Transcript January 26

Review Question
What is retrograde motion?
Review Question
Explain why the phases of Venus proved
to Galileo that Venus must orbit the Sun
when the Moon also has phases and it
orbits the Earth.
Review Question
What are Kepler’s laws?
Review Question
What are Newton’s laws?
Discussion
Consider an object in uniform circular motion.
That is, an object traveling in a circle with a
constant speed. Is there a force acting on this
object? Why or why not?
Discussion
Consider an object in uniform circular motion.
That is, an object traveling in a circle with a
constant speed.
How is the velocity of the object changing and
how must the force on the object be directed to
change its velocity in this way?
Discussion
Is there a force acting on the Moon? How
can you tell?
Speed and distance
The faster you throw something, the farther it
goes before it hits the ground.
The Moon is falling
The nearly circular orbit of the Moon is
constantly accelerating toward the Earth.
The Moon is constantly falling toward the
Earth.
Discussion
Newton’s third law tells us that the force of
the Sun on the Earth is the same as the force
of the Earth on the Sun. Why then does the
Earth orbit the Sun instead of the other way
around?
The Sun is more Massive
The force of the Earth on the Sun is the same
as the force of the Sun on the Earth.
But, the Sun is 333,000 times more massive
than the Earth. From Newton’s second law the
Earth will be accelerated 333,000 time more
than the Sun.
Orbits
Because of Newton’s third law, it is not exactly
correct to say that the Earth orbits the Sun or
the Moon orbits the Earth. Instead, both
objects, the Earth and the Sun or the Moon
and the Earth, orbit a common point called the
center of mass.
Discussion
If two stars with equal mass are held
together by gravity, describe their orbits.
Newton’s Universal Law of
Gravity
• Every mass attracts every other mass
through a force called gravity
• The force is directly proportional to the
product of their masses
• The force is inversely proportional to the
square of the distance between them
Newton’s law of gravity
M 1M 2
F G
2
d
Discussion
Consider the gravitational force between two
objects with mass M1 and M2 separated by a
distance d. How would the gravitational force
change if the distance between them increases
to 3  d. How will it change in the distance in
decreased to 0.1  d?
Discussion
Suppose a new planet is discovered out in
the Kuiper belt. This planet has twice the
mass of the Earth but is also twice the size.
Is the surface gravity of this new planet
greater than, less than or the same as the
surface gravity of the Earth?
Gravitational forces between
spherical masses
d
The distance to use is the distance between the
two spheres centers.
Discussion
You dig a very deep mine shaft. As you get
closer to the center of the Earth, does your
weight increase or decrease? Why?
(Hint: consider what the force of gravity will be
at the very center of the Earth.)
Escape Velocity
If an object is thrown up with a high
enough velocity it will leave Earth forever.
For Earth this velocity is about 11 km/sec.
Discussion
If the Earth were the same mass but
twice the radius, would the escape velocity
be greater or less than 11 km/sec? Why?
Where does it come from?
For a planet to orbit the Sun, it must constantly
accelerate toward the Sun, otherwise it would
fly off in a straight line at a constant velocity.
Discussion
If I drop two balls at exactly the same time and
from exactly the same height, with each ball
exactly same shape and size but very different
masses, which ball hits the ground first?
Discussion
If I drop two balls at exactly the same time and
from exactly the same height, with each ball
exactly same shape and size but very different
masses, which ball has the greater force acting
on it?
Why proportional to the mass?
All objects, regardless of their mass, fall with
the same acceleration. Because F = ma,
a
Fgrav
m
To keep the acceleration
constant, the force must vary
proportional to the mass.
Discussion
If I swing a ball in a circle over my head
with a short string and a long string with
each ball moving at the same speed,
which ball has the greater force acting on
it? Explain why.
Why the square of the distance?
An inverse square central force law is
required to get stable orbits that are conic
sections, i.e. orbits that are elliptical.
Kepler’s 1st law
The planets move in ellipses with the Sun
at one focus.
Conic section orbits are a natural outcome
of the 1/d2 nature of the gravitational
force, in fact the only force law that gives
stable orbits.
Discussion
A ball held on a string is coasting around in a
large horizontal circle. The string is then
pulled so the ball coasts in a smaller circle.
When coasting the smaller circle its speed is
a) Greater
b) Less
c) Unchanged
Kepler’s 2nd law
A line drawn from the planet to the Sun
sweeps out equal areas in equal intervals
of time.
Just another way of saying angular
momentum is conserved which comes
from Newton’s 2nd law of motion.
Conservation of angular
momentum
Any object that is spinning or orbiting has
angular momentum which is equal to the
mass × velocity × radius.
In the absence of an external torque, or
twisting force, the angular momentum
will remain constant.
Discussion
If angular momentum is conserved and I
decrease the radius what happens?
Newton’s form of Kepler’s 3rd law

 3
4
P 
a
G
(
m

m
)
1
2 

2
2
Discussion
The Moon’s mass (consider it all at the center of
the Moon) attracts every atom on the Earth. If
every atom has exactly the same mass, is the
gravitational attraction of the Moon the same
on each atom on the Earth? Explain.
Tidal Forces
Different distances from a mass will experience
different forces and therefore different
accelerations.
Discussion
Consider yourself sitting on the center ball,
number 2 in the previous diagram. How will
you perceive the motion of the other two
balls relative to you?
Spring Tides and Neap Tides
The Sun also contributes to the Earth’s tides.
When the Sun and Moon line up to produce
higher tides, this is called spring tides.
Neap tides occur when the Moon and Sun
Partially cancel each other.
What phases of the Moon do spring and
neap tides occur?
Tidal Forces
Tidal forces act to stretch things out along
the direction of a gravitating source and
squeeze them in the middle.
Discussion
Does it matter that all the atoms on the earth
have the same mass? Or that all three billiard
balls have the same mass? Why or why not?
Tidal Friction
Discussion
Due to tidal friction, what happens to the
rotation rate of the Earth over time?
The Earth-Moon system
The Moon exerts a force on the tidal
bulges on Earth, trying to twist the tidal
bugles back to face the Moon. This torque
slows the rotation of the Earth.
Discussion
The Moon is slowing the Earth’s rotation
and the Earth is losing angular momentum.
Where does that angular momentum go?
Orbit of the Moon
Due to conservation of angular momentum, the
slowing of Earth’s rotation requires the Moon to
move faster in its orbit. Because the force of
Earth’s gravity does not change, this means the
Moon is getting farther away, by about 2 inches
per year.
Discussion
What do you think will be the ultimate fate
of the Earth Moon system?
Fate of the Earth Moon System
Over billions of years the length of the month
and the length of the day will be the same,
about 47 days. The Moon will stationary in the
sky and will be too small for total solar eclipses.
Discussion
Will the Moon still have phases?
Discussion
What would happen to the Moon if its
orbital period was less than the Earth’s day?