Transcript Gravity: Orbits and Tides

Lecture 7
• More on gravity and its consequences
– Orbits
– Tides and tidal forces
– The Three Kepler laws revisited
• Assigned reading: Down to end of Chapter
5.2 (no Relativity)
Announcements
• Late-enrolled students who still have to
pass HW1 should start to wrap up their
work.
– I will post solutions to HW1 as soon as all
Homework papers have been handed over.
Gravity
• What keeps us on the rotating Earth?
• Why don’t planets move in straight lines, but
orbit around the Sun instead?
The Universal Law of Gravity
m
The strength of the force only depends on M, m and r
mh
D
r
F=-
Mm
G 2
r
(G is the Universal
constant of gravity.
Mother Nature has
set its value, and we
cannot change it)
M
The Universal Law of Gravity
m
Now suppose that Earth shrinks in size, but keeps its mass
(gravitational collapse).
Would the force of gravity on the Moon change?
Would the force of gravity on the human change?
mh
r
D’
F=-
Mm
G 2
r
(G is the Universal
constant of gravity.
Mother Nature has
set its value, and we
cannot change it)
M
The Universal Law of Gravity
m
Now suppose that Earth shrinks in size, but keeps its mass
(gravitational collapse).
Would the force of gravity on the Moon change?
No, same M and m, and same r
Would the force of gravity on the humans change?
Yes, same M, same mh, but different D
mh
r
D’
F=-
Mm
G 2
r
(G is the Universal
constant of gravity.
Mother Nature has
set its value, and we
cannot change it)
M
… so what is an orbit?
Suppose you dig a hole through the
center of Earth and pump all the air
off.
And then toss an object into the
hole.
Can you visualize the motion of that
stone?
… so what is an orbit?
Suppose you dig a hole through the
center of Earth and pump all the air
off.
And then toss an object into the
hole.
Can you visualize the motion of that
stone?
The object will move up and down,
periodically, for ever.
Its speed will be highest when it
transits at the center of Earth…
… and the slowest in proximity of
the surface, when it stops and
reverts its motion
THAT IS AN ORBIT!
(with velocity always having the
same direction and alternating
sense)
Now, let’s make
an orbit whose
velocity does
change direction
If I do not give
the object enough
speed, its
trajectory will
eventually
intersect the
ground
V=8km/s
At the right speed, namely the ORBITAL VELOCITY,
the vertical downward motion and the horizontal
outward motion combine to produce the circular orbit
Suppose you
shrink Earth
keeping the same
mass.
Now the
trajectory of even
the slowest moving
object does bot
intersect ground.
That is an elliptic
orbit!
The circular orbit
is only a special
case of an orbit
where the speed
is always the same
In no circular
orbits the speed
changes along the
orbit
V=8km/s
At the right speed, namely the ORBITAL VELOCITY,
the vertical downward motion and the horizontal
outward motion combine to produce the circular orbit
Important: an
object in orbit is
free falling.
Thus, it feels no
gravity, since it
is falling onto
the source of
gravity
V=8km/s
At the right speed, namely the ORBITAL VELOCITY,
the vertical downward motion and the horizontal
outward motion combine to produce the orbit
Geosynchronous Orbits
… so why don’t planets just fall
into the sun?
M1
M2
… because they miss (that is, they
have enough tangential velocity to
always miss)
v
Fg
Fg
M1
This is the concept of an orbit.
M2
Why doesn't the earth fall to
the sun?
• It has a velocity
and it has
inertia!
• Force of gravity
causes change in
the direction of
velocity --acceleration.
• The earth is
falling towards
the sun all the
time!
Orbital Velocity
• Another way to understand orbits: in orbit,
force of gravity and centrifugal force
balance each other:
– mv2/r = GMm/r2
• Solving for v gives:
•v=
1/2
[GM/r]
• For example, in the case of the Moon:
• v = 1.02 km/s ~ 3,600 km/h
Quiz


Astronauts inside the space shuttle float
around because ____
they are falling in the same way as the
space shuttle.
If you are in a free-falling elevator, you
are massless. (true or false)
false
You are a shuttle astronaut returning after
attempting to fix the ISS with a hammer. As you
are jetting back to your shuttle, your lifeline
breaks, your jets run out of fuel, your radio goes
dead, and you miss the shuttle. To get back
safely, you should:
• use a swimming motion with your arms and legs
• throw the hammer at the shuttle to get
someone’s attention
• throw the hammer away from the shuttle
• make a hammering motion in the direction of
the shuttle
• make a hammering motion away from the
shuttle
Escape velocity:
how can I free
myself from the
pull of gravity of
a given body?
V=8km/s
Can I accelerate myself at such a speed that I will
start spiraling out and eventually abandon the planet
instead of keep orbiting it? How fast is such speed?
Gravity: depends on mass and on distance
So, what is escape velocity?
m
Now suppose that Earth shrinks in size, but keeps its mass
(gravitational collapse).
Would the force of gravity on the Moon change?
No, same M and m, and same r
Would the force of gravity on the humans change?
Yes, same M, same mh, but different D
mh
r
D’
F=-
M
Mm
G 2
r
You can change the
force of gravity either by
varying the mass(es), or
by varying the distance,
or both
Such an escape velocity will have to depend how
far away I am from the center of gravity of the
body that generate the gravity field.
Escape Velocity
• Kinetic Energy (energy due to motion):
• Ek = ½ m v2
• Potential Energy (energy due to position):
• Eg = GMm/r
• To escape, Kinetic Energy has to be larger
(or at least equal) than Potential Energy:
• ½ m v2 >= GMm/r
• Solving for v:
• vesc = [2GM/r]1/2
• For example, to escape Earth:
• vesc = 11.2 km/s = 40,320 km/h
Tides
• Tides occur because of the gravitational
pull of the Moon on the Earth.
• The Moon pulls more strongly the closer
side of Earth than the one further away.
• It literally stretches Earth
• Water (and air) get stretched much more
easily than rock.
• This, in essence, is what makes tides
• Note that the Sun does the same, too
Let’s build this one step at a time
low tide
high tide
Looking down
on the Earth
low tide
high tide
Moon
Exaggerated view
of tides
We have two high tides because of
the stretching action
Moon
The Moon exerts a stronger gravitational pull
on the near side of the Earth than on the far
side of the Earth. This difference in pull
causes the Earth to stretch!
low tide
Tides
high tide
high tide
low tide
Rotation of Earth
Exaggerated view
of tides
The tides aren’t quite aligned with the EarthMoon line because it takes time for the water
to slosh over.
Friction drags the tidal bulges eastward
out of the direct earth-moon line
Friction wastes energy, and this energy comes
at the expense of Earth’s rotational energy
Earth's rotation slows down by 0.0023 s/100 years as a result.
Only 900 million years ago, Earth' day was 18 hrs long.
The moon's orbit is growing larger by about 4 cm/yr.
Acceleration of the Moon’s
Orbital Motion
Earth’s tidal bulges are
slightly tilted in the direction
of Earth’s rotation.
Gravitational force pulls
the moon slightly
forward along its orbit.
Spring and Neap Tides
The Sun is also
producing tidal
effects, about
half as strong as
Spring tides
the Moon.
• Near Full and
New Moon,
those two effects
add up to cause
spring tides.
• Near first and
third quarter, the
two effects work
at a right angle,
Neap tides causing neap
tides.
Discussion Question
• Why does the Moon always show the same
face to the Earth? (hint: think of the tidal
pull of the Earth on the Moon)
• The friction of changing Moon’s tidal bulge
dissipated rotational energy, and put
Moon’s rotation to such a value that the
tidal bulge does not move any longer: the
synchronous rotation with orbit
Moon
Earth
The near face
is pulled harder
than the far face.
Earth
Survey Question
M 1M 2
Fg  G
2
d
If our Sun mysteriously turned into a black hole of
the same mass but 10 times smaller diameter,
what would change about the Earth’s orbit?
1) it would be 10 times smaller in radius
2) it would spiral into the black hole
3) nothing would change
4) it would spiral away from the black hole
5) it would be 10 times larger in radius
Angular Momentum
• Depends on the geometry, the mass, and the rotational
velocity of an object.
• Angular momentum is conserved.
– A spinning wheel wants to keep spinning.
– A stationary wheel wants to keep still.
• Angular momentum is also a vector quantity – this means
that the direction of the axis of rotation is significant
and resistant to change. Its INTENSITY is:
• P = m·v·r
• m is the mass, v the speed, and r the distance to
the center of rotation
Everyday Examples of the
Conservation of Angular Momentum
• Riding a bike
• Spinning a basketball on your finger
• A spinning ice skater
Kepler’s Laws of Planetary Motion Revisited
1
The orbits of the planets are ellipses, with the Sun
at one focus of the ellipse.
2
Planets move proportionally faster in their orbits
when they are nearer the Sun.
3
More distant planets take proportionally longer to
orbit the Sun
Kepler’s Three Laws of Orbits
1. The orbit of each planet about the Sun is an
ellipse with the Sun at one focus.
Kepler’s Three Laws of Orbits
2. As a planet moves around it’s orbit, it sweeps
out equal areas in equal times.
1 month
Figuring out orbital velocities with
angular momentum
• The angular momentum of an object (like a
planet) moving in a circle (like an orbit!) is:
L = m·v·r
Angular Momentum is always conserved
v
m
m = mass of planet
v = velocity of planet
r = orbital radius of planet
r
Kepler’s Three Laws of Orbits
3. A planet’s Period (the time it takes to
complete one orbit) is related to its
average distance to the sun.
(orbital period in years)2 = (average distance in AU)3
P2 = a3
Notice that there is nothing stated about the
planet’s or Sun’s mass here!
Newton's laws of motion imply Kepler's Laws.
In orbit, centrifugal force balances gravitational force
Fc = mv2/r
+
Fg = GmM/r2
v = 2pr/P
v2 = 4p2r2/P2
mv2/r = GmM/r2
----> 4p2r2/P2 m/r = GmM/r2
----> r3= G/4p2 M P2
If you express P in years and r in AU, then the term G/4p2
cancels out and you have Kepler Third Law.