9.4 – Orbital Motion & Kepler’s 3rd Law

Orbital Motion in the Solar System

Kepler’s LawsIn the early 1600s Kepler summarized the carefully collected data of Tycho Brahe and proposed three laws:

1st Law: The Law of ellipsesThis just said that planets moved in ellipses around the sun.

The ends and length of the string in the picture is fixed and the pencil draws out an ellipse.

2nd Law: The Law of equal areasThe area swept out by the orbit in a certain time is constant.

You don’t need to know these, only the 3rd Law!

Use this simulation link to investigate Kepler’s Third Law: http://bit.ly/keplers3rdlaw or CLICK ON THE PICTURE

3rd Law: Time Period and Radius of orbit relationship

Derivation of Kepler’s 3rd Law

The 3rd Law: The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit

When something is in orbit, Centripetal Force is caused by Gravitational Force.

𝑚𝑣2

𝑟=𝐺𝑀𝑚

𝑟 2𝑣=

2𝜋𝑟𝑇

+

𝑚( 2𝜋𝑇 )2

𝑟=𝐺𝑀𝑚𝑟 2

𝑇2

𝑟 3= 4𝜋

2

𝐺𝑀

𝑇 2∝𝑟3

Learn This!!

Energy of an orbiting body - derivationsLearn This!!

1. Remember: 𝑃𝐸=−𝐺𝑀𝑚𝑟

𝐾𝐸=12𝑚𝑣2and

2. Remember for objects in orbit: FC = FG

𝑚𝑣2

𝑟=𝐺𝑀𝑚

𝑟 2

12𝑚𝑣2=

𝐺𝑀𝑚2𝑟

Centripetal Force

Gravitational Force

KE -

𝐸𝑇=𝐾𝐸+𝑃𝐸=𝐺𝑀𝑚2𝑟

−𝐺𝑀𝑚𝑟

For objects in orbit their KE is -

𝐸𝑇=−𝐺𝑀𝑚2𝑟

For objects in orbit their total Energy is

𝐾𝐸=𝐺𝑀𝑚2𝑟

Graphs of KE, PE and Total Energy

𝐾𝐸=𝐺𝑀𝑚2𝑟

𝑃𝐸=−𝐺𝑀𝑚𝑟

Graphs of KE, PE and Total Energy

Graphs of KE, PE and Total Energy

𝐾𝐸=𝐺𝑀𝑚2𝑟

𝑃𝐸=−𝐺𝑀𝑚𝑟

Energy, Orbits & Escape SpeedTry these two simulations to help your understanding:

1. Energy & Orbits 2. Energy & Escape Speed

http://webphysics.davidson.edu/physlet_resources/bu_semester1/c17_orbits.html

http://webphysics.davidson.edu/physlet_resources/bu_semester1/c17_escape.html BLUE = KE PINK = PE BLACK = Total Energy

Weightlessness

What’s happening?

What happened to gravity?

Weightlessness

The plane follows a strange parabolic flight path that gives about 30 seconds of weightlessness

WeightlessnessIf everything is accelerating downwards at the same rate then we get the illusion of weightlessness.

Because the space station is orbiting the earth, everything in it is being accelerated towards the centre of the earth.

This is what happens with all circular motion.