Download Newton*s Theory of Gravity and Planetary Motion

Newton’s Theory of Gravity
and
Planetary Motion
AP Physics C: Mechanics
Geocentric View Points
• Aristotle (Greek) 384-322 BC
• Ptolemy (Alexandrian Greek) 85-65AD
Heliocentric Viewpoints
• Aristarchus (Greek)310-230 BC
• Copernicus (Poland and Italy) 1473-1543
• Galileo Galilei (Italian) 1564-1642
Tycho Brahe (Danish)
1546-1601
Tycho Brahe
Compromise Theory: The Sun orbits the
Earth, but the remaining planets orbit the
Sun.
Brahe passed along his observations to
Johannes Kepler, his assistant.
Brahe’s Compromise Theory
What’s Wrong With This?
• As we all know, Brahe’s theory is not accurate
for multiple reasons, but two should be
immediately noticeable.
– The sun is the CENTER of our solar system. ALL
planets, including Earth, revolve around the sun
with a certain period that is related to its distance
to the sun.
– The orbits are circular! We know that the planet’s
orbits are not circular but elliptical.
Johannes
Kepler
(1571-1630)
German Astronomer
Kepler is credited with
developing three laws
that govern planetary
motion:
Kepler’s Laws of Planetary Motion
Kepler’s First Law of Planetary Motion
1. Planets move in elliptical orbits, with the sun at
one focus of the ellipse.
Slide 13-20
Kepler’s Laws of Planetary Motion
Kepler’s Second Law of Planetary Motion
2. A line drawn between the sun and a planet
sweeps out equal areas during equal intervals of
time.
WHY?
Slide 13-21
Kepler’s 2nd Law (The Law of Areas)
A line from the sun to a planet sweeps out
equal areas in equal lengths of time.
Eccentricity of an Ellipse
• e=c/a
• For a circle e=0
• Pluto has the highest e=0.25 in our
system.
• Earth’s e=0.017
Kepler’s Laws of Planetary Motion
Kepler’s Third Law of Planetary Motion
3. The square of a planet’s orbital period (or the
amount of time it takes to complete one
revolution) is proportional to the cube of the
semimajor-axis length (distance from sun.
T2 ~ r3
Slide 13-22
Kepler’s Third Law
• Kepler’s Third Law describe how long it will
take a planet to revolve around the sun based
on its semi-major axis. In order to calculate
said period, we use the equation:
*** G is the universal gravitational constant =
6.67 x 10-11 Nm2 / kg2
Kepler’s Third Law
• Since distances are normally very large, it is helpful to
measure the distances in Astronomical Units (AU).
• 1 AU is the average radius of the orbit of the Earth
about the sun. The period for 1 AU is approximately 1
year.
• However, this depends on the type of problem we use.
We can use either one, but the AU is normally used to
comparisons to the Earth. Remember, we use ratios to
compare!
EXAMPLE:
• An asteroid revolves around the sun
with a mean orbital radius of three times
that of the earth. What is the period of
the asteroid in earth years?
• Answer: 5.2y
Example:
• Astronomers have only recently seen evidence of
planets orbiting nearby stars. These are called
extrasolar planets. Suppose a planet is observed
to have a 1200 day period as it orbits a star at the
same distance as Jupiter is from the sun. What is
the mass of the star? Assume r = 7.78x1011
meters.
• 2.59 x 1031 kg
Kepler’s Laws of Planetary Motion
A circular orbit is a special case of an elliptical orbit.
Slide 13-23
Isaac Newton
Newton’s Law of Universal Gravitation
Isaac Newton, 1642–1727.
 Legend has it that Newton
saw an apple fall from a tree,
and it occurred to him that
the apple was attracted to
the center of the earth.
 If the apple was so attracted,
why not the moon?
 Newton posited that gravity
is a universal attractive force
between all objects in the
universe.
Slide 13-24
Newton’s
Law
Universal Gravitation
Newton’s
Law
of of
Gravity
The moon is in free fall around the earth.
Slide 13-25
Newton’s Law of Gravity
Newton’s Law of Universal Gravitation
The gravitational constant is a universal constant with
the value G = 6.67  1011 N m2/kg2.
Slide 13-26
QuickCheck 13.1
Newton’s Law of Universal Gravitation
The force of Planet Y on Planet X is ___ the magnitude
of X on Y .
A.
B.
C.
D.
E.
One quarter.
One half.
The same as.
Twice.
Four times.
Slide 13-29
QuickCheck 13.1
Newton’s Law of Universal Gravitation
The force of Planet Y on Planet X is ___ the magnitude
of X on Y .
A.
B.
C.
D.
E.
One quarter.
One half.
The same as.
Twice.
Four times.
Slide 13-30
Little g and Big G
Newton’s Law of Universal Gravitation
 An object of mass m sits on the
surface of Planet X.
 According to an observer on
the planet, the gravitational
force on m should be
FG = mgsurface.
 According to Newton’s law of
gravity, the gravitational force
on m should be
.
 These are the same if:
Slide 13-36
QuickCheck 13.4
Newton’s Law of Universal Gravitation
Planet X has free-fall acceleration 8 m/s2 at the surface.
Planet Y has twice the mass and twice the radius of
planet X. On Planet Y
A.
B.
C.
D.
E.
g = 2 m/s2.
g = 4 m/s2.
g = 8 m/s2.
g = 16 m/s2.
g = 32 m/s2.
Slide 13-37
QuickCheck 13.4
Newton’s Law of Universal Gravitation
Planet X has free-fall acceleration 8 m/s2 at the surface.
Planet Y has twice the mass and twice the radius of
planet X. On Planet Y
A.
B.
C.
D.
E.
g = 2 m/s2.
g = 4 m/s2.
g = 8 m/s2.
g = 16 m/s2.
g = 32 m/s2.
Slide 13-38
gravity above the Earth’s surface
• r = RE + h
g
GM E
 RE  h 
2
Note:
• g decreases with increasing altitude
• As r , the weight of the object
approaches zero
Decrease of g with Distance
Gravity Depends on Height
Slide 13-40
Speeds of Satellites
• In order to find the
speed of a satellite, or
any object revolving
around the Earth, we
can use the principles of
centripetal force and
Newton’s Law of
Universal Gravitation to
find it. Let’s take a look.
Gravitational Potential Energy
• As a particle moves from A to B, its gravitational
potential energy changes by:
U  U f  U i  W
rf
U f  U i    F (r ) dr
ri
Gravitational Potential Energy of the
Earth-particle system
• The reference point is chosen at infinity
where the force on a particle would approach

zero. Ui = 0 for ri =
GM E m
U (r)  
r
• ∞This is valid only for r > = RE and not valid for r < RE
• U is negative because the object is getting closer to Earth.
Gravitational Potential Energy of any
two particles
Gm1m2
U 
r
The negative is used
because the distance
is decreasing, gravity
is bringing the object
closer!
This is the same potential
energy as mgh. However,
this is used as a whole,
while the other is used
when we are closer to the
surface of the earth.
QuickCheck 13.6
Gravitational Potential Energy of any two particles
Which system has more (larger absolute value)
gravitational potential energy?
A. System A.
B. System B.
C. They have the same gravitational potential energy.
Slide 13-46
QuickCheck 13.6
Gravitational Potential Energy of any two particles
Which system has more (larger absolute value)
gravitational potential energy?
A. System A.
B. System B.
C. They have the same gravitational potential energy.
Slide 13-47
Systems with Three or More Particles
(Configuration of Masses)
• The total gravitational
potential energy of the
system is the sum over
all pairs of particles
• Gravitational potential
energy obeys the
superposition principle
Systems with Three Particles
U total  U12  U13  U 23
 m1m2 m1m3 m2m3 
 G 



r13
r23 
 r12
• The absolute value of Utotal represents
the work needed to separate the
particles by an infinite distance.
• Remember energy is a scalar quantity.
Ex #31
• A system consists of three particles, each of
mass 5.00g, located at the corners of an
equilateral triangle with sides of 30.0cm.
a) Calculate the potential energy of the system.
b) If the particles are released simultaneously,
where will they collide?
Ans: a) -1.67x10-14 J
Energy in a Circular Orbit
1 2
Mm
E  mv  G
2
r
GM
Tangential v 
r
GMm
E
2r
Note: Energy in a Circular Orbit
• K>0 and is equal to half the absolute
value of the potential energy.
• |E| = binding energy of the system.
• The total mechanical energy is
negative.
Energy in a Circular Orbit
GM
Tangential v 
r
GMm
E
2r
GMm
E =2a
Circular Orbit
2r = diameter of orbit
Elliptical Orbit
2a = semi-major axis
Energy in a Circular Orbit
A satellite of mass m is in the elliptical orbit shown
below around the Earth (radius rE, mass M).
1) Determine the speed of the satellite at perigee.
Write your answer in terms of r1, r2, M, and G.
2) Determine the speed of the satellite when it’s
at the point X in the figure.
Escape Speed
• In order to find the escape speed of a rocket or
any object from a planet, we will use the
principles of energy conservation.
• Before the rocket leaves the planet, it has both
kinetic energy and gravitational potential energy
due to gravity trying to bring it back down.
• Once the rocket leaves the planet is has no
gravitational potential energy and no kinetic.
Example 13.2 Escape Speed
Slide 13-48
Note: For a Two Particle Bound System
• Both the total energy
and
• the total angular momentum are constant.
Compare the Kinetic Energy and
Angular Momentum of a Satellite at
orbit 1 and 2
1
Earth
2
How does the speed of a satellite at
position 2 compare to the speed at
position 1. The distance r2 =2r1.
(Hint: Use conservation of angular
momentum)
2
Earth
1