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| − | [http://www.karnatakaeducation.org.in/KOER/en/index.php/Science:_History The Story of Science]
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| − | |style=" width:10%; border:none; border-radius:5px;box-shadow: 5px 5px 5px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|[http://www.karnatakaeducation.org.in/KOER/en/index.php/Science:_Philosophy Philosophy of Science]
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| − | [http://www.karnatakaeducation.org.in/KOER/en/index.php/Science:_Pedagogy Teaching of Science]
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| − | [http://www.karnatakaeducation.org.in/KOER/en/index.php/Science:_Curriculum_and_Syllabus Curriculum and Syllabus]
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| − | [http://www.karnatakaeducation.org.in/KOER/en/index.php/Science:_Topics Topics in School Science]
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| − | [http://www.karnatakaeducation.org.in/KOER/en/index.php/Textbooks#Science_-_Textbooks Textbooks]
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| − | [http://www.karnatakaeducation.org.in/KOER/en/index.php/Science:_Question_Papers Question Bank]
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| | While creating a resource page, please click here for a resource creation [http://karnatakaeducation.org.in/KOER/en/index.php/Resource_Creation_Checklist '''checklist'''] | | While creating a resource page, please click here for a resource creation [http://karnatakaeducation.org.in/KOER/en/index.php/Resource_Creation_Checklist '''checklist'''] |
| − | = Concept Map = | + | ===Concept Map=== |
| − | <mm>[[Gravitation.mm|Flash]]</mm>
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| − | __FORCETOC__
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| | | | |
| − | = Textbook =
| + | [[File:Gravitation_-_Concept_Map_October_2020.mm|Flash]] |
| − | To add textbook links, please follow these instructions to:
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| − | ([{{fullurl:{{FULLPAGENAME}}/textbook|action=edit}} Click to create the subpage])
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| | | | |
| − | = Additional information = | + | ===Additional Resources=== |
| − | ==Useful websites== | + | ====Reference books==== |
| − | ==Reference Books== | + | #[http://www.conceptualphysics.com/ Conceptual Physics, Paul Hewitt, 10th Edition] |
| | + | #Physics for Scientists and Engineers, Douglas C. Giancoli, 3rd Edition |
| | + | #Resnick & Halliday with Jearl Walker, 8th Edition |
| | + | #NCERT Class 9 textbook [http://ncert.nic.in/NCERTS/textbook/textbook.htm?iesc1=8-14 Motion] |
| | + | #NCERT Class 9 textbook [http://ncert.nic.in/NCERTS/textbook/textbook.htm?iesc1=9-14 Laws of Motion] |
| | + | #NCERT Class 9 textbook [http://ncert.nic.in/NCERTS/textbook/textbook.htm?iesc1=10-14 Gravitation] |
| | | | |
| | + | ====Additional web resources==== |
| | + | #[[Image:Walter-Levin.png|80px|left]]This is a college lecture by Prof Walter Levin explaining gravitation. Click [https://upload.wikimedia.org/wikipedia/commons/4/4f/Newtonslawofgravity.ogv here] to see the video.<br><br><br><br> |
| | + | #[http://www.conceptualphysics.com/ Illustrations and videos to accompany the textbook Conceptual Physics] |
| | + | #[http://www.physicsclassroom.com Physics Classroom] |
| | | | |
| − | = Teaching Outlines =
| + | __FORCETOC__ |
| | | | |
| − | .<br> | + | === Teaching Outlines === |
| − | <br>
| + | |
| | + | =====Notes for teachers===== |
| | + | ''These are short notes that the teacher wants to share about the concept, any locally relevant information, specific instructions on what kind of methodology used and common misconceptions/mistakes.'' |
| | | | |
| − | ==Concept #1 - Contact and Non-Contact Forces== | + | #Forces can act at a distance and are called non-contact forces. To understand gravitation, we need to understand inertial and gravitational properties of mass. Gravity is one of the four fundamental forces. This document further discusses the concept of acceleration due to gravity and gravitational potential energy. Gravity is responsible for planetary motion and introduces the Kepler's laws of planetary motion. |
| − | ===Learning objectives=== | + | #In everyday life, gravitation is most familiar as the cause due to which masses fall to the ground. Gravitation causes dispersed matter to coalesce, and coalesced matter to remain intact, thus accounting for the existence of the Earth, the Sun, and most of the macroscopic objects in the universe. Gravitation is responsible for keeping the Earth and the other planets in their orbits around the Sun; for keeping the Moon in its orbit around the Earth; for the formation of tides; for natural convection, by which fluid flow occurs under the influence of a density gradient and gravity; for heating the interiors of forming stars and planets to very high temperatures; and for various other phenomena observed on Earth and the Universe. These are some of the questions we will explore here. |
| | + | #Gravitational force acts over a distance through a field and is one of the four fundamental forces. Gravitational force is a very small force. Idea of a force field and that forces can act at a distance. The distance over which the effect of a force can be felt is called its field. Gravitation explains several phenomena |
| | + | ##the force that allows us to move on Earth |
| | + | ##the motion of the moon around the Earth |
| | + | ##the motion of the planets around the Sun |
| | + | ##the tides due to the moon and the sun |
| | + | |
| | + | ====Concept #1 - Contact and Non-Contact Forces==== |
| | + | =====Learning objectives===== |
| | #Forces can act in contact | | #Forces can act in contact |
| | #Forces can act over a distance | | #Forces can act over a distance |
| − | ===Notes for teachers=== | + | =====Activities ===== |
| − | ''These are short notes that the teacher wants to share about the concept, any locally relevant information, specific instructions on what kind of methodology used and common misconceptions/mistakes.''
| + | #Activity 1 - [http://karnatakaeducation.org.in/KOER/en/index.php/Gravitation_Activity_1 The various forces we see] |
| − | | + | #Activity 2 - [http://karnatakaeducation.org.in/KOER/en/index.php/Gravitation_Activity_2 More about contact forces] |
| − | Forces can act at a distance and are called non-contact forces. To understand gravitation, we need to understand inertial and gravitational properties of mass. Gravity is one of the four fundamental forces. This document further discusses the concept of acceleration due to gravity and gravitational potential energy. Gravity is responsible for planetary motion and introduces the Kepler's laws of planetary motion.
| + | #Activity 3 - [https://karnatakaeducation.org.in/KOER/en/index.php/Activity_3_Mass_has_an_inertial_property Understanding mass] |
| − | | + | #Activity 4 - [https://karnatakaeducation.org.in/KOER/en/index.php/Gravitation_Acitivity_4_An_example_of_force_at_a_distance Understanding forces acting at a distance] |
| − | In everyday life, gravitation is most familiar as the cause due to which masses fall to the ground. Gravitation causes dispersed matter to coalesce, and coalesced matter to remain intact, thus accounting for the existence of the Earth, the Sun, and most of the macroscopic objects in the universe. Gravitation is responsible for keeping the Earth and the other planets in their orbits around the Sun; for keeping the Moon in its orbit around the Earth; for the formation of tides; for natural convection, by which fluid flow occurs under the influence of a density gradient and gravity; for heating the interiors of forming stars and planets to very high temperatures; and for various other phenomena observed on Earth and the Universe. These are some of the questions we will explore here.
| + | ====Concept #2 - Gravitation is a field force==== |
| − | ===Activity No # 1 - The various forces we see === | + | =====Learning objectives===== |
| − | {| style="height:10px; float:right; align:center;"
| + | #Gravitational force acts at a distance - this force is due to the gravitational property of mass |
| − | |<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;">
| + | #Weight is due to force called gravity |
| − | ''[http://www.karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
| + | =====Activities===== |
| − | |}
| + | # Activity 5 - [https://karnatakaeducation.org.in/KOER/en/index.php/Gravitation_Activity_5_See_Gravity_in_Action See Gravity in Action] |
| − | * '''Estimated Time'''- 30 minutes
| + | # Activity 6 - [http://karnatakaeducation.org.in/KOER/en/index.php/Gravitation_Activity_6 Introducing the idea of weight] |
| − | * '''Materials/ Resources needed''' - Discussions
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| − | * '''Prerequisites/Instructions, if any'''
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| − | * '''Multimedia resources'''
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| − | The following pictures have been sourced from the NCERT Class 7 textbook.
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| − | <gallery>
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| − | Image:force3.png
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| − | Image:force2.png
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| − | Image:force1.png
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| − | | |
| − | </gallery>
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| − | | |
| − | * '''Website interactives/ links/ simulations'''
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| − | * '''Process (How to do the activity)'''
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| − | #This is in the form of a discussion
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| − | #Students explore words like force, push, pull, contact, distance, etc
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| − | #Let the students look at the picture and describe the force and the effect
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| − | * '''Developmental Questions (What discussion questions)'''
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| − | #What do we mean when we say force?
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| − | #What does it do? (Produces an acceleration/ change in state - they can start with push and pull)
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| − | #For force to be there, do I have to touch? (Contact)
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| − | #When have you seen non-contact forces? (Electric shock, magnetism)
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| − | #Force can act at a distance; it is called field
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| − | * '''Evaluation (Questions for assessment of the child)'''
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| − | #How will you define the field of a force?
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| − | #Does the field have a definite size?
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| − | * '''Question Corner'''
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| − | | |
| − | ===Activity No # 2 More about contact forces===
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| − | {| style="height:10px; float:right; align:center;"
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| − | |<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;">
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| − | ''[http://www.karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
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| − | |}
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| − | * '''Estimated Time'''- 30 minutes
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| − | * '''Materials/ Resources needed''' - Discussions
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| − | * '''Prerequisites/Instructions, if any'''
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| − | Students must have been introduced to the Newton's laws.
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| − | * '''Multimedia resources'''
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| − | * '''Website interactives/ links/ simulations'''
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| − | * '''Process (How to do the activity)'''
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| − | #This is in the nature of discussions | |
| − | #Present scenarios for children to analyse
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| − | * '''Developmental Questions (What discussion questions)'''
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| − | #When you hang a bucket from a rope what are the forces acting? ''(Tension force is the force that is transmitted through a string, rope, cable or wire when it is pulled tight by forces from opposite ends. The tension force is directed along the length of the wire and pulls equally on the objects on the opposite ends of the wire.)''
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| − | #In which direction does tension act?
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| − | #Is this the same as friction? When do you notice friction? ''(Frictional force is the force exerted by a surface as an object moves across it or makes an effort to move across it. When two objects are in contact but are not moving relative to one another then the frictional force between the two surfaces is called static friction. When the two objects are moving relative to one another, the friction between the surfaces is called the kinetic friction.)''
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| − | #When you push against an object, any surface, what do you feel? Do you feel a push back? Normal force is the perpendicular force that is exerted upon an object that is in contact with another stable object. If a book is resting upon a surface, then the surface is exerting an upward force upon the book in order to support the weight of the book. A normal force can also be exerted horizontally. If a person leans against a wall, the wall pushes back on the person.
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| − | * '''Evaluation (Questions for assessment of the child)'''
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| − | [[Image:force4.png|200px|left]] Click [http://phet.colorado.edu/en/simulation/friction here] for friction simulation | |
| − | <br>
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| − | Answer the following questions
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| − | #What do the moving green and yellow circles indicate? | |
| − | #When you are rubbing the yellow book on the green book what happens? How do you know that the heat is being generated?
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| − | #When the temperature goes very high, the yellow circles fly off. What does this mean physically?
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| − | #Is there any force acting in this picture between the books?
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| − | #Is gravitational force acting?
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| − | * '''Question Corner'''
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| − | | |
| − | ==Concept #== | |
| − | ===Learning objectives=== | |
| − | ===Notes for teachers===
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| − | ''These are short notes that the teacher wants to share about the concept, any locally relevant information, specific instructions on what kind of methodology used and common misconceptions/mistakes.''
| |
| − | ===Activity No # === | |
| − | {| style="height:10px; float:right; align:center;"
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| − | |<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;">
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| − | ''[http://www.karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
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| − | |}
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| − | * '''Estimated Time'''
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| − | * '''Materials/ Resources needed'''
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| − | * '''Prerequisites/Instructions, if any'''
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| − | * '''Multimedia resources'''
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| − | * '''Website interactives/ links/ simulations'''
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| − | * '''Process (How to do the activity)'''
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| − | * '''Developmental Questions (What discussion questions)'''
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| − | * '''Evaluation (Questions for assessment of the child)'''
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| − | * '''Question Corner'''
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| − | | |
| | | | |
| − | ===Activity No # ===
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| − | {| style="height:10px; float:right; align:center;"
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| − | |<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;">
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| − | ''[http://www.karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
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| − | |}
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| − | * '''Estimated Time'''
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| − | * '''Materials/ Resources needed'''
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| − | * '''Prerequisites/Instructions, if any'''
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| − | * '''Multimedia resources'''
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| − | * '''Website interactives/ links/ simulations'''
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| − | * '''Process (How to do the activity)'''
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| − | * '''Developmental Questions (What discussion questions)'''
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| − | * '''Evaluation (Questions for assessment of the child)'''
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| − | * '''Question Corner'''
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| | | | |
| − | = Project Ideas =
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| | | | |
| − | = Fun corner =
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| − | '''Usage'''
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| | | | |
| | Create a new page and type <nowiki>{{subst:Science-Content}}</nowiki> to use this template | | Create a new page and type <nowiki>{{subst:Science-Content}}</nowiki> to use this template |
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| − | | + | [[Category:Gravitation]] |
| − | = Concept Map =
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| − | | |
| − | <br><br>
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| − | | |
| − | [[Image:Gravitation%20for%20wiki_html_m4dc3444b.jpg]] <br> | |
| − | | |
| − | = Weight =
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| − | == Concept flow ==
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| − | | |
| − | *Every particle has mass; weight is a force acting on a mass due to the gravitational pull.
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| − | | |
| − | *This force experienced by an object due to the gravitational pull of the Earth is what we call the weight. Weight is nothing but the force exerted on a mass due to the gravitational pull of the Earth.
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| − | | |
| − | === How do we perceive this weight? ===
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| − | | |
| − | When you stand on a surface, the force of the Earth's gravity is acting upon you downwards and there is a normal force exerted by the surface on which you stand. Since you stand on a firm surface and there is no acceleration, the normal force is equal to the gravitational force and this is equal to mg. If an object is suspended from a spring, the gravitational force will be balanced by the tension force in the string.
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| − | | |
| − | Weight is that supporting force felt by an object in equilibrium; this opposes and balances the gravitational pull of the Earth. Thus, humans experience their own body weight as a result of this supporting force, which results in a normal force applied to a person by the surface of a supporting object, on which the person is standing or sitting. In the absence of this force, a person would be in free-fall, and would experience weightlessness. It is the transmission of this reaction force through the human body, and the resultant compression and tension of the body's tissues, that results in the sensation of weight.
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| − | | |
| − | When an object is in equilibrium, it only experiences the gravitational and restoring force/ Weight is mass multiplied by the acceleration due t gravity
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| − | | |
| − | '''Measured weight can change:'''
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| − | *when acceleration due to gravity changes
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| − | | |
| − | *when the object is accelerating (non-inertial frame)
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| − | | |
| − | When used to mean force, magnitude of weight (a scalar quantity), often denoted by an italic letter W, is the product of the mass, m, of the object and the magnitude of the local gravitational acceleration g;. thus: W = mg. When considered a vector, weight is often denoted by a bold letter W. The unit of measurement for weight is that of force, which in the International System of Units (SI) is the newton.
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| − | | |
| − | For example, an object with a mass of one kilogram has a weight of about 9.8 newtons on the surface of the Earth, about one-sixth as much on the Moon, and very nearly zero when in deep space far away from all bodies imparting gravitational influence.
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| − | <br><br>
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| − | '''Earlier concepts of weight'''
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| − | | |
| − | Concepts of heaviness (weight) and lightness (levity) date back to the ancient Greek philosophers. These were typically viewed as inherent properties of objects. Plato described weight as the natural tendency of objects to seek their kin. To Aristotle weight and levity represented the tendency to restore the natural order of the basic elements: air, earth, fire and water. He ascribed absolute weight to earth and absolute levity to fire. Archimedes saw weight as a quality opposed to buoyancy, with the conflict between the two determining if an object sinks or floats. The first operational definition of weight was given by Euclid, who defined weight as: "weight is the heaviness or lightness of one thing, compared to another, as measured by a balance.".
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| − | Satellites and weightlessness
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| − | | |
| − | It is a very common misconception that when astronauts are in orbit they are weightless because they are somehow far enough from the earth that the force of earth's gravity does not pull on them. This is totally incorrect. If they were that far away, earth's gravity would not pull on the shuttle either and it would be impossible for it to be in orbit around the earth.
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| − | Gravity (a force we call weight) is actually responsible for keeping the space craft and the astronaut in orbit around the earth. Gravity is still pulling on the astronaut. The feeling of weight;ess is no differenet than when in ree fall. What they are not experiencing is the normal force, which is the opposing force. When that force is gone, we feel say we feel "weightless." In fact, whenever a person is in freefall they feel weightless even though gravity is still causing them to have weight. While in orbit, the space shuttle does not have to push on the astronaut (or anything else in the cabin) to keep him up. The space shuttle and the astronaut are in a constant state of freefall around the earth.<br><br>
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| − | | |
| − | = Significance of the gravitational force =
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| − | | |
| − | == Discovery of planets ==
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| − | | |
| − | Accurate measurements on the orbits of the plantes indicated that they did not precisely follow Kepler's laws. Slight deviations from perfectly elliptical orbits were observed. Newton was aware that this was to be expected from the Law of Universal Gravitation. The derivation of perfectly elliptical ignores the forces due to the other planets. These deviations called perturbations are observed and led to the discovery of Neptune and Pluto. Planets around distant stars were also inferred from the regular wobble of each star due to the gravitational attraction of the revolving plant.
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| − | | |
| − | == Ocean tides ==
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| − | | |
| − | Ocean tides are caused by differences in the gravitational pull between the Moon and the Earth on the opposite sides of the Earth. Gravitational force is stronger on the side of the Earth nearer to the Moon and is weaker on the side of the Earth farther from the Moon. The bulge that is caused in the Earth's oceans due to this gravitational pull results in two sets of tides on the Earth.
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| − | === Activity 2 – Observe a freely falling body ===
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| − | | |
| − | '''Objective: To observe the behaviour of a freely falling object'''
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| − | | |
| − | '''Procedure:'''
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| − | Ask a child to drop a piece of chalk from terrace, Start the stop clock as soon as the child drops it. Put off the clock as soon as the chalk touches the ground, note down the time taken , Repeat the same expt with a stone,& calculate the time, Time taken will be same in both the cases. Inference : All bodies accelerate equally irrespective of their mass.
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| − | | |
| − | === Activity 3 : Thought Experiment ===
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| − | | |
| − | '''Objective: To understand the nature of gravitational force'''
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| − | | |
| − | '''Procedure:'''
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| − | | |
| − | Ask the children to think about what will happen if we had a hollow tunnel running through the centre of the Earth and we dropped a ball into it. What will happen to the ball?<br><br>
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| − | | |
| − | = Projectile and Satellite Motion =
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| − | == Concept flow ==
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| − | | |
| − | *A projectile motion of a body thrown is due to the gravitational force.
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| − | | |
| − | *Satellites are projectiles that are continuously falling in the orbit around planets
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| − | Let us study this picture below and analyze what happens in each of the cases.
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| − | [[Image:Gravitation%20for%20wiki_html_545339b5.gif]] <br>
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| − | | |
| − | In the first case, the ball is just dropped from the cliff and it falls down in a straight line, subject to the force of gravity. In the second and third instances, the ball is thrown upwards, reaches a certain height and still falls down. In the third case, the ball covers a horizontal range as well.
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| − | In all these cases, gravity is the only force acting. Without gravity, we could throw a rock upwards at an angle and it follow a straight line path. Because of gravity, however, the path curves.
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| − | | |
| − | Such an object, when thrown/ projected and continues its motion on its own inertia is called a projectile. A projectile will have two components to ots velocity – the horizontal and the vertical. The horizontal component is similar to an object rolling on a plane/ along a straight line. The vertical component of the velocity is subject to the acceleration due to gravity. A projectile moves horizontally as it moves downwards or upwards.
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| − | Imagine throwing a ball straight up. It will fall down to the same place. In this case the ball has a velocity in the vertical direction which changes with time as the force due to gravity causes an acceleration in the downward direction all the time. But suppose on were to throw a ball with only a horizontal velocity. The ball will now move with a velocity that has two components to it - one the horizontal velocity which remains unchanged as long as there is no force such as air resistance acting on it and a vertical velocity that is continually changing. This vertical velocity starts at zero - we threw the ball horizontally - and keeps increasing with an acceleration g.
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| − | The resultant velocity is a combination of the two. This is what causes objects to follow a parabolic path when they are thrown with a combination of horizontal and vertical velocities.
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| − | The greater the horizontal component the farther the ball will travel. For short distances, and small velocities, the curvature of the Earth will make no difference.
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| − | But suppose that we throw it so hard that the horizontal distance is very large and we can no longer ignore the curvature of the Earth?
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| − | Suppose we throw it so hard that that ball will continue to fall but will never reach the ground - the curvature of the fall of the ball is greater than the curvature of the Earth? Then the ball will become a satellite! It will move round and round the Earth - constantly falling but never reaching the ground!
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| − | The satellite and everything in it are constantly falling towards the Earth but will never reach it. Since they are all falling with the same velocity, the satellite does not exert any force on the objects or people inside. The people inside, therefore feel weightless. remember we have a sense of weight because of the Normal force. Here the Normal force is zero and so we feel weightless.
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| − | This is very similar to the sense of loss of weight in a lift that is accelerating downwards - except that here the acceleration is the acceleration due to gravity.<br><br>
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| − | == Satellite ==
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| − | | |
| − | An Earth satellite is simply a projectile that falls around the Earth rather than into it. That means the horizontal falling distance matches the Earth''s curvature. Geometrically, the curvature of the surface is that its surface drops a vertical distance of 5 metres for every 8000 metres tangent to the surface.
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| − | Therefore, if we throw a rock or a ball at a high enough speed (about 29000 km/s), it would follow the curvature of the Earth. But at this speed, atmospheric friction (due to air drag) would burn up everything. This is why satellites are launched at an altitude high enough for the air drag to be negligible.
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| − | Satellite motion was understood by Newton who reasoned that the Moon was simply a projectile that was circling the Earth. <br><br>
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| − | === Activity 4 - Demonstration of satellite motion using simulation ===
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| − | The following simulation can illustrate how satellite motion takes place. <br>
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| − | [http://phet.colorado.edu/sims/gravity-and-orbits/gravity-and-orbits_en.jnlp Click to run PhET simulation]
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| − | = Kepler's Laws =
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| − | == Concept flow ==
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| − | *The key concept to understand here is that gravitational forces play an important role in planetary motion.
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| − | *Three laws of planetary motion that describe the motion of the planets have been postulated based on detailed astronomical observations
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| − | == Laws of Planetary Motion ==
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| − | We now know that satellites are continually falling towards the Earth following a curved path whose curvature is greater than that of the curvature of the Earth. The Moon is just such a satellite that moves around the Earth. In a similar way, all the planets that move around the Sun are satellites of the Sun. The motion described in such a situation is not strictly circular - it is elliptical.
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| − | Johannes Kepler, working with data painstakingly collected by Tycho Brahe without the aid of a telescope, developed three laws which described the motion of the planets across the sky.
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| − | 1. The Law of Orbits: All planets move in elliptical orbits, with the sun at one focus.<br><br>
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| − | 2. The Law of Areas: Each planet moves so that an imaginary line drawn from the Sun to the planet sweeps out equal areas in equal periods of time.<br><br>
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| − | 3. The Law of Periods: The square of the period of any planet is proportional to the cube of the semimajor axis of its orbit.<br><br>
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| − | Kepler's laws were derived for orbits around the sun, but they apply to satellite orbits as well.<br><br>
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| − | === The Law of Orbits ===
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| − | All planets move in elliptical orbits, with the sun at one focus. An ellipse is a closed curve such that the sum of the distances from any point P on the curve to two fixed points (called the foci, F1 and F2) remains constant. <br><br>
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| − | '''Orbit eccentricity'''
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| − | The semi major axis of the ellipse is a and represents the planet's average distance from the Sun. The eccentricity, “e” is defined so that “ea” is the distance from the centre to either focus. A circle is a special case of an ellipse where the two foci coincide. The Earth and most of the other planets have nearly circular orbits. For Earth, “e” = 0.017.<br><br>
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| − | [[Image:Gravitation%20for%20wiki_html_m75870f64.gif]] <br>
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| − | | |
| − | === The Law of Equal Areas ===
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| − | Kepler's second law states that each planet moves so that an imaginary line drawn from the Sun to the planet sweeps out equal areas in equal periods of time.
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| − | This can be shown to be true using the law of conservation of angular momentum.
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| − | [[Image:Gravitation%20for%20wiki_html_m4a5ede30.png]] <br>
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| − | If “v” is the velocity of the planet, in time “dt” the planet moves a distance vdt and sweeps out an area equal to the area of a triangle of base “r” and altitude vdt sinα.
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| − | Hence dA = ½ (r) (“v” x “dt” x sinα)
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| − | dA/ dt = ½ rv sinα
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| − | The magnitude of the angular momentum of the planet about the Sun is L = mvr sinα.
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| − | dA/ dt = (½)L/m
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| − | Because the angular momentum is conserved, the rate of change of area covered is constant. This means that the planets move with different velocities depending upon their position in the orbits.<br><br>
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| − | | |
| − | === The Law of Periods ===
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| − | The ratio of the squares of the periods of any two planets revolving about the Sun is equal to the ratio of the cubes of their semi-major axes.
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| − | Can you derive this?
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| − | [[Image:Gravitation%20for%20wiki_html_m290004f7.gif]] <br>
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| − | G m1 Ms / r12 = m1 (v12)/ r1
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| − | v1 = 2πr1/T1
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| − | | |
| − | Substituting and rearranging we get
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| − | T12/ r13 = 4π2 / G Ms
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| − | Deriving this for another planet, we can arrive at the third law.<br><br>
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| − | = Additional resources: =
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| − | 1. Gravity is more than a name - This link gives an overview of what gravity is.
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| − | 2.
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| − | 4. Animation of the Cavendish experiment
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| − | 5. www.hyperphysics.com - From Classical Mechanics to General Relativity - This is a good description of the geometry of Newtonian gravity and how to move from classical mechanics to relativity.
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| − | 6. The Value of "g" (http://www.physicsclassroom.com/Class/circles/U6L3e.cfm) - This is a good resource to study the variation of “g” at various distances above the Earth's atmosphere.
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| − | 7. http://science.nasa.gov/science-news/science-at-nasa/2004/06may_lunarranging/ - This link examines the Galileo experiment and discusses if there are other possible explanations.
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| − | 8. http://www.physicsclassroom.com/Class/circles/U6L4b.cfm -This website describes the mathematics of orbital motion.
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| − | 9. http://spaceflight.nasa.gov/gallery/images/station/crew-9/html/iss008e21996.html
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| − | <br>
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| − | = Keywords =
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| − | Mass, Inertial, Gravitational, Force field, Universal law of gravitation, Acceleration due to gravity, “g”, weight, weightlessness
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