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Satellite Motion
Warm Up:
Given that the centers of the Earth
and Moon are 3.85 x 108 m apart and
the mass of the moon is 7.36 x 1022
kg,
a) calculate the force of gravity
between them.
b) find the gravitational field strength
of the moon felt at the center of the
earth.
This is a huge amount of force! Why
doesn’t the Moon
get
pulled
into the
Yeah,
that's
great,
but how does it
Earth???
get into orbit?
The moon is in orbit around the
Earth.
Orbit: perpetual motion, usually
elliptical, around a celestial body.
Consider this:
If you throw a baseball at different
speeds it will travel forwards, but
eventually fall to the ground.
Newton's Cannon:
Now imagine that you are standing on
top of a very high mountain (ignore air
friction) and you just happened to bring
your trusty baseball cannon.
The baseball is still bei
Notice that at all times the
baseball is being pulled towards
the Earth by gravity.
Because of this we say that
objects in orbit are in free fall –
always falling towards the center
of the Earth, but never reaching
it.
Centripetal Force
Recall that Fc is any force that keeps
objects moving in a circle.
In the case of a satellite or orbiting
spaceship (or moon), gravity provides
the centripetal force.
F c= F g
Orbital Speed
F c= F g
This relationship
allows us to find the
speed necessary for
an object to achieve
orbit:
2
mv G M m
=
2
r
r
GM
2
v =
r
√
GM
v=
r
Orbital Speed
Ex 1. So just how fast does your
cannonball have to go? Let's say the
peak of your mountain is at an altitude
of 3.0 km above Earth's surface.
Orbital Speed
Ex 2.The International Space Station
orbits at an altitude of 350 km. What is
its orbital speed?
Orbital Acceleration
Recall: Orbital
velocity is can also
be found using:
d 2 πr
v= =
t
T
We can put our formula for v together with
our formula for ac:
Orbital Acceleration
Ex 3. Find centripetal acceleration
necessary to keep Neptune orbiting
around the Sun every 164.97 Earth
years.
Two Orbital Accelerations?
Recall: Orbital
acceleration is also
the acceleration due
to gravity:
GM
g= 2 = ac
r
Combining our two formulas for ac:
2
G M 4π r
ac = 2 =
2
r
T
Orbit
Ex 4. How long does it take the ISS to
orbit the earth
(alt = 350 km)?
Orbit
Ex 5. Calculate the radius of the earth's
orbit around the sun?
Formula Recap
F g= G
m1 m2
r
2
F g= m g
T=
GM
g= 2 = a c
r
v 4π r
ac= =
2
r
T
2
mv
F c=
r
2
√
2
4π r
GM
√
2 3
GM
v=
r
d 2 πr
v= =
t
T
Try It:
1) Find the period of Pluto's orbit
around the sun.
2) Find Mercury's speed as it orbits the
sun.
3) Find strength of the Sun's gravitation
field on Venus.