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Board Work
1. A satellite revolves
around its planet in a
perfectly circular orbit at a
constant speed.
a. At each of the four
positions, draw a vector
representing the net force
on the satellite. Label all
the force vectors F.
b. At each position, draw a vector to represent the
satellite’s velocity. Label each vector v.
Board Work
c. Are all F vectors the same
magnitude?
d. Are all v vectors the same
magnitude?
e. What is the angle between
F and v at any position of
the satellite?
f. Is any component of F ever
parallel to v?
g. Is the KE of the satellite varying or constant?
Gravity
The laws of physics are universal
§ 12.1–
12.3
What’s the point?
• Gravity is one of the fundamental forces.
Historical Background
• Earthly and celestial objects thought
fundamentally different
– Earthly objects seek a lowly position
– Celestial objects move in perfect circles
Newton’s Insight
• The force pulling us to the ground is the
same force curving the moon’s path:
gravity!
Poll Question
Which is greater?
A.
B.
C.
D.
The pull of gravity from the earth on the moon.
The pull of gravity from the moon on the earth.
Both forces are equally strong.
Cannot tell without more information.
Force of Gravity
We have seen
F = mg
(g = gravitational field)
This is true, but:
• g varies with location
• both in direction and magnitude
Newton’s Law of Universal
Gravitation
Gravitational force between two particles:
m1m2
F = –G
r
2
r
G: universal gravitational constant,
6.672  10–11 Nm2/kg2
m1, m2: masses of the particles
r: distance between particles
Properties of Gravity
m1m2
F=G
r2
• direction is toward the other object
• magnitude decreases as (distance)2
increases
• never becomes zero
• infinite at zero separation (?!)
Gravity between Real Objects
..
The actual force acting on an object is the
sum of all forces on all its particles from all
the particles of the other object!
A Lucky Break
The gravitational field around a sphere is the same
as would be around a point particle of the same
mass at the center of the sphere.
Poll Question
The planet Saturn is about 100 times more
massive than Earth, and about 10 times farther
from the Sun. How does the gravitational
attraction between Saturn and the Sun compare to
that between Earth and the Sun?
A.
B.
C.
D.
E.
Saturn is attracted with 1/100 the force as Earth.
Saturn is attracted with 1/10 the force as Earth.
Their attractions to the Sun are about the same.
Saturn is attracted with 10 times the force as Earth.
Saturn is attracted with 100 times the force as Earth.
Poll Question
The planet Saturn is about 100 times more
massive than Earth, and about 10 times farther
from the Sun. How does the acceleration of
Saturn toward the Sun compare to that of Earth
toward the Sun?
A.
B.
C.
D.
E.
Saturn accelerates 1/100th as much as Earth.
Saturn accelerates 1/10th as much as Earth.
Their accelerations are about the same.
Saturn accelerates 10 times as much as Earth.
Saturn accelerates 100 times as much as Earth.
Poll Question
When two objects are separated by a distance D,
the gravitational force between them is F. If they
move closer, to 1/2 the distance (D/2), the force
becomes
F
F
D
A.
B.
C.
D.
F/2.
2F.
D/2
4F.
Cannot tell without more information.
Gravitational Field
• Gives the force acting on an object of
mass m
GM
F = m · field
field = g = 2
r
• M = mass of object creating the field
• r = distance from the object
• field is a vector!
Gravity at Earth’s Surface
•
•
•
•
G = 6.670  10–11 Nm2/kg2
Earth’s mass = 5.976  1024 kg = M
Earth’s radius = 6378174 m = d
so the field g is
g = (6.670  10–11 Nm2/kg2 )
= 9.8 N/kg
look familiar?
(5.976  1024 kg )
(6378174 m )2
Group Work
Find the tangential speed v of an object in
circular orbit a distance r from an attractor of
mass M.
– Use the fact that the centripetal force is the
force of gravity.
Find the square of the orbital period T2 as
well.
– Use the fact that v = 2pr/T.
Kepler’s Laws
of planetary motion
1. Planets travel in elliptical paths with one
focus at the Sun.
2. A planet’s path traces out equal areas in
equal times.
3. The square of a planet’s orbital period is
directly proportional to the cube of the
semi-major axis of the orbit.
Kepler’s Laws
2. Equal-area law
3. T2  a3
– For planets, y and AU are easy units
Kepler’s laws
• Work for other systems too (planets and
moons, etc.)
Example Problem
Calculate the orbital distance of the Moon
based on the observation that the Moon
orbits the Earth once every 27.3 days. The
mass of the Moon is 7.35  1022 kg and the
mass of the Earth is 6.01024 kg.
Double Systems
• Both objects orbit system’s center of
mass
• Orbital radii < separation
• Centripetal force = gravity