Transcript acceleration

Lecture 5
Modern Astronomy
Announcements
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Test on Wednesday – Details to follow
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Homework 2 Due Now
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Homework 3 Due next Monday, but it is a
good idea to finish before the Test
Wednesday.
Test Information: Materials
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Required
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Recommended
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Pencil/pen
Equation sheet
Calculator
Scratch paper
Test Coverage
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Units 1, 2, 5, 6, 9, 11, 12
Test Information: Materials
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Forbidden
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Cell phone (not even for use as calculator)
Communication with anyone other than me
Textbook or any other reference material not written
by you.
Equation Sheet
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Single page (front and back) HAND WRITTEN notes,
equations, or any information you want to bring to
the test.
Max size 8.5 x 11 inches.
Will be turned in – counts 10% of your test grade.
Test Information: Format
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85 minutes Wednesday (5-6:25 pm)
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Approximate Test Format:
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20 multiple-choice questions (2 points each)
10 True/False questions (2 points each)
6 Short answer/problem questions (5 points
each)
Equation sheet (10 points)
Review
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3000+ BC – Use Sun to predict seasons
1000 BC – Babylonian calendar, predict planet motion
700 BC – Start of classical revolution
500 BC – Earth is a sphere
360 BC – Plato Postulates Perfection
350 BC – Eudoxos/Aristotle – Geocentric system
310 BC – Aristarchus – Heliocentric (not accepted)
180 BC – Hipparchus – Star mapping
185 AD – Ptolemy – Most advance Geocentric model
1543 AD – Copernicus – Heliocentric model
1577 AD – Tycho – Modified Geocentric model
Johannes Kepler (1571-1630)
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Student of Tycho Brahe
Uses Tycho’s observations
to support Copernicus.
Deduces three laws that
govern planetary motion.
Explains planetary motion
better than Ptolemy’s
model.
Kepler’s Big Idea
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Found a consistent description by
abandoning both
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Circular motion and
Uniform motion.
Planets move around the sun on elliptical
paths, with non-uniform velocities.
Kepler’s Three Laws
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First Law: All planets move around the
sun in elliptical orbits, with the sun at one
focus.
Second Law: All planets sweep out equal
areas of their orbits in equal times.
Third Law: The square of a planet’s
orbital period is equal to the cube of its
average distance from the sun.
Kepler’s First Law
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All planets move around the sun in elliptical
orbits, with the sun at one focus.
In an ellipse:
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a + b = a’ + b’
Eccentricities of Ellipses
1)
2)
e = 0.02
3)
e = 0.1
e = 0.2
5)
4)
e = 0.4
e = 0.6
Eccentricities of Planetary Orbits
Orbits of (major and minor) planets are
virtually indistinguishable from circles:
Most extreme example:
Earth: e = 0.0167
Pluto: e = 0.248
Kepler’s Second Law
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All planets sweep out equal areas of their orbits
in equal times.
Means the closer a planet is to the sun, the
faster it moves in its orbit.
Conservation Of Angular Momentum
Kepler’s Third Law
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The square of a planet’s orbital period is equal
to the cube of its average distance from the sun.
Written Mathematically As:
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P2 = a3
P = orbital period of the planet in years.
a = average distance between the planet and the sun
in Astronomical Units (AU).
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Giving something’s distance from the sun in AU is the same
as giving its distance as “this many times further from the
sun than the Earth.”
Example: Jupiter is 5.2 AU from the sun means Jupiter
orbits the sun 5.2 times farther away than the Earth does.
The Beginning of the End:
Galileo Galilei (1564-1642)
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Contemporary with Kepler
and an avid Copernican
Contributions were useful
in helping to discredit
Aristotelian concepts of
motion, and he was one of
the first astronomers to
record telescopic
observations of the celestial
objects
Promoted the modern view
of science: Transition from
a faith-based “science” to
an observation-based
science.
Galileo’s Telescopic Observations
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Craters on the moon
Moons of Jupiter
Saturn
Phases of Venus
Sunspots
Milky way
Variable planet size
Major Discoveries of Galileo
• Moons of Jupiter
(4 Galilean moons)
(What he really saw)
• Rings of Saturn
(What he really saw)
Major Discoveries of Galileo
• Surface structures on the moon; first estimates
of the height of mountains on the moon
Major Discoveries of Galileo
• Sun spots (proving
that the sun is not
perfect!)
Major Discoveries of Galileo
• Phases of Venus (including “full Venus”),
proving that Venus orbits the sun, not the Earth!
Galileo’s Observations Conflict
With Catholic Dogma
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Valleys and craters on moon
Sunspots on sun
“Belts” on Jupiter
“Ears” on Saturn
All in conflict with the Catholic “perfect
heavens.”
Christiaan Huygens (1629-1695)
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Good early telescope
maker.
“Discovered” Saturn’s
rings.
Lots of contributions to
optics.
First estimate of distance
to another star (Sirius at
27,000 AU).
Daily Grade 5 – Question 1
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What is the difference between a
heliocentric cosmology and a
geocentric cosmology?
Daily Grade 5 – Question 2
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The seventh and eighth planets out from
the sun are Uranus and Neptune. Which
one moves more slowly in its orbit?
The Birth of Modern Physics:
Sir Isaac Newton (1643-1727)
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One of the most
famous scientists of
all time.
Building on the work
of Galileo and Kepler,
devised the Laws of
motion and Law of
Universal Gravitation
that provide the
physics behind the
heliocentric model.
Other Major Accomplishments
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Invented Calculus as a necessary tool to solve
mathematical problems related to motion
Major advances in optics and reflecting telescopes
Newton’s Laws
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Newton asked: When and Why do things move?
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Can something be moving when no force is
being applied to it?
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Things move when you apply a force to them.
A force is something that pushes or pulls on
something else.
Yes, it can.
But, can something’s motion change when no
force is being applied to it?
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No, it can’t!
Newton’s First Law of Motion
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Newton’s First Law:
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A body at rest tends to stay at rest unless acted upon
by an outside force.
A body in motion tends to stay in motion unless acted
upon by an outside force.
Consequences:
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You need to apply a force to make a stationary object
start moving.
You need to apply a force to stop a moving object.
You need to apply a force to change the motion of
an object:
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Speed it up
Slow it down
Change it’s direction
Acceleration
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We call any change in how an object moves an
acceleration.
Examples of acceleration:
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When your car speeds up after you hit the gas pedal
(your direction of motion is the same, but you are
going faster than before)
When your car slows down because you hit the
brakes (same direction of motion, less speed than
before)
Whenever your car turns, even if it doesn’t speed
up or slow down (your direction of motion changes,
but your speed stays the same).
Forces And Acceleration
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Notice how in all three
examples you would
feel a force…
Car speeds up: you feel a
force pressing you back
into your seat.
Car slows down: you feel
a force pushing you
forward toward the
windshield.
Car turns: you feel
“pulled” toward the side
of the car.
Newton’s Second Law of Motion
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Describes the relationship between force
and acceleration.
Newton’s Second Law:
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The acceleration of a body is directly
proportional to the force applied to that body.
The acceleration of a body is inversely
proportional to the mass of the body.
Newton’s Second Law – The
Equation!
 Let’s say I wanted to give an object an acceleration of 10 m/s2 (free-fall).
You can write Newton’s Second Law as a
 Themathematical
more massive the equation:
object, the more force I’d have to apply to get
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that acceleration.
F
=
m
×
a
 0.1 kg × 10 m/s = 1 N
 A 0.1 kg object, like an apple, would require a “push” of only 1 N:
2
 A 4 kg object, like a gallon jug of milk, would require a “push” of 40 N:
In this equation:
2 = 40 N
 4F kg
10 m/s
= ×the
force,
measured in a unit called Newtons
(N) kg object, like a car, would require a “push” of 10,000 N!
 A 1,000
m = the mass,2 measured in kilograms (kg)
 1,000 kg × 10 m/s = 10,000 N
a = the acceleration, measured in m/s2
 More mass means I need more force to get the same effect. Mass is an
object’s resistance to a change in its motion.
Acceleration of Gravity
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Acceleration of gravity
is independent of the
mass (weight) of the
falling object!
Newton’s second law
still applies! Since the
acceleration is the
same and the mass is
different, the force is
more for the heavier
object.
Acceleration of Gravity
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Light objects (like
feathers) appear to
fall slower on Earth
due to air
resistance.
In a vacuum (such
as on the moon), a
hammer and a
feather fall at the
same rate.
Daily Grade 5 – Question 3
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If we drop a feather and a hammer at the
same moment and from the same height, 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. Feathers are made mostly of air.
And Now, On To The Most Famous
Of Newton’s Laws… The Law For
Rocket Science!
Newton’s Third Law
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The Action-Reaction Law:
For every action there is an equal and
opposite reaction.
Examples:
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If you punch a wall, you will exert a force on the wall.
The wall will exert an equal and opposite force back
on your hand (which is why it hurts)!
In a rocket engine, the engine “pushes” superheated
gas down out of the bottom of the rocket. The gas
exerts an equal and opposite force up on the rocket,
“pushing” the rocket upward.
Newton’s Third Law
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Newton’s Third Law means forces always
come in opposing pairs: equal in strength
and opposite in direction.
Newton’s Laws
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Newton’s three laws of motion explain
how everything in the universe moves,
from cars …
…to colliding galaxies!
Newton’s Law of Gravitation
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After devising his three laws of motion, Newton
became interested in studying the actual force
that caused objects to fall.
Newton realized that according to his laws of
motion, some force must be keeping the moon
in orbit around the Earth, and the Earth in orbit
around the Sun.
In a flash of insight, Newton hypothesized that
the force that held the moon in its orbit was the
same force that caused apples to fall to the
ground, gravity.
Newton’s Law of Gravitation
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Newton knew gravity could reach the
tallest mountain, but could it reach all the
way to the moon?
Newton made a few assumptions about
how gravity worked:
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Gravity should get weaker the farther away
you got from the Earth’s center.
That it got weaker at a rate that was
proportional to the distance squared from
the Earth’s center.
The Inverse Square Law
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Gravity’s strength getting weaker at a rate
proportional to the distance squared is an
example of an inverse square law.
At the Earth’s surface, you are 6,370 km from
the Earth’s center. At Earth’s surface, gravity
accelerates you downward at 9.8 m/s2.
If you move to 6,370 km above Earth’s surface,
you have doubled your distance from the
Earth’s center. So gravity should get weaker by
a factor of 22 = 4. At 6,370 km above Earth’s
surface, gravity should accelerate you downward
at
9.8 / 4 = 2.45 m/s2
Can Gravity Hold The Moon In
Orbit?
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Newton wondered: Was gravity strong enough
to hold the moon in orbit?
He calculated that the moon would have to be
accelerated at a rate of 0.0027 m/s2 to stay in
an orbit around the Earth.
It was already known that the moon is 60 times
farther from the Earth’s center than the Earth’s
surface is.
So the force of gravity at the moon should be
602 = 3600 times weaker than at the Earth’s
surface.
9.8 / 3600 = 0.0027 m/s2
So Newton determined that gravity
was indeed what held the moon in
orbit around the Earth!
Daily Grade 5 – Question 4
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Why did Newton conclude that some
force had to pull the moon toward Earth?
A. The moon's motion is a straight line away
from Earth.
B. Some force causes the moon to change its
phases.
C. The moon does not seem to go anywhere.
D. The moon follows a curved path around
Earth.
Gravity
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Newton also realized that, according to his third
law, gravity had to be mutual – if the Earth
pulls on the moon, the moon pulls on the Earth
with an equal and opposite force.
Also, larger objects seem to be able to pull with
much more force: the Sun pulls on the Earth
harder than Earth pulls on the Moon.
So gravity’s strength must also depend on
mass!
Gravity
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If gravity is mutual, its strength must
depend on the masses of both objects.
Gravity increases proportionally with mass.
Gravity decreases proportionally with
distance squared.
Led Newton to his famous equation for the
Law of Gravity:
2
F = GMm/r
The Universal Law of Gravitation
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In this equation:
F = GMm/r2
M = the mass of the more massive object (kg).
m = the mass of the less massive object (kg).
r = the distance between the centers of each object
(m)
G = the gravitational constant (6.67 × 10-11)
F = the force with which gravity pulls on the two
objects (N).
How To Use It?
1.
2.
3.
Using your calculator, first multiply the masses
of the two object together. Then multiply the
answer by 6.67 × 10-11. Write down the
resulting number.
Next, use your x2 or xy button to get the
square of the distance between the centers of
the two objects.
Finally, take your result from 1 and divide it by
your result from 2. That’s the force of gravity
between the two objects.
The Universal Law of Gravitation
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Is the culmination of the Copernican
revolution
United terrestrial & celestial mechanics
Balancing the force (called “centripetal
force”) necessary to keep an object in
circular motion with the gravitational force
= an expression equivalent to Kepler’s
third law
Gravitation And Orbits
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If an object moves fast enough, its path will
match the curvature of the Earth, and it will
never hit the ground - it goes into orbit.
Circular orbital velocity for a low Earth orbit is
about 5 miles per second.
If the object's velocity is greater than 5
miles/sec but less than 7 miles/sec, its orbit
will be an ellipse.
Velocities greater than 7 miles/sec reach
escape velocity, and the object moves in a
curved path that does not return to Earth.
Orbital Motion
In order to stay on a
closed orbit, an object
has to be within a
certain range of
velocities:
Too slow => Object falls
back down to Earth
Too fast => Object escapes
Earth’s gravity
Orbital Motion
Geosynchronous
Orbits
For Next Time
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Test on Wednesday
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Read Units 22 and 28 for next Monday
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Students may also find the following sections
useful:
Unit
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 Unit
 Unit
 Unit
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21.1
26.1
27.2
29.2
30.1