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## Chapter 13 Universal Gravitation Examples

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### Example 13.2 The Density of the Earth (Example 13.3 Text Book) Using the last result, find the . average density. of the Earth: From Example 13.1. mg = – PowerPoint PPT presentation

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Title: Chapter 13 Universal Gravitation Examples

1
Chapter 13Universal Gravitation
Examples
2
Example 13.1 Weighing Earth
• Weighing Earth! Determining mass of Earth.
• ME mass of Earth (unknown)
RE 6,370 km 6.37 x
106 m
• Assume the Earth is a uniform, perfect sphere.
• m mass of object on the surface of Earth
(known)
• The weight of m is Fg mg
• The gravitational force on m is Fg
G(MEm)/RE2
• But these two are the same!!!
• mg G (mME)/RE2 ?
• g G ME/RE2 ?
• Solving for ME
• ME g RE2/G 5.98 ? 1024kg

m
RE
ME
3
Example 13.2 The Density of the Earth (Example
13.3 Text Book)
• Using the last result, find the average density
of the Earth
• From Example 13.1
• mg G (mME)/RE2 g G ME/RE2 ?
• ME g RE2/G
• Since ?E ME /VE VE (4/3)? RE3 ?

4
Example 13.3 Spacecraft at 2RE
• Find Fg acting on a 2000 kg spacecraft at 2RE
• RE 6,380 km ME 5.98 ? 1024 kg
• FG G(mME/r2)
• At surface (r RE)
• FG mg G(mME/RE2)
• At r 2RE
• FG G(mME)/(2RE)2 ?
• FG G(mME)/4RE2 ?
• FG ¼ G(mME/RE2) ?
• FG ¼mg 4900 N

5
Example 13.4 Geosynchronous Satellite
• A Geosynchronous Satellite appears to remain over
the same point on the Earth
• The gravitational force supplies a centripetal
force
• A satellite of mass m is moving in a circular
orbit around the Earth at a constant speed v and
at an altitude h above the Earths surface. Find
the speed (v) of the satellite in terms of G, h,
RE and ME

6
Example 13.4 Geosynchronous Satellite, final
• Assume m is moving around the Earth in a circular
orbit. Applying Newtons 2nd Law
• Taking (1) (2)
• Since r RE h

7
Example 13.5The Change in Potential
Energy(Example 13.6 Text Book)
• A particle of mass m is displaced through a small
vertical displacement ?y near the Earths
surface. Show that the general expression for the
change in gravitational potential energy
(Equation 13.12) reduces to the familiar
relationship ?U mg?y
• From Equation 13.12
• Since ri and rf are to close to the Earths
surface, then
• rf ri ?y and rfri RE2
• Remember r is measured from center of the Earth
• ?U mg?y

8
Material for the Final Exam
• Examples to Read!!!
• Example 13.1 (page 376)
• Material from the book to Study!!!
• Objective Questions 2-4-9
• Conceptual Questions 1-6-7
• Problems 3-6-12-25-27-29