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Gravitation Part II
One of the very first telescopic observations ever was
Galileo’s discovery of moons orbiting Jupiter. Here two
moons are visible, along with the shadow of another on
the surface of the planet.
Orbits
When in a circular orbit, an object is continually
falling ( under the influence of the earth’s gravity).
However, it is continuing to move tangent to the earth,
so it continues in a circular path at a constant speed.
Orbit Speed
Notice how the
projectile must
have enough
speed so that it
can continually
fall around the
earth.
When just
enough speed is
reached, 8 km/s,
a circular orbit
will result.
The first man made object to accomplish
this task of orbiting the earth was Sputnik,
launched on October 4th 1957. Sputnik II
even carried a dog as a passenger!
The audio is a
transmission from
Sputnik
Gravity Changes Direction
Fc
gravity
Tangential
velocity
Notice that gravity does not pull the satellite forward or
backward. Gravity simply acts as the centripetal force
to keep it going in a circular orbit.
Gravity Equals Centripetal Force
Since the centripetal force is provided by gravity, we
can equate the two forces:
FG  Fc
mM mv
G 2 
r
r
M
G  v2
r
GM
v
r
2
Notice the mass of
the satellite cancels
out!
The speed of an
circular orbiting
satellite depends only
on the radius,
gravitational constant
and mass of the earth!
Correct Distance Value
The radius used in the previous equation is measured
from the center of the orbit ( center of the earth). Shell
theorem!
Don’t just plug in the distance above the surface of the
earth!
Importance of Mass
This means that any mass satellite will have the same
orbital speed for any particular radius.
A giant orbiting satellite will have the same speed as a
tiny satellite in the same orbit.
However, it can be much more difficult to get that large
satellite into orbit in the first place…
Energy Costs
It turns out that it takes 62,000,000 J of energy to put
1kg outside of the Earth’s orbit. (62 MJ)
This is a large amount of energy, which is why it is
so costly and difficult to put people and objects into
space!
Elliptical Orbits
If an object is fired faster than 8 km/s, then it will follow
an elliptical orbit.
< 8km/s
8 km/s
> 8 km/s
Perigee =
largest velocity
Apogee =
smallest velocity
foci
In an elliptical orbit, the sum of the distances from the
foci is constant. As the foci get closer together, the orbit
becomes more circular, and less elliptical.
Energy is still conserved at any point in the orbit,
KE+PE=constant
Escape Speed
If an object travels fast enough, it may have sufficient
kinetic energy to overcome the gravitational potential
energy of its location.
For this situation, the speeding object would not fall
back to the surface of the planet. Instead, it would
escape the surface and even escape orbiting the planet!
Escape Speed Derivation
The escape speed of any body can be found if the
gravitation potential energy balances out the kinetic
energy. This total would equal zero.
KE  PEgrav  0
We have formulas for both of these quantities.
2
mv
Gm1m2

0
2
d
Canceling out the mass, and rearranging a bit, we
can get a formula for the velocity needed to escape
the gravity of a body.
Escape Speed Result
2
v Gm1

0
2
d
2
v
Gm1

2
d
2Gm1
2
v 
d
2Gm1
v
d
Notice the mass of the
satellite cancels out! The
remaining mass is the
planet being escaped.
Solve for v
This is the speed needed
to completely escape any
orbit of a body.
Weight and Weightlessness
You can feel weightless even though gravity is
acting on you.
Astronauts in free fall are still being pulled around
the earth by gravity.
Gravity Effects
When astronauts live long time periods in space, this
impacts their bodies. Bones may weaken, muscles
may lose mass, etc...
In the future, humans may design space ships that
create “artificial” gravity through circular motion...
Rotating Space Habitats
We can’t create a gravitational force, but we can use
centripetal force to act like gravity.
If a round space ship is large enough, and spins at
the correct rate, the centripetal force would simulate
gravity.
Kepler’s 1st Law:
Each planet moves in an elliptical orbit with the sun at
one focus of the ellipse.
foci
Kepler’s 2nd Law:
The line from the sun to any planet sweeps out equal
areas of space in equal time intervals. This is a
restatement of conservation of angular momentum.
An equal area covered
in a different month of
time.
Area covered in
1 month of time.
Kepler’s 3rd Law:
The squares of the times of revolutions (periods) of the
planets are proportional to the cubes of their average
distances from the sun.
2
T ~r
3
 4π  3
r
T  
 GM 
2
2
The second format gives the correct units and scaling.
Questions?
Your physics
assignment is:
Page 353+
P # 23,43,44,52