Download Kepler`s first and second law state that

Location of Lecture 28 Material in the 7th Edition of your
Textbook
 = Torque: Pages 248-250;   l  F (right hand rule for vector product)
I = Moment of Inertia: pages 261-262; I is a scalar
L = Angular Momentum: pages 271-274; L  I
Vector Nature of:
 , L,  ,  .
page 237, Fig. 8.17 for  and  .
For  see page 276, Eq. 9.1 and problem 9.9 on page 279.
Newton’s 2nd law for rotational motion; Eq. 9.7 on page 262.
Updated Kepler’s Laws
1. The Law of Orbits: All planets move in elliptical orbits, with the Sun at one of the
foci.
2. The Law of Areas: A line that connects a planet to the Sun sweeps out equal areas
in the plane of the planet’s orbit in equal times: that is the rate
dA / dt  constant  L / 2mp
L = angular momentum of the planet
mp= mass of the planet
L=I1ω1 = I2ω2  constant of the eliptic orbit, where v  r
This is the consequence of the fact that no net torque is exerted on the planet
( net  0)
3. The Law of Periods: The square of the period of any planet is proportional to the
cube of the semi major axis of its orbit
T 2 /a 3 = constant
For a circle where e = 0, a = r
4 2
T /r 
GM s
2
3
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The shape of an ellipse is determined by its eccentricity “e”.
e
ab
ab
Where a and b are the semi major and minor axis of the ellipse.
This is tabulated for all the planets. It is largest for Pluto (eP=0.248) and Mercury
(eM = 0.24). This results in a dramatic variation in velocity as Mercury executes a
revolution along the ellipse.
X Perigee
Apogee X
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Material from the last Lecture for the Final Exam
No questions on impulse and eclipse
Chapter 9: 1, 9, 57, 58 Conceptual questions 1 and Example 15
Understand Equations: 9.7 and 9.10
Understand the figures: 9.28 and 9.29
Understand that   0 on a planet in elliptical orbit.
Understand that from Newton’s 2nd law for rotation.

t
  I  I
If  =0 ,  =
(Eq. 9.7 page 262)

 0    constant,
t
thus the angular momentum of a planet in elliptical orbit.
L  I
(Eq. 9.10 page 272)
is also constant.
The final exam will cover homework set 13 to 15.
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