Transcript Slide 1

Lecture 9
ASTR 111 – Section 002
Outline
• Exam Results
• Finish Chapter 4
– Kepler’s Laws Review
– Newton’s Laws
Kepler proposed elliptical paths for the
planets about the Sun
• Using data collected by
Brahe, Kepler deduced
three laws of planetary
motion:
1. the orbits are ellipses
2. a planet’s speed varies
as it moves around its
elliptical orbit
3. the orbital period of a
planet is related to the
size of its orbit
Lingering questions
• Kepler’s laws are not so “clean”
• Need to explain
– Why orbits of planets are elliptical
– Why distance from Sun is related to orbital period
– Why planet velocity changes during orbit
• Also want a recipe that gives good predictions of
when eclipses will occur, where the planets will
be in the future.
Lingering questions
• Kepler’s laws are not so “clean”
• Need to explain
–
–
–
–
Why orbits of planets are elliptical
Why distance from Sun is related to orbital period
Why planet velocity changes during orbit
Why people on the south pole don’t fall into space …
• Also want a recipe that gives good predictions of
when eclipses will occur, where the planets will
be in the future.
Isaac Newton
Isaac developed
three principles,
called the laws of
motion, that apply
to the motions of
objects on Earth
as well as in
space
Isaac Newton
Isaac was a little
nutty – See short
biography “Newton”
by James Gleick
Newt’s “Principles” (Laws of Motion)
1. The law of inertia: a body remains at rest,
or moves in a straight line at a constant
speed, unless acted upon by a net outside
force
2. F = m x a: the force on an object is directly
proportional to its mass and acceleration,
provided the mass does not change
3. The principle of action and reaction:
whenever one body exerts a force on a
second body, the second body exerts an
equal and opposite force on the first body
Group Question
• An object at rest tends to stay at rest.
An object in motion tends to stay in
motion.
–What is wrong with this statement?
–Why don’t we observe “objects in
motion tending to stay in motion”
more often?
Newton’s Law of Universal
Gravitation
A number
(T.B.D.)
 m1m2 
Force  G 2 
 r 
r
Mass m1
Mass m2
• Mass and Weight are not the
same
–Mass refers to how much stuff is
in an object (atoms, molecules,
etc).
–Weight refers to how much that
stuff will push down on a scale.
This depends on what planet
you are on.
Newton’s Law of Universal
Gravitation
Mass m1
 m1m2 
Force  G 2 
 r 
A spring
Mass m2
Weight is a number that
tells you about how
much this spring will
compress. It depends
on m1 and r.
How to get Weight = mass x gravity
 m1m2 
Force  G 2 
 r 
Mass of Earth
 Gm1 
Force  m2  2   m2 g
 r 
2
Radius of Earth
m/s
g  9.8
What about Bob Beamon?
• The law of universal gravitation
accounts for planets not falling
into the Sun nor the Moon
crashing into the Earth
v
v
m2
m2
 m2v
Force  
 r
2



(You will need
to take my
word on this
equation)
 m2v
Force  
 r
v
m1
m2
2


v
Now suppose Earth
m2
provides “pull” instead
of string and arm
 m1m2 
Force  G 2 
 r 
 m1m2 
Force  G 2 
 r 
(Force that can be provided)
 m2v

r

 m2v
Force  
 r
(Force needed to keep it in orbit)

 m1m2 
  G 2 
r



Gm1
2
v 
r
2
2



Is this right?
• G = 6.7 x 10-11 N.m2/kg2
• m1 = 2 x 1030 kg
• Mars
– Orbital velocity = 24 km/s
– Distance from Sun = 228 x 109 km
• Earth
– Orbital velocity = 30 km/s
– Distance from Sun = 150 x 109 km
Gm1
v 
r
2
Is this right?
• G = 6.7 x 10-11 N.m2/kg2
• m1 = 2 x 1030 kg
• Mars
– Orbital velocity = 24 km/s
– Distance from Sun = 228 x 109 km
• Earth
– Orbital velocity = 30 km/s
– Distance from Sun = 150 x 109 km
Gm1
v 
r
2
Gm1
v 
r
2
6.7 10 x 2 10
24,000 
228,000 109
-11
30
2
-11
30
6.7

10
x
2

10
2
30,000 
9
150,000 10
Compare
• Kepler’s 3rd law relates orbital speed and
radius
• Newton’s law of gravitation was used to
derive a relationship between orbital
speed and radius
• Both will give the same answer. Which is
“better”?
To get something in orbit, you need a
special horizontal velocity
• The law of universal
gravitation accounts for
planets not falling into the
Sun nor the Moon crashing
into the Earth
• Paths A, B, and C do not
have enough horizontal
velocity to escape Earth’s
surface whereas Paths D,
E, and F do.
• Path E is where the
horizontal velocity is
exactly what is needed so
its orbit matches the
circular curve of the Earth
Question
• How far would you have to go from Earth
to be completely beyond the pull of
gravity?
• Suppose the Earth was 2x its current
radius (with the same mass). How would
your mass change? How would your
weight change?
• Given that Earth is much larger and more massive than the
Moon, how does the strength of the gravitational force that
the Moon exerts on Earth compare to the gravitational force
that Earth exerts on the Moon? Explain your reasoning.
• Consider the following debate between two students about
their answer to the previous question. Do you agree or
disagree with either or both students? Explain.
– Student 1: I thought that whenever one object exerts a force on the
second object, the second object also exerts a force that is equal in
strength, but in the other direction. So even if Earth is bigger and
more massive than the Moon, they still pull on each other with a
gravitational force of the same strength, just in different directions.
– Student 2: I disagree. I said that Earth exerts the stronger force
because it is way bigger than the Moon. Because its mass is bigger,
the gravitational force Earth exerts has to be bigger too. I think that
you are confusing Newton’s third law with the law of gravity.
• How would the strength of the force between the Moon and
Earth change if the mass of the Moon were somehow made
two times greater than its actual mass?
Earth
Mars
• In the picture, a spaceprobe traveling from Earth to
Mars is shown at the halfway point between the two
(not to scale).
• On the diagram, clearly label the location where the
spaceprobe would be when the gravitational force by
Earth on the spacecraft is strongest. Explain.
• On the diagram, clearly label the location where the
spaceprobe would be when the gravitational force by
Mars on the spacecraft is strongest. Explain your
reasoning.
• When the spacecraft is at the halfway point, how does
the strength and direction of the gravitational force on
the spaceprobe by Earth compare with the strength
and direction of the gravitational force on the
spaceprobe by Mars. Explain your reasoning.
Earth
Mars
• Where would the spaceprobe experience the
strongest net (or total) gravitational force
exerted on it by Earth and Mars? Explain
your reasoning.
Earth
Mars
• If the spaceprobe had lost all ability to control its
motion and was sitting at rest at the midpoint
between Earth and Mars, would the spacecraft stay
at the midpoint or would it start to move.
– If you think it stays at the midpoint, explain why it would
not move.
– If you think it would move, then (a) Describe the direction
it would move; (b) describe if it would speed up or slow
down; (c) describe how the net (or total) force on the
spaceprobe would change during this motion; and (d)
identify when/where the spaceprobe would experience
the greatest acceleration.
Earth
Mars
• Imagine that you need to completely stop the motion
of the spaceprobe and have it remain at rest while you
perform a shutdown and restart procedure. You have
decided that the best place to carry out this procedure
would be at the postion where the net (or total)
gravitational force on the spaceprobe by Mars and
Earth would be zero. On the diagram, label the
location where you would perform this procedure.
(Make your best guess; there is no need to perform
any calculations here.) Explain the reasoning behind
your choice.
• Your weight on Earth is simply the gravitational force
that Earth exerts on you. Would your weight be more,
less, or the same on the Mars. Explain your
reasoning.
5
Earth
1
2
3
Sun
4
Orbital Mechanics
Tides
http://en.wikipedia.org/wiki/Tide