principles, making the resources available for others to use and build upon via Creative Commons licensing.
Workshops and Educator Engagement: There is activity in supporting educators using workshops and online resources to develop their own content for their classrooms.
In-house Resource Development: There is reference to teachers creating their own in-house resources, leveraging the EJS framework, and networking opportunities for teacher-led initiatives.
Use within Student Learning Space (SLS): There are many references to the simulations and tools being used within the Student Learning Space (SLS) in Singapore.
Analytics Integration: There is reference to projects that include data analytics in the simulations.

Cast of Characters:

Anne Cox: Listed as a contributor to the "Credits" section, likely involved in the development of the simulation or related resources. Her specific role is not detailed in the text.
Wolfgang Christian: Listed as a contributor to the "Credits" section, likely involved in the development of the simulation or related resources. His specific role is not detailed in the text. He is also credited on other models, specifically the 3D Wave Machine.
Francisco Esquembre: Listed as a contributor to the "Credits" section, likely involved in the development of the simulation or related resources. Also listed as supporting several workshops, and credited with other simulations.
Loo Kang WEE Lawrence aka lookang: User who has uploaded and created the content and the main maintainer of this website.

Frequently Asked Questions about Gravitational Potential

What is gravitational potential (ϕ), and what are its units? Gravitational potential (ϕ) is defined as the work done per unit mass in moving a point mass from infinity to a specific point in a gravitational field. It's a measure of the potential energy per unit mass at that location. The units for gravitational potential are Joules per kilogram (J/kg).
How is gravitational potential (ϕ) related to gravitational potential energy (U)? The gravitational potential energy (U) of a mass (m) at a point is directly related to the gravitational potential (ϕ) at that point by the equation: U = mϕ. This means that the gravitational potential is the potential energy "per unit mass". If you know the potential at a point, multiplying by the mass you are interested in gives you the potential energy for that mass.
Is gravitational potential (ϕ) a vector or a scalar quantity? Gravitational potential (ϕ) is a scalar quantity. Unlike vector quantities, it has no direction. The negative sign associated with gravitational potential simply means it is less than zero; it doesn't denote a direction. Scalar quantities like gravitational potential only have magnitude.
Why is gravitational potential (ϕ) typically negative? By convention, the gravitational potential at an infinite distance from a mass is defined as zero, which is its maximum value. As you move closer to a source of gravity, work is done by the gravitational field. Because the gravitational field does positive work when the point mass is moved from infinity, the potential of the field at a finite distance is therefore less than the potential at infinity. The potential at infinity is zero by definition, thus potential at finite distances is negative.
Does the mass of the test object affect the value of gravitational potential (ϕ)? No, gravitational potential (ϕ) is independent of the mass of the test object. It only depends on the source mass (M) and the distance (r) from that mass. The equation ϕ = -GM/r shows that the potential is determined only by G, M, and r, none of which involve the test mass "m". This means that the potential at a given point due to source mass M will be the same regardless of what the mass of the object you put at that location.
How does gravitational potential (ϕ) change as the distance from the source mass increases? As the distance (r) from the source mass increases, gravitational potential (ϕ) also increases (becomes less negative), following an inverse relationship (ϕ = -GM/r). This means the absolute value of potential decreases, approaching zero, as the test mass moves away from the source mass.
How is the gravitational potential (ϕ) related to gravitational field strength (g)? Both gravitational potential (ϕ) and gravitational field strength (g) are related to gravity, but they measure different aspects. Gravitational field strength is a vector quantity that describes the gravitational force per unit mass (g = GM/r²), while gravitational potential is a scalar quantity describing gravitational potential energy per unit mass (ϕ = -GM/r). They are both defined in terms of the gravitational field created by a source mass M, but one is a measure of the force, and the other the energy.
Can the concepts of gravitational potential and potential energy be modeled and explored using simulations? Yes, the source describes that simulations are valuable tools for exploring these concepts, with interactive applets that can be modeled and run. These simulations allow users to see the relationship between gravitational potential and various parameters like mass and distance, as well as how it relates to gravitational potential energy. They can even input their own models to compare how well they fit the theoretical curve.