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Orbital Motion A. Orbits B. Orbital Velocity C. Calculating Escape Velocity D. Kepler's Laws Re-examined E. Newton's Version of Kepler's Third Law F. Astronomy After ... – PowerPoint PPT presentation

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Title: Note that the following lectures include animations and PowerPoint effects such as fly ins and transitions that require you to be in PowerPoint's Slide Show mode (presentation mode).


1
Note that the following lectures include
animations and PowerPoint effects such as fly ins
and transitions that require you to be in
PowerPoint's Slide Show mode (presentation mode).
2
Newton, Einstein, and Gravity
  • Chapter 5

3
Guidepost
Astronomers are gravity experts. All of the
heavenly motions described in the preceding
chapters are dominated by gravitation. Isaac
Newton gets the credit for discovering gravity,
but even Newton couldnt explain what gravity
was. Einstein proposed that gravity is a
curvature of space, but that only pushes the
mystery further away. What is curvature? we
might ask. This chapter shows how scientists
build theories to explain and unify observations.
Theories can give us entirely new ways to
understand nature, but no theory is an end in
itself. Astronomers continue to study Einsteins
theory, and they wonder if there is an even
better way to understand the motions of the
heavens. The principles we discuss in this
chapter will be companions through the remaining
chapters. Gravity is universal.
4
Outline
I. Galileo and Newton A. Galileo and Motion B.
Newton and the Laws of Motion C. Mutual
Gravitation II. Orbital Motion A. Orbits B.
Orbital Velocity C. Calculating Escape
Velocity D. Kepler's Laws Re-examined E.
Newton's Version of Kepler's Third Law F.
Astronomy After Newton III. Einstein and
Relativity A. Special Relativity B. The General
Theory of Relativity C. Confirmation of the
Curvature of Space-Time
5
A New Era of Science
Mathematics as a tool for understanding physics
6
Isaac Newton (1643 - 1727)
  • Building on the results of Galileo and Kepler
  • Adding physics interpretations to the
    mathematical descriptions of astronomy by
    Copernicus, Galileo and Kepler

Major achievements
  1. Invented Calculus as a necessary tool to solve
    mathematical problems related to motion
  1. Discovered the three laws of motion
  1. Discovered the universal law of mutual gravitation

7
Velocity and Acceleration
Acceleration (a) is the change of a bodys
velocity (v) with time (t)
a
a Dv/Dt
Velocity and acceleration are directed quantities
(vectors)!
v
Different cases of acceleration
  1. Acceleration in the conventional sense (i.e.
    increasing speed)
  1. Deceleration (i.e. decreasing speed)
  1. Change of the direction of motion (e.g., in
    circular motion)

8
Acceleration of Gravity
Acceleration of gravity is independent of the
mass (weight) of the falling object!
Iron ball
Wood ball
9
Newtons Laws of Motion (1)
  1. A body continues at rest or in uniform motion in
    a straight line unless acted upon by some net
    force.

An astronaut floating in space will continue to
float forever in a straight line unless some
external force is accelerating him/her.
10
Newtons Laws of Motion (2)
  1. The acceleration a of a body is inversely
    proportional to its mass m, directly proportional
    to the net force F, and in the same direction as
    the net force.

a F/m ? F m a
11
Newtons Laws of Motion (3)
  1. To every action, there is an equal and opposite
    reaction.

M 70 kg
V ?
The same force that is accelerating the boy
forward, is accelerating the skateboard backward.
m 1 kg
v 7 m/s
12
The Universal Law of Gravity
  • Any two bodies are attracting each other through
    gravitation, with a force proportional to the
    product of their masses and inversely
    proportional to the square of their distance

Mm
F - G
r2
(G is the Universal constant of gravity.)
13
Understanding Orbital Motion
The universal law of gravity allows us to
understand orbital motion of planets and moons
Example
  • Earth and moon attract each other through
    gravitation.

Dv
  • Since Earth is much more massive than the moon,
    the moons effect on Earth is small.

v
v
  • Earths gravitational force constantly
    accelerates the moon towards Earth.

Moon
F
  • This acceleration is constantly changing the
    moons direction of motion, holding it on its
    almost circular orbit.

Earth
14
Center of Mass
(SLIDESHOW MODE ONLY)
15
Orbital Motion (2)
In order to stay on a closed orbit, an object has
to be within a certain range of velocities
Too slow gt Object falls back down to Earth
Too fast gt Object escapes Earths gravity
16
Orbital Motion (3)
Geosynchronous Orbits
17
Newtons Cannon
(SLIDESHOW MODE ONLY)
18
Geosynchronous Orbit
(SLIDESHOW MODE ONLY)
19
Keplers Third Law Explained by Newton
Balancing the force (called centripetal force)
necessary to keep an object in circular motion
with the gravitational force ? expression
equivalent to Keplers third law,
Py2 aAU3
20
Einstein and Relativity
Einstein (1879 1955) noticed that Newtons laws
of motion are only correct in the limit of low
velocities, much less than the speed of light.
? Theory of Special Relativity
Also, revised understanding of gravity
? Theory of General Relativity
21
Two Postulates Leading to Special Relativity (1)
  1. Observers can never detect their uniform motion,
    except relative to other objects.

This is equivalent to
The laws of physics are the same for all
observers, no matter what their motion, as long
as they are not accelerated.
22
Two Postulates Leading to Special Relativity (2)
  1. The velocity of light, c, is constant and will be
    the same for all observers, independent of their
    motion relative to the light source.

23
Basics of Special Relativity
The two postulates of special relativity have
some amazing consequences. Consider thought
experiment
Motion of stationary observer
Assume a light source moving with velocity v
relative to a stationary observer
v
v
v
c Dt
c Dt
Light source
c Dt
v Dt
Seen by an observer moving along with the light
source
Seen by the stationary observer
24
Basics of Special Relativity (2)
Now, recall that the velocity of light, c, is the
same for all observers.
? The times Dt and Dt must be different!
Then, the Pythagorean Theorem gives
(cDt)2 (cDt)2 (vDt)2
or Dt (Dt)/g where g 1/(1 v/c2)1/2 is
the Lorentz factor.
c Dt
c Dt
v Dt
This effect is called time dilation.
25
Other Effects of Special Relativity
  • Length contraction Length scales on a rapidly
    moving object appear shortened.
  • Relativistic aberration Distortion of angles
  • The energy of a body at rest is not 0. Instead,
    we find
  • E0 m c2

26
General Relativity
A new description of gravity
Postulate Equivalence Principle Observers can
not distinguish locally between inertial forces
due to acceleration and uniform gravitational
forces due to the presence of massive bodies.
27
Another Thought Experiment
Imagine a light source on board a rapidly
accelerated space ship
Time
Time
a
Light source
a
a
a
g
As seen by a stationary observer
As seen by an observer on board the space ship
28
Thought Experiment (2)
For the accelerated observer, the light ray
appears to bend downward!
Now, we cant distinguish between this inertial
effect and the effect of gravitational forces
Thus, a gravitational force equivalent to the
inertial force must also be able to bend light!
29
Thought Experiment (Conclusion)
This bending of light by the gravitation of
massive bodies has indeed been observed
During total solar eclipses The positions of
stars apparently close to the sun are shifted
away from the position of the sun.
? New description of gravity as curvature of
space-time!
30
Another manifestation of bending of light
Gravitational lenses
A massive galaxy cluster is bending and focusing
the light from a background object.
31
Other Effects of General Relativity
  • Perihelion advance (in particular, of Mercury)
  • Gravitational red shift Light from sources near
    massive bodies seems shifted towards longer
    wavelengths (red).

32
New Terms
natural motion violent motion acceleration of
gravity momentum mass acceleration velocity invers
e square law field circular velocity geosynchronou
s satellite center of mass closed orbit escape
velocity open orbit angular momentum energy joule
(J)
special relativity general theory of relativity
33
Discussion Questions
1. How did Galileo idealize his inclines to
conclude that an object in motion stays in motion
until it is acted on by some force? 2. Give an
example from everyday life to illustrate each of
Newtons laws.
34
Quiz Questions
1. According to Aristotle, where is the proper
place of the classical elements earth and water
that is, what location do they seek? a. The
center of Earth. b. The center of the
Universe. c. The Heavens. d. Both a and b
above. e. Both b and c above.
35
Quiz Questions
2. According to the principles of Aristotle, what
part of the motion of an arrow that is fired
vertically upward is natural motion and what part
is violent motion? a. Both the upward and
downward parts are natural motion. b. Both the
upward and downward parts are violent motion. c.
The upward part is natural motion and the
downward part is violent motion. d. The upward
part is violent motion and the downward part is
natural motion. e. Neither the upward nor the
downward parts are natural or violent motion.
36
Quiz Questions
3. If we drop a feather and a hammer at the same
moment and from the same height, on Earth we see
the hammer strike the ground first, whereas on
the Moon both strike the ground at the same time.
Why? a. The surface gravity of Earth is
stronger than the gravity of the Moon. b. In
strong gravity fields heavier objects fall
faster. c. The is no air resistance effect on the
Moon. d. Both a and b above. e. All of the above.
37
Quiz Questions
4. Which statement below best describes the
difference between your mass and your weight? a.
Your mass is constant and your weight varies
throughout your entire life. b. Your mass is a
measure of the amount of matter that you contain
and your weight is a measure of the amount of
gravitational pull that you experience. c. Your
mass is a measure of your inertia, whereas your
weight is a measure of the amount of material you
contain. d. The only difference is the unit used
to measure these two physical quantities. Mass
is measured in kilograms and weight is measured
in pounds. e. There is no difference between your
mass and your weight.
38
Quiz Questions
5. Which of the following is true for an object
in uniform circular motion? a. The velocity of
the object is constant. b. The acceleration of
the object is zero. c. The acceleration of the
object is toward the center of motion. d. The
angular momentum of the object is zero. e. The
speed of the object is changing.
39
Quiz Questions
6. If a 1-kilogram rock and a 6-kilogram rock are
dropped from the same height above the Moon's
surface at the same time, they both strike the
Moon's surface at the same time. The
gravitational force with which the Moon pulls on
the 6-kg rock is 6 times greater than on the 1-kg
rock. Why then do the two rocks strike the
Moon's surface at the same time? a. The
acceleration of each rock is inversely
proportional to its mass. b. The Moon's surface
gravity is one-sixth the surface gravity at
Earth's surface. c. The 1-kg rock is attracted
less by the nearby Earth. d. Both a and b
above. e. All of the above.
40
Quiz Questions
7. Why did Newton conclude that some force had to
pull the Moon toward Earth? a. The Moon's
orbital motion is a curved fall around Earth. b.
The Moon has an acceleration toward Earth. c. The
force and acceleration in Newton's second law
must have the same direction. d. Both b and c
above. e. All of the above.
41
Quiz Questions
8. What did Newton determine is necessary for the
force exerted by the Sun on the planets to yield
elliptical orbits? a. The force must be
attractive. b. The force must be repulsive. c.
The force must vary inversely with distance. d.
The force must vary inversely with distance
squared. e. Both a and d above.
42
Quiz Questions
9. Which of Kepler's laws of planetary motion is
a consequence of the conservation of angular
momentum? a. The planets orbit the Sun in
elliptical paths with the Sun at one focus. b. A
planet-Sun line sweeps out equal areas in equal
intervals of time. c. The orbital period of a
planet squared is proportional to its semimajor
axis cubed. d. Both b and c above. e. All of the
above.
43
Quiz Questions
10. How did Galileo slow down time in his falling
body experiments? a. He performed the
experiments near the speed of light. b. He
measured the time objects took to fall through
water. c. He used a stopwatch. d. He rolled
objects down inclines at low angles. e. He began
each fall with an upward toss.
44
Quiz Questions
11. Which of Newton's laws was first worked out
by Galileo? a. The law of inertia. b. The net
force on an object is equal to the product of its
mass and its acceleration. c. The law of action
and reaction. d. The law of universal mutual
gravitation. e. Both c and d above.
45
Quiz Questions
12. According to Newton's laws, how does the
amount of gravitational force on Earth by the Sun
compare to the amount of gravitational force on
the Sun by Earth? a. The amount of force on
Earth by the Sun is greater by the ratio of the
Sun's mass to Earth's mass. b. The amount of
force on the Sun by Earth is negligible. c. The
amount of force on the Sun by Earth is the same
as the amount of force on Earth by the Sun. d.
The amount of force on the Sun by Earth is
greater by the ratio of the Sun's mass to Earth's
mass. e. It is impossible to compare these two
vastly different amounts of force.
46
Quiz Questions
13. Suppose that Planet Q exists such that it is
identical to planet Earth yet orbits the Sun at a
distance of 5 AU. How does the amount of
gravitational force on Planet Q by the Sun
compare to the amount of gravitational force on
Earth by the Sun? a. The amount of the two
forces is the same. b. The amount of force on
Planet Q is one-fifth the force on Earth. c. The
amount of force on Planet Q is 5 times the force
on Earth. d. The amount of force on Planet Q is
one twenty-fifth the force on Earth. e. The
amount of force on Planet Q is 25 times the force
on Earth.
47
Quiz Questions
14. Newton's form of Kepler's law can be written
as (Msun Mplanet) Py2 aAU3, where the masses
of the Sun and planet are in units of solar
masses, the period is in units of years, and the
semimajor axis in astronomical units. Why is
Kepler's form of his third law nearly identical
to Newton's form? a. Both forms are very similar
in that they have periods and semimajor axes in
units of years and astronomical units
respectively. b. The mass of the Sun plus the
mass of a planet is nearly one. c. The mass of
each planet is very large. d. Both b and c
above. e. All of the above.
48
Quiz Questions
15. How does the orbital speed of an asteroid in
a circular solar orbit with a radius of 4.0 AU
compare to a circular solar orbit with a radius
of 1.0 AU? a. The two orbital speeds are the
same. b. The circular orbital speed at 4.0 AU is
four times that at 1.0 AU. c. The circular
orbital speed at 4.0 AU is twice that at 1.0
AU. d. The circular orbital speed at 4.0 AU is
one-half that at 1.0 AU. e. The circular orbital
speed at 4.0 AU is one-fourth that at 1.0 AU.
49
Quiz Questions
16. In the 1960s television program "Space 1999"
an accident on the Moon causes the Moon to be
accelerated such that it escapes Earth and
travels into interstellar space. If you assume
that the Moon's orbit was nearly circular prior
to the accident, by what minimum factor is the
Moon's orbital speed increased? a. The Moon's
speed must be increased by a factor of 4 to
escape Earth. b. The Moon's speed must be
increased by a factor of pi to escape Earth. c.
The Moon's speed must be increased by a factor of
2 to escape Earth. d. The Moon's speed must be
increased by a factor of 1.4 to escape Earth. e.
It cannot be determined from the given
information.
50
Quiz Questions
17. Just after a alien spaceship travels past
Earth at one-half the speed of light, a person on
Earth sends a beam of light past the ship in the
same direction that the ship is traveling. How
fast does an alien on the ship measure the light
beam to be traveling as it zips past the
spaceship? a. At the speed of light, or 300,000
km/s. b. At one-half the speed of light, or
150,000 km/s. c. At one and one-half the speed of
light, or 450,000 km/s. d. At twice the speed of
light, or 600,000 km/s. e. The measured speed
depends on the method of measurement.
51
Quiz Questions
18. Who first proposed that gravity is the
bending of space-time due to the presence of
matter? a. Tycho Brahe (1546 - 1601) b. Johannes
Kepler (1571 - 1630) c. Galileo Galilei (1564 -
1642) d. Isaac Newton (1642 - 1727) e. Albert
Einstein (1879 - 1955)
52
Quiz Questions
19. What major orbital problem of the late 1800s
is solved by general relativity? a. The reason
for the elliptical shape of planetary orbits. b.
The relationship between circular and escape
velocity. c. The periods of parabolic and
hyperbolic orbits. d. The excess precession of
Mercury's perihelion. e. The three-body problem.
53
Quiz Questions
20. What is significant about the May 29, 1919
solar eclipse? a. It was an annular eclipse
visible from South America, the South Atlantic,
and central Africa in 1919. b. The bending of
light by gravity was observed, thus verifying
general relativity. c. The Moon was at New phase
and at one node of its orbit during this
eclipse. d. It marked the end of the first
complete Saros cycle of the 20th century. e. It
was not predicted.
54
Answers
1. d 2. d 3. c 4. b 5. c 6. a 7. e 8. e 9. b 10. d
11. a 12. c 13. d 14. b 15. d 16. d 17. a 18. e 19
. d 20. b
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