Download The Force of gravity

Gravity and Universal
Gravitation
How does gravity affect “the universe”
Objectives
• Describe gravity
– As a force
– As an acceleration
• Universal Gravitation
– Linking Newton’s Apple with
Kepler’s planetary motion
The earth was at its closest approach to the sun on Sunday Jan 4, This distance is known as
perihelion. On Jan 4th the sun is about 91.4 million miles from the sun. On July 4th the earth is at its
farthest point, which is about 94.5 million miles. Too weird…it is closest to the sun when it is winter
in New Jersey! How can this be?
Story by AccuWeather.com senior Meteorologist Brett Anderson
What is Gravity?
• We know what it does.
– It makes your Vitamin water “keep
falling”
– It makes what goes up…fall down
Gravity is the “thing” that makes objects fall to earth
Universal principles
• In physics
– We want to explain what happens
in terms of “universal” principles
– These are principles that explain
many phenomena (things that
happen) in a consistent manner
Need to understand
• Cause
• Source
• Implication of gravity on the
structure and motion of objects
in the universe
What do we “know”
• Gravity is a force that exists
between earth and objects near
the earth
• Force of gravity Fgrav
How does gravity “act”?
• Slows us down as we go
“upward” (away from the earth)
• Speeds us up as we travel
downward (toward the earth)
The Force of gravity (Fgrav) causes an acceleration (g)
Force vs. Acceleration
• Two different things!
– Acceleration due to gravity (g) in
units of (m/s/s) is the change in
velocity experienced by an object
when only the force of gravity
(Fgrav) in units of Newtons acts
upon it.
Acceleration (g)
• Value is 9.8 m/s/s
• Is independent of mass.
• If no other force is acting on an
object, the acceleration due to
gravity for ALL objects is the
same.
Planetary Movement
and Gravity
• 1600s
– Kepler analyzed movement of the
planets.
• RESULT
– Kepler’s Laws of Planetary Motion
Kepler’s Laws
• Planets move in an ellipse around the sun, with the
center of the sun being located at one focus. (The Law of
Ellipses)
• An imaginary line drawn from the center of the sun to the
center of the planet will sweep out equal areas in equal
intervals of time. (The Law of Equal Areas)
• The ratio of the squares of the periods of any two planets
is equal to the ratio of the cubes of their average
distances from the sun. (The Law of Harmonies)
Why do we care?
• All planetary motion can be
described using these laws.
• But…really no explanation for
WHY this path existed.
Kepler theorized
• Some sort of interaction between
the planets and the sun
• Kepler thought they were driven to
interact due to magnetism of the
sun.
Newton and Gravity
• Newton needed an
“explanation” for the planet’s
elliptical movement
• Felt that the moon’s orbiting
around the earth (circular) was
linked as well
What did Newton know
• He knew that there had to be an
inward source (toward the sun for
planets and toward the earth for the
moon) that kept the paths “circular”
• A CENTRIPETAL FORCE!!! But…what
caused it?
The Midrash states
• Newton discovered the force while
sitting under an apple tree!
• Whether myth or fact…Newton did
relate the cause for heavenly
motion, to the cause for earthly
motion (apple falling) which
eventually produced …..
The Theory of
Universal
Gravitation
How a Cannon proves
Universal Gravitation
• What happens when you shoot a
cannon ball
– Does it continue on a straight
path?
– Why?
– What happens when it is shot with
higher velocity?
• Now suppose that there is a
speed at which the cannonball
could be fired such that the
trajectory of the falling
cannonball matched the
curvature of the earth. What
would happen?
• What would happen if the speed
was faster, but not fast enough
to totally break gravity?
• Let’s watch the four paths
Launch Speed less than 8000 m/s
Projectile falls to Earth
B.
Launch Speed less than 8000 m/s
Projectile falls to Earth
Launch Speed equal to 8000 m/s
Projectile orbits Earth - Circular Path
D.
Launch Speed greater than 8000 m/s
Projectile orbits Earth - Elliptical
Path
A.
C.
What can we conclude?
• Path C is similar to the moon
• Path D is simlar to a planet
How do Satellites work
• The earth drops 5 m for every 8000 meters
of surface (it is round). Therefore if you
launch at 8000m/s, it will “go into orbit”.
Why do planets have a
“different path”
• Distance is key
• The acceleration of the moon toward
the earth was known (0.00272 m/s/s)
• The acceleration of an apple toward
the earth was known (9.8 m/s/s
Ratios were common
• gmoon = 0.0072m/s/s = 1___
gearth =9.8 m/s/s
3600
Distance from apple to center of earth is 1/60 the distance
From the moon to the center of the earth.
He knew squares and square roots well…if acceleration was linked
to distance, then the moon experiences a force of gravity which is
1/(602) that of the apple…. This is an inverse
square relationship
In pictures
The Inverse Square Law
• Distance is in the denominator
of this relationship,
– SO the force of gravity is inversely
related to the distance.
• Distance is raised to the second
power,
– SO the force of gravity is inversely
related to the square of the
distance.
The Inverse Square Law
• Provides sufficient evidence for
– Newton's explanation of why
gravity can be credited as the
cause of
• the falling apple's acceleration
• AND the orbiting moon's acceleration.
• AND the elliptical orbits of the
planets
But…do other factors
exist?
• F = ma but we took into
account only the ratios of the
accelerations
gmoon = 0.0072m/s/s =
gearth = 9.8 m/s/s
1___
3600
Force is dependent on mass AND acceleration. Mass must play a role.
It does!
• The larger the mass, the larger
for the force
• So…the larger the planets (or
sun) the stronger the force.
Magnitude of Fgrav is
Putting it all together
Proportionality and
Equality
• Right now our equation is a
proportionality.
• We can multiply the right side of
the equation with a constant to
change “proportional” to equal
Universal gravitation
Constant
• The precise value was
determined by Cavendish AFTER
Newton died.
• G (capital G is NOT the same as
g)
• G = 6.673 x 10-11 Nm2/kg2
Why is this helpful
• If we know G, we can calculate
the force of gravitational
attraction between ANY two
objects of known mass and
known separation distance.
• Determine the force of gravitational
attraction between the earth (m =
5.98 x 1024) and a 70 kg student if
the student is in an airplane at
40000 feet above the earth’s surface
(or 6.36 x 106 meters from the
earth’s center.
• Don’t forget to use G. It looks
really complicated, but add the
exponents (in the numerator),
but the hardest thing is tracking
the exponents.
What can we observe?
• Fgrav is less above the earth than on
the earth
– ie. the larger the distance the smaller
the force
– Weight change is very small (2 N)
• Fgrav = m*g
• BECAUSE g= (earth mass/r2)*G
The Univesality of
Gravity
• All objects with mass are attracted
to other objects
• You are attracted to the person next
to you, to ME, and to the desk
• But forces are very SMALL so are
only recognizable with large things
like planets, moons, stars and tides
Conclusion
• On a planet…
g = G*Mplanet
Rplanet2
The mass of the object is insignificant as compared to the
mass of the planet…
Value of g and location
This demonstrates the
inverse relationship of distance
and g (acceleration due to
gravity)
Planetary g
Planet
Radius (m)
Mass (kg)
g (m/s2)
Mercury
2.43 x 106
3.2 x 1023
3.61
Venus
6.073 x 106
4.88 x1024
8.83
Mars
3.38 x 106
6.42 x 1023
3.75
Jupiter
6.98 x 107
1.901 x 1027
26.0
Saturn
5.82 x 107
5.68 x 1026
11.2
Uranus
2.35 x 107
8.68 x 1025
10.5
Neptune
2.27 x 107
1.03 x 1026
13.3
Pluto
1.15 x 106
1.2 x 1022
0.61
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And Now ….
• BACK TO KEPLER’s LAWS
Elliptical orbits cause planetary motion to vary orbital speed.
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