PHYS16 – Lecture 27

Gravitation November 10, 2010

This Week

• Newton’s law of Gravity• Gravitational Potential Energy• Satellites• Kepler’s Laws of Planetary Motion– Orbital Shape– Orbital Areas– Orbital Period

Satellites

Orbital Energy

• Gravitational potential energy increases as distance increases

• Kinetic energy decreases as distance increases• So, why do higher orbitals have more energy?

rGMmE

rGMm

rGMm

rGMmmvE

UKE

2

221 2

Example Question: Satellite

• A Satellite orbiting the Earth wants to go to a lower orbit. What should the satellite do?

A) Nothing. It is falling toward the earth.B) Turn on rocket thrusters to accelerate, moving to lower orbital.C) Turn on rocket thrusters to decelerate, moving to a lower orbital

Escape velocity

• Minimum velocity needed to “escape” the gravitational force

RGMv

RGMmmv

UUKKE

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i

ifif

2

0)0()210(

0)()(

2

Trivia: Launching Rockets

• When launching a rocket to escape the Earth’s gravity, does the mass of the rocket matter? Why or why not?

The escape velocity does not depend on mass. Butthe kinetic energy does. Thus, a more massive rocketrequires more fuel.

Trivia: Launching Rockets

• When launching a rocket to escape the Earth’s gravity, does the mass of the rocket matter? Why or why not?

• Which planet has the smallest escape velocity?

Mercury!

Trivia: Launching Rockets

• When launching a rocket to escape the Earth’s gravity, does the mass of the rocket matter? Why or why not?

• Which planet has the smallest escape velocity?

• Does the direction of launch from the earth matter? Why or why not?

Want to launch eastward to take advantage of the Earth’srotational velocity. Also want to launch along Earth’s orbitalpath, as well.

Trivia: Launching Rockets

• When launching a rocket to escape the Earth’s gravity, does the mass of the rocket matter? Why or why not?

• Which planet has the smallest escape velocity?• Does the direction of launch from the earth

matter? Why or why not?• What would be the escape velocity of a black

hole?

Infinity!

Discovery Launch

• http://www.youtube.com/watch?v=4FROxZ5i67k

http://www.intothemusic.biz/wp-content/uploads/2010/04/space_shuttle_launch.jpg

Geostationary Satellites

• Satellites that have an orbital radius such that their period is the same as the period of the earth

http://www.splung.com/kinematics/images/satellites/geostationary.png

Earth above km 36,000

km 000,424

4

32

2

2

3

2

GMTr

GMrT

Which satellites would you want to be geostationary?

• Spy satellites• Television satellites• GPS satellites• Cell phone satellites• Weather forecasting satellites

Kepler’s Laws

Orbits are elliptical

• Eccentricity = 0, then circular

• Eccentricity = 1 then linear

• Most planets have an eccentricity close to 0

Equal areas in equal times

• The area of the triangle swept out by the object is equal for equal amounts of time

• The further the object is from the sun the lower the speed.

constantdtdA

Period-squared is proportional to a3

constant

4

v

3

2

32

2

22

2

aT

rGM

T

rGMr

rGM

Example Question: Earth Speed

• Where is the speed of the earth the greatest?

D

A C

B

Example Question: Seasons

• If the earth spends two less days in the winter half of the orbit than the summer half of the orbit (for N. hemisphere), what is its speed in winter vs. summer?

A) FasterB) SameC) SlowerD) Not enough information

Winter half is shorter and speed in winter is faster…

Main Points

• Satellites– Escape velocity depends on the gravitational field

of planet– Geostationary orbits 35,000 km from Earth

• Kepler’s Laws– Orbits are elliptical– Orbits sweep out equal area in equal time– Orbital period-squared is proportional at a3