Download Gravity and Orbits

Gravity and Orbits
Announcements
n  Homework # 2 is available in SPARK/OWL. It is due on
Friday, October 7th.
n 
EXAM 1 will take place next Thursday, October 6th.
¨  IMPORTANT:
Bring your student ID. IDs will be checked
at the end of the exam, when you give back your exam
sheet. No ID = No EXAM!
¨  The exam will consist of ~40 multiple-choice questions;
you have 1 hr and 10 minutes (plenty of time!).
¨  The exam will follow the open-notes/open-book policy. Not
permitted: talking to or copying from your neighbor,
computers, PDAs or any device with wireless/wired
connection (Violation = 0 points for the Exam).
¨  Covered: all lectures (including the lecture of October 4th)
and Units 1-3, 5-20).
Today s Goals
1) Discuss the universal law of gravity.
2) Discuss how the law of gravity defines
the orbital motion of celestial bodies.
Assigned Reading
n 
Units 16 (second half), 17, and 18.
Gravity
n  What
force is responsible for motions in
the universe?
n  What force makes objects fall?
n  What keeps us on the rotating Earth?
n  Why don’t planets move in straight lines,
but orbit around the Sun instead?
Gravity
n  We
can summarize the universal law of
gravitation with the following statements:
§ 
Every mass attracts every other mass through the
force of gravity.
§ 
If mass #1 exerts force on mass #2, and mass#2
exerts force on mass#1, the force must depend on
both masses, and is directly proportional to them.
§ 
The force of attraction is inversely proportional to the
square of the distance between the masses.
The Law of Gravity
Near Earth s
surface
M 1M 2
Fg = G
2
d
G=
6.67x10-11
Fg = gM 2
m3/kg/s2
M1
g =G 2
d
2
= 9.8m/s
d
M1
M2
M 1M 2
Fg = G
Survey Question
d2
Two equal masses, m, separated by a distance,
d, exert a force, F, on each other due to their
gravitational attraction. How large would their
gravitational attraction be if the distance
between them was doubled?
1) ¼ F
2) ½ F
3) F
4) 2F
5) 4F
M 1M 2
Fg = G
2
Survey Question
d
Two equal masses, m, separated by a distance,
d, exert a force, F, on each other due to their
gravitational attraction. How large is the
gravitational force between an object of mass
m and an object of mass 2m separated by the
same distance d?
¼F
½F
F
2F
4F
M 1M 2
Fg = G
Survey Question
d2
If aliens magically turned our Sun into a black hole of
the same mass but 10 times smaller in 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
A Special Consequence of the
Law of Gravitation
n  If
we combine Newton s law of gravity
with Newton s 2nd law, we see that the
acceleration of an object due to a
gravitational force is independent of the
accelerating object s mass.
M1 M 2
Fg = G
d2
so
!
and Fg = M 1a1
M2
a1 = G 2
d
(on Earth surface a1=g)
So… why don t planets just fall
into the sun?
M1
M2
… because they miss it (that is,
they have enough tangential
velocity to always miss)
v
Fg
Fg
M1
This is the concept of an orbit.
M2
Is the earth falling into the sun?
n 
n 
n 
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!
V=8km/s
Newton s `thought experiment!
Survey Question
Why do astronauts float around inside the space
shuttle?
1) there is no gravity in space
2) they are falling at the same rate as the
space shuttle
3) they are above the Earth s atmosphere
4) their mass is smaller
5) more than one of these
n  The
§ 
§ 
§ 
Tethered Rock
Tie a rock to a string, and start rotating it
overhead.
You feel the rock trying to `pull away, but if
you hold on the string, you oppose a force to
the `pulling away force of the string.
What is happening?
§ 
§ 
The `rotating rock has an inertia, and the rotating
motion imparts it an acceleration (change of
direction). If the rock could break free, it would
move in a straight line (a tangent of the circle).
Mass x acceleration = Force
Your tether opposes a Force (Centripetal Force) to
that Force
Centripetal Force
n 
n 
n 
n 
Planets orbiting the Sun are subject to centripetal force,
the same force that keeps a weight tethered to a string
from `flying away when you rotate it.
The centripetal force is the force that keeps objects on a
curved path. In the case of planets, the centripetal force
is due to gravity, and compensates the planet s inertia
and constant acceleration [Newton s Third Law])
The larger the velocity, the
larger the force
The farther away the weight,
the smaller the force
Fc = m v2 / R
Centripetal Force = Gravitational Force:
In an orbit, the gravitational force must equal the centripetal
force
F c = Fg
Fc = mv2/R
mv2/R = GmM/R2
Fg = GmM/R2
or: v2 = GM/R
In a circular orbit: v = 2πR/P
and:
R3= G/4π2 M P2
If you express P in years and R in AU, then the term GM/4π2 cancels out and
you have Kepler’s Third Law: R3 = P2.
Kepler s Third Law of Orbits:
Revisited
3. 
A planet s Period (the time it takes to
complete one orbit) is related to its
average distance to the sun.
G/4π2 M P2 = R3
(orbital period in years)2 = (average distance in AU)3
P2 = a3
§ Notice that there is nothing stated about the
planet s mass here!
§ The period/radius is independent of the
planet s mass!
The same formula:
R3= G/4π2 M P2
can be used to derive the mass of the
Sun (if R and P are measured in MKS units):
M = 4π2/G (R3/P2 ) = 2.0 E+30 kg
R=Earth-Sun distance
P=Earth s orbital period
Orbital Velocity
To remain in orbit, an
object s gravity and
centripetal force must
balance each other:
§  Mv2/R = GMm/R2
n  Solving for v gives:
vorb = √GM/R
n  For the Earth:
vorb = 7.9 km/s ~
28,440 km/h =
17,665 miles/h
n 
Escape Velocity
It is the velocity needed by an
object (a rocket) to escape a
planet s gravitational pull
n  To escape, you must
have a velocity that
compensates for the
gravitational pull:
n 
2GM
v=
d
Vesc, Earth = 11.2 km/s
Vesc, Moon = 2.4 km/s
Escape Velocity
The velocity to compensate for the
gravitational pull must be calculated from
the surface of the planet.
It is the same for the rocket and
for the cannon ball!