Kepler, Newton and Gravitationcf.linnbenton.edu/.../physci/rajabza/upload/6_Gravity.pdfIntro to...
Transcript of Kepler, Newton and Gravitationcf.linnbenton.edu/.../physci/rajabza/upload/6_Gravity.pdfIntro to...
Kepler, Newton and Gravity 1 Intro to Stars
Kepler, Newton and
Gravitation
Kepler, Newton and Gravity 2 Intro to Stars
Using the unit of distance
1 AU = Earth-Sun distance
PLANETS COPERNICUS MODERN
Mercury 0.38 0.387
Venus 0.72 0.723
Earth 1.00 1.00
Mars 1.52 1.52
Jupiter 5.22 5.20
Saturn 9.17 9.54
Kepler, Newton and Gravity 3 Intro to Stars
Johannes Kepler - 1600 A.D.
• built on successes and failures
• demanded that his model’s predictions be
at least as accurate as the observations (1
arcmin error)
foundation for modern cosmological concepts
“I have discovered among the celestial movements
the full nature of harmony.”
Kepler, Newton and Gravity 4 Intro to Stars
Kepler - the Ellipse
Kepler, Newton and Gravity 5 Intro to Stars
Kepler - the Ellipse two focal points -
foci
focal distance
Kepler, Newton and Gravity 6 Intro to Stars
Kepler - the Ellipse
minor axis
semi-minor axis - half the minor axis
Kepler, Newton and Gravity 7 Intro to Stars
Kepler - the Ellipse
major axis
minor axis
semi-major axis - half the major axis (a)
semi-minor axis - half the minor axis
Kepler, Newton and Gravity 8 Intro to Stars
Kepler - the Ellipse two focal points -
foci major axis
minor axis
focal distance
semi-major axis - half the major axis (a)
semi-minor axis - half the minor axis
eccentricity = focal distance
major axis
Kepler, Newton and Gravity 9 Intro to Stars
Kepler - the Ellipse
major axis
focal distance
eccentricity = focal distance
major axis
e = 0 means ????
e = 1 means ????
Kepler, Newton and Gravity 10 Intro to Stars
Kepler’s Three Laws
I. Law of Ellipses
Each planet’s orbit is an ellipse with
the Sun at one of the foci.
implication: distance of the planet to
the Sun varies
Kepler, Newton and Gravity 11 Intro to Stars
Kepler’s Three Laws
II. Law of Equal Areas
A line drawn from a planet to the Sun
sweeps out equal areas in equal times.
implication: orbital speeds are non-uniform
yet vary in a regular way
Closer a planet is the the Sun, the faster
it moves in its orbit
(force ????)
Kepler, Newton and Gravity 12 Intro to Stars
Kepler’s Three Laws
III. Harmonic Law
Planet p (Earth years) a (AU)
============================
Mercury 0.24 0.39
Venus 0.62 0.72
Earth 1.0 1.0
Mars 1.9 1.5
Jupiter 12 5.2
Saturn 29 9.5
Kepler, Newton and Gravity 13 Intro to Stars
Kepler’s Three Laws
III. Harmonic Law
Square of the orbital period is proportional
to the cube of the average distance.
p2 = k a3
implication: planets with large orbits
move slowly
Proportion holds for all planets =>
A PHYSICAL CAUSE !!
Kepler, Newton and Gravity 14 Intro to Stars
Kepler’s Third Law
p2 = k a3
k = proportionality constant
p (years)
a (AU’s)
=> k = 1
True for any body orbiting the Sun, even spacecraft!
“I contemplate its beauty with incredible
and ravishing delight.
Kepler, Newton and Gravity 15 Intro to Stars
Kepler
made perhaps the greatest leap in scientific thinking
predictions were 10 times more accurate than either
Ptolemaic model (geocentric)
Copernican model (heliocentric)
gave birth to astronomy as a physical science
Kepler, Newton and Gravity 16 Intro to Stars
Orbital Eccentricities
Planet Orbital Eccentricity
Mercury 0.206
Venus 0.007
Earth 0.017
Mars 0.094
Jupiter 0.048
Saturn 0.054
Uranus 0.048
Neptune 0.007
Pluto 0.253
Kepler, Newton and Gravity 17 Intro to Stars
GALILEO, NEWTON,
and GRAVITY
Kepler, Newton and Gravity 18 Intro to Stars
Mechanics: the study of falling bodies
speed: how fast you are going
velocity: how fast you are going
in a specific direction
Kepler, Newton and Gravity 19 Intro to Stars
Albany to Portland:
speed = 100 km/hr
velocity =
Portland to Albany:
speed = 80 km/hr
velocity =
?????? 100 km/hr north
?????? 80 km/hr south
Kepler, Newton and Gravity 20 Intro to Stars
• change in speed
• change in direction
• change in both speed and direction
velocity: speed and direction
acceleration: CHANGE in velocity
Kepler, Newton and Gravity 21 Intro to Stars
• You are stopped at a red light.
You step on the gas.
Your speed changes from 0 to 15 km/hr.
Are you accelerating?
speed: measured in km/hr
velocity: measured in km/hr in a direction
acceleration: measured in km/hr
unit of time
Three Cases
Yes, it is your speed that is changing.
Kepler, Newton and Gravity 22 Intro to Stars
• You are driving in a circle around a racetrack
at a constant 20 km/hr.
Are you accelerating?
Yes. Your direction is changing.
• You are braking from 25 km/hr to 15 km/hr.
Are you accelerating?
Yes. Your speed is changing.
Kepler, Newton and Gravity 23 Intro to Stars
Galileo and Motion
downward motion:
gravity pulling downward
forced motion
attractive force
horizontal motion:
would continue forever if no forces acted
natural motion
Kepler, Newton and Gravity 24 Intro to Stars
Galileo’s Experiment
inclined plane, perfectly smooth
perfectly smooth ball
Kepler, Newton and Gravity 25 Intro to Stars
Galileo’s Experiment
Kepler, Newton and Gravity 26 Intro to Stars
Galileo’s Experiment
Kepler, Newton and Gravity 27 Intro to Stars
Galileo’s Experiment
Kepler, Newton and Gravity 28 Intro to Stars
Galileo
inertia: natural tendency for a body in motion
to remain in motion or natural tendency
for a body at rest to remain at rest
Kepler, Newton and Gravity 29 Intro to Stars
acceleration due to gravity
• constant
• velocity changes at a constant rate
• at earth’s surface g = 9.8 m/s
Galileo also experimented with falling bodies
falling is NOT a natural motion
motion due to the force of gravity
s
We can use 10 m/s/s
Kepler, Newton and Gravity 30 Intro to Stars
DROPPED ROCK
How fast is it going after
5 seconds of falling?
10 m/s
30 m/s
Starts at
0 m/s
How fast is it going after
1 second of falling?
How fast is it going after
3 seconds of falling?
50 m/s
Kepler, Newton and Gravity 31 Intro to Stars
PARACHUTIST
How fast is she going after
5 seconds of falling?
10 m/s
30 m/s
Starts at
0 m/s
How fast is she going after
1 second of falling?
How fast is she going after
3 seconds of falling?
50 m/s
Ignore air resistance
Kepler, Newton and Gravity 32 Intro to Stars
ALL falling bodies
fall with the
SAME acceleration !!
Kepler, Newton and Gravity 33 Intro to Stars
Newton 1600 A.D.
Principia - Physics of motion
and Concept of Gravitation
Concept of Force
force produces an acceleration
acceleration is in the same direction
as the force
Kepler, Newton and Gravity 34 Intro to Stars
Increase the force
=> greater acceleration
=> object reaches a greater velocity
What kind of relationship is this?
ACCELERATION
is directly proportional to
FORCE
Kepler, Newton and Gravity 35 Intro to Stars
Increase the MASS
apply the original force
=> less acceleration
What kind of relationship is this?
ACCELERATION
is inversely proportional to
MASS
Kepler, Newton and Gravity 36 Intro to Stars
Can you come up with a Pot O’ Gold type
relationship for acceleration, force & mass?
a = F
m
Forces have direction (same direction
as the acceleration)
Kepler, Newton and Gravity 37 Intro to Stars
Newton clarified these definitions:
• mass : the measure of an object’s resistance
to a change in motion
• velocity : how fast an object moves in a
particular direction
• acceleration : how much the velocity (or
direction) changes with time
Kepler, Newton and Gravity 38 Intro to Stars
Newton’s Three Laws of Motion
• INERTIAL LAW
A body at rest remains at rest unless
acted on by an outside force.
A body in motion at a constant velocity
along a straight line remains in motion
unless acted on by an outside force.
Kepler, Newton and Gravity 39 Intro to Stars
Newton’s Three Laws of Motion
INERTIAL LAW
Implication:
If we see an acceleration, we know
there’s a net force acting on the
body in question (that is, change
in speed direction, or both)
Kepler, Newton and Gravity 40 Intro to Stars
Newton’s Three Laws of Motion
• FORCE LAW
rate of change in a body’s velocity,
due to an applied force (in other words,
a body’s acceleration) is
– in the same direction as the force
– proportional to the force
– inversely proportional to its mass
Kepler, Newton and Gravity 41 Intro to Stars
Newton’s Three Laws of Motion
FORCE LAW
Implication: can apply a force, measure
acceleration and infer the mass
of an object
Kepler, Newton and Gravity 42 Intro to Stars
Newton’s Three Laws of Motion
• REACTION LAW
For every applied force, a force of
equal size by opposite direction arises.
Implication: forces act in pairs
Kepler, Newton and Gravity 43 Intro to Stars
AN EXAMPLE
You are an astronaut out in space.
You push on your space capsule.
What is the equal and opposite force?
Since the forces are the SAME,
exactly what is different?
Spaceship pushes back on you.
Kepler, Newton and Gravity 44 Intro to Stars
You Spaceship
F F
Kepler, Newton and Gravity 45 Intro to Stars
You Spaceship
F F
m M
Kepler, Newton and Gravity 46 Intro to Stars
You Spaceship
F F
m M
A a
Kepler, Newton and Gravity 47 Intro to Stars
You Spaceship
F F
m M
A a
Which has the greater velocity?
Kepler, Newton and Gravity 48 Intro to Stars
NEWTON’S GREAT INSIGHT
Gravitation is an interaction
between two (or more)
bodies, such as the Sun and
the planets.
Kepler, Newton and Gravity 49 Intro to Stars
Newton’s Law of Gravitation
• What direction does the force
of gravity act? (nature of the force)
• What is the amount of force?
(physical properties that determine
the strength of the force)
Kepler, Newton and Gravity 50 Intro to Stars
LAWS OF MOTION
+
Kepler’s Planetary Laws
Law of Universal Gravitation
Kepler, Newton and Gravity 51 Intro to Stars
GRAVITATION
• central force : type of force that causes
elliptical orbits; force directed towards
the center of motion
• planets moving in orbits under the influence
of a central force followed Kepler’s
Second Law (Law of Areas)
• from the geometric properties of ellipses,
force is described by a specific type of
force law => re-derived Kepler’s Third
Law using this force law
Newton’s Laws and Kepler’s Laws were in
total agreement
Kepler, Newton and Gravity 52 Intro to Stars
• Direction - changing? or not?
Consider the moon in orbit:
curved path => changing
Newton’s First Law:
direction is changing
there is an acceleration
Therefore, there is a force acting
on the moon
Kepler, Newton and Gravity 53 Intro to Stars
Force is directed towards the center
of the Earth. We call a center-directed
force a centripetal force.
At every point in its orbit, a centripetal
force acts on the moon to keep it
bound to Earth.
Kepler, Newton and Gravity 54 Intro to Stars
Billiard Ball Analogy
• Ball makes a
collision with
side
• Ball changes
direction
• Ball accelerates
from applied
force of the
wall
Force is
directed towards
the center of the
table
Kepler, Newton and Gravity 55 Intro to Stars
Pentagonal
Table
Force still
points to
the center Angle of strike and rebound
is smaller
Kepler, Newton and Gravity 56 Intro to Stars
Hexagonal
Table
Force still
points to
the center Angle of strike and rebound
is even smaller
Kepler, Newton and Gravity 57 Intro to Stars
Circular
Table
Infinite number
of sides, angle
of strike and
rebound is
zero
Force points to
center at every
point
Kepler, Newton and Gravity 58 Intro to Stars
Moon in its ‘circular’ orbit: at every
point in its orbit, a centripetal force
acts on the moon to keep it bound to
Earth.
GRAVITY !!
Kepler, Newton and Gravity 59 Intro to Stars
Law of Gravitation
Every body in the Universe attracts
every other body with a gravitational
force.
Kepler, Newton and Gravity 60 Intro to Stars
Consider two objects only
• amount of gravitational force depends DIRECTLY
on the amount of material each object has (mass)
What would happen if you doubled the mass
of one of the objects?
Kepler, Newton and Gravity 61 Intro to Stars
The force between them was F.
After the pink object doubles, the gravitational
force is 2 times F or 2F
TWICE AS MUCH
Kepler, Newton and Gravity 62 Intro to Stars
The force between them was 2F
After the purple object doubles, the gravitational
force is 2 times 2F or 4F
4 TIMES the original
amount of force
Now what happens
if we double the mass
of the other object?
Kepler, Newton and Gravity 63 Intro to Stars
To express that in algebra:
Fg is directly proportional to
mpurple x masspink
or
Fg m1 x m2
Kepler, Newton and Gravity 64 Intro to Stars
• objects at greater separations have less gravitational
force between them than those closer together
Decrease of force with
distance happens in a
special way
Force is inversely
proportional to the
square of the distance.
1 m
When the distance
is 1 meter, the force
between them is F.
Kepler, Newton and Gravity 65 Intro to Stars
1 m
Start with the objects
1 meter apart.
What happens if we move
them to 2 meters apart? (that is,
we double the distance)
2 m
Kepler, Newton and Gravity 66 Intro to Stars
2 m
The force, when they were 1 meter apart, was
F - now at 2 meters, the force is
Less is it 1/2 as much?
is it 1/4 as much?
Force is proportional to 1
(distance) 2 Force is as much as it was. 1 4
Kepler, Newton and Gravity 67 Intro to Stars
1 m 3 m
What happens to the force if we move
them 3 meters apart?
Force is then 1/9 as much as it was
Kepler, Newton and Gravity 68 Intro to Stars
F is proportional to m1
F is proportional to m2
Force is proportional to 1
(distance) 2
Force m1 m2
(distance) 2 or
m1 m2
R 2
Kepler, Newton and Gravity 69 Intro to Stars
Constant of proportionality is
G - Universal Gravitational Constant
G = 6.67 x 10-11
Force (newtons)
masses (kgs)
R (meters)
m1 m2
R 2
Force = G
Kepler, Newton and Gravity 70 Intro to Stars
Law of Gravitation
Fgravity = G m1m2
R2
Interpretation:
Force due to gravity is directly
proportional to the masses of
the objects involved.
Kepler, Newton and Gravity 71 Intro to Stars
Law of Gravitation
Fgravity = G m1m2
R2
Interpretation:
Force due to gravity is inversely
proportional to the distance
between them squared.
1 over
R2 Law
Kepler, Newton and Gravity 72 Intro to Stars
Newton worked this out mathematically
(it’s a model) - how did he test this?
1 Re
60 Re Moon is 60 times
farther from the center
of the Earth than the
apple is.
How much less gravitational
force is felt at the location of
the Moon ?
Kepler, Newton and Gravity 73 Intro to Stars
Force at Moon is 1
(60)2
= 1
3600
Newton’s mathematical model predicted
the force (acceleration) should be
3600 times less.
Comparing to measured observations of
the Moon in its orbit, this was
‘pretty nearly’ the same.
Kepler, Newton and Gravity 74 Intro to Stars
Newton’s Concept Extension
• Earth’s gravity keeps the moon
swinging around the Earth
• Sun’s gravity keeps the planets
swinging around the Sun
Kepler, Newton and Gravity 75 Intro to Stars
Newton Revised
Kepler’s Third Law
p2 = k a3
Kepler, Newton and Gravity 76 Intro to Stars
Newton Revised
Kepler’s Third Law
p2 = k a3
p2 = 4 2
G(Msun + mplanet)
a3
Kepler, Newton and Gravity 77 Intro to Stars
Kepler’s Third Law Revised
p2 = 4 2
G(Msun + mplanet)
a3
TREMENDOUSLY
IMPORTANT !!
Can use this to find the
mass of the Sun !!
Kepler, Newton and Gravity 78 Intro to Stars
p2 = 4 2
G(Msun + mplanet)
a3
• measure the period of the Earth’s orbit
• measure the average distance from the Sun
• approximate
Msun + mearth » Msun
• look up the value of G
• calculate the mass of the Sun !!
Kepler, Newton and Gravity 79 Intro to Stars
We can find the mass of the
Sun using any body orbiting
the Sun.
We can find the mass of the
Earth using any body orbiting
the Earth.
Kepler, Newton and Gravity 80 Intro to Stars
What would you do to find
the mass of a planet that
has no orbiting body?
Kepler, Newton and Gravity 81 Intro to Stars
Center of Mass
center of mass : balance point of a group of objects
Where’s the balance point?
Kepler, Newton and Gravity 82 Intro to Stars
1 AU (150 million km)
What’s your best
guess as to the location
of the center of mass
between the Earth
and the Sun ?
Earth
A mere 500 km
from the center of
the Sun !!!
Kepler, Newton and Gravity 83 Intro to Stars
5 AU (750 million km)
What’s your best
guess as to the location
of the center of mass
between Jupiter and
the Sun ?
Jupiter
Just barely
outside the
surface of the Sun
Kepler, Newton and Gravity 84 Intro to Stars
Newton’s Successful Predictions
• return of Halley’s comet
• discovery of planet Neptune
• binary star systems follow Kepler’s
Laws
Kepler, Newton and Gravity 85 Intro to Stars
Newton’s Accomplishments
• found the physical interaction between
the Sun and the planets
• revised Kepler’s Third Law so that it
became a tool for calculating masses of
distant objects
• answered the question of how planets move
• made predictions far more accurate than any
before
• physical support for the heliocentric model
Kepler, Newton and Gravity 86 Intro to Stars
Newton’s Law of Gravitation
Finally, a universal law, that is,
one that is the same for the
heavens and the Earth.
Kepler, Newton and Gravity 87 Intro to Stars
Escape Speed:
minimum speed an object
must have to break free of
gravity
11 km/sec escape speed from
Earth
Kepler, Newton and Gravity 88 Intro to Stars
Mass and radius of planet
determines the escape speed.
vescape = 2 Gm
R
Kepler, Newton and Gravity 89 Intro to Stars
Newton’s Cosmology
cosmology : study of the origin, the nature and
the evolution of the Universe
Universe is infinite in extent
If it were not infinite, but finite, gravitation would
eventually pull all the matter in the Universe back
together => one large mass !
Kepler, Newton and Gravity 90 Intro to Stars
In an infinite universe, there would be
an infinite number of small blobs of matter.
=> exactly the universe we see !
Kepler, Newton and Gravity 91 Intro to Stars
One Small Teeny, Tiny Problem
Orbit of Mercury had an unexplained ‘wobble’
41 arcsec per century
Could not be explained with Newton’s physics
Hypothesis of a planet, hidden behind the Sun,
never proved to be true (Vulcan)