Transcript Presentation Lesson 10 Universal Gravitation

Universal Gravitation
Chin-Sung Lin
Isaac Newton
1643 - 1727
Newton & Physics
Universal Gravitation
Newton’s Law of Universal Gravitation
states that gravity is an attractive
force acting between all pairs of
massive objects.
Gravity depends on:
 Masses of the two objects
 Distance between the objects
Universal Gravitation - Apple
Universal Gravitation - Moon
Universal Gravitation - Moon
Universal Gravitation
Newton’s question:
Can gravity be the force
keeping the Moon in its
v
orbit?
v
v
v
Newton’s approximation: Moon
is on a circular orbit
Even if its orbit were perfectly
circular, the Moon would still
be accelerated
The Moon’s Orbital Speed
radius of orbit: r = 3.8 x 108 m
Circumference: 2pr = ???? m
orbital period:
T = 27.3 days = ???? sec
orbital speed:
v = (2pr)/T = ??? m/sec
The Moon’s Orbital Speed
radius of orbit: r = 3.8 x 108 m
Circumference: 2pr = 2.4 x 109 m
orbital period:
T = 27.3 days = 2.4 x 106 sec
orbital speed:
v = (2pr)/T = 103 m/sec = 1 km/s
The Moon’s Centripetal Acceleration
The centripetal acceleration of the moon:
orbital speed:
orbital radius:
v = 103 m/s
r = 3.8 x 108 m
centripetal acceleration: Ac = v2 / r = ???? m/s2
The Moon’s Centripetal Acceleration
The centripetal acceleration of the moon:
orbital speed:
orbital radius:
v = 103 m/s
r = 3.8 x 108 m
centripetal acceleration: Ac = v2 / r
Ac = (103 m/s)2 / (3.8 x 108 m) = 0.00272 m/s2
The Moon’s Centripetal Acceleration
At the surface of Earth (r = radius of Earth)
a = 9.8 m/s2
At the orbit of the Moon (r = 60x radius of Earth)
a =0.00272 m/s2
What’s relation between them?
The Moon’s Centripetal Acceleration
At the surface of Earth (r = radius of Earth)
a = 9.8 m/s2
At the orbit of the Moon (r = 60x radius of Earth)
a =0.00272 m/s2
9.8 m/s2 / 0.00272 m/s2 =
3600 / 1 =
602 / 1
LineAcceleration
The Moon’sBottom
Centripetal
r
2r
3r
4r
5r
6r
60r
g
1
g
4
g
9
g
16
g
25
g
36
g
3600
LineAcceleration
The Moon’sBottom
Centripetal
If the acceleration due to gravity is
inverse proportional to the square of the
distance, then it provides the right
acceleration to keep the Moon on its orbit
(“to keep it falling”)
LineAcceleration
The Moon’sBottom
Centripetal
If the acceleration due to gravity is
inverse proportional to the square of the
distance, then it provides the right
acceleration to keep the Moon on its orbit
(“to keep it falling”)
The moon is falling as the apple does
!!! Triumph for Newton !!!
Bottom
Line Law
Gravity’s Inverse
Square
The acceleration due to gravity is inverse
proportional to the square of the distance
Ac ~
2
1/r
The gravity is inverse proportional to the
square of the distance
Fg = Fc = m Ac
Fg ~Ac Fg ~ 1/r2
Bottom
Line Law
Gravity’s Inverse
Square
Gravity is reduced as the inverse square
of its distance from its source increased
Fg ~
r
Fg
1
2
1/r
2r
3r
4r
5r
6r
60r
Fg
4
Fg
9
Fg
16
Fg
25
Fg
36
Fg
3600
Bottom
Line Law
Gravity’s Inverse
Square
Fg ~ 1/r2
Bottom
Line Law
Gravity’s Inverse
Square
Bottom
Line Law
Gravity’s Inverse
Square
Gravity decreases with altitude, since
greater altitude means greater distance
from the Earth's centre
If all other things being equal, on the top of
Mount Everest (8,850 metres), weight
decreases about 0.28%
Bottom
Line Law
Gravity’s Inverse
Square
Astronauts in orbit are NOT weightless
At an altitude of 400 km, a typical orbit of
the Space Shuttle, gravity is still nearly
90% as strong as at the Earth's surface
Bottom
Line Law
Gravity’s Inverse
Square
Location
Distance from
Earth's center
(m)
Value of g
(m/s2)
Earth's
surface
6.38 x 106 m
9.8
6000 km
above
1.24 x 107 m
2.60
1000 km
above
7.38 x 106 m
7.33
7000 km
above
1.34 x 107 m
2.23
2000 km
above
8.38 x 106 m
5.68
8000 km
above
1.44 x 107 m
1.93
3000 km
above
9.38 x 106 m
4.53
9000 km
above
1.54 x 107 m
1.69
4000 km
above
1.04 x 107 m
3.70
10000 km
above
1.64 x 107 m
1.49
5000 km
above
1.14 x 107 m
3.08
50000 km
above
5.64 x 107 m
0.13
Bottom
Line
Law of Universal
Gravitation
Newton’s discovery
Newton didn’t discover gravity. In stead, he
discovered that the gravity is universal
Everything pulls everything in a beautifully
simple way that involves only mass and
distance
Bottom
Line
Law of Universal
Gravitation
Universal gravitation formula
Fg = G m1 m2 / d2
Fg:
G:
m1:
m2:
d:
gravitational force between objects
universal gravitational constant
mass of one object
mass of the other object
distance between their centers of mass
Bottom
Line
Law of Universal
Gravitation
d
m1
Fg
Fg
m1m 2 
Fg  G  2 
 d 
m2
p.83
Bottom
Line
Law of Universal
Gravitation
Fg = G m1 m2 / d2
Gravity is always there
Though the gravity decreases rapidly with
the distance, it never drop to zero
The gravitational influence of every object,
however small or far, is exerted through all
space
Bottom
Line Example
Law of Universal
Gravitation
Mass 1
Mass 2
Distance
Relative Force
m1
m2
d
F
2m1
m2
d
m1
3m2
d
2m1
3m2
d
m1
m2
2d
m1
m2
3d
2m1
2m2
2d
Law of Universal Gravitation Example
Mass 1
Mass 2
Distance
Relative Force
m1
m2
d
F
2m1
m2
d
2F
m1
3m2
d
3F
2m1
3m2
d
6F
m1
m2
2d
F/4
m1
m2
3d
F/9
2m1
2m2
2d
F
Universal Gravitational Constant
The Universal Gravitational Constant (G)
was first measured by Henry Cavendish
150 years after Newton’s discovery of
universal gravitation
Henry Cavendish
1731 - 1810
Universal Gravitational Constant
Cavendish’s experiment
 Use Torsion balance (Metal thread, 6-foot
wooden rod and 2” diameter lead sphere)
 Two 12”, 350 lb lead spheres
 The reason why Cavendish measuring the G
is to “Weight the Earth”
 The measurement is accurate to 1% and his
data was lasting for a century
Cavendish’s Experiment
Universal Gravitational Constant
G = Fg d2 / m1 m2 = 6.67 x 10-11 N·m2/kg2
Fg = G m1 m2 / d2
Calculate the Mass of Earth
G = 6.67 x 10-11 N·m2/kg2
Fg = G M m / r2
The force (Fg) that Earth exerts on a mass (m) of
1 kg at its surface is 9.8 newtons
The distance between the 1-kg mass and the
center of Earth is Earth’s radius (r), 6.4 x 106 m
Calculate the Mass of Earth
G = 6.67 x 10-11 N·m2/kg2
Fg = G M m / r2
9.8 N = 6.67 x 10-11 N·m2/kg2 x 1 kg x M / (6.4 x 106 m)2
where M is the mass of Earth M = 6 x 1024 kg
Universal Gravitational Force
G = 6.67 x 10-11 N·m2/kg2
Fg = G m1 m2 / d2
Gravitational force is a
VERY WEAK FORCE
Universal Gravitational Force
G = 6.67 x 10-11 N·m2/kg2
Gravity is is the weakest of the presently
known four fundamental forces
Universal Gravitational Force
Force
Strong
Electromagnetic
Strength
1
1/137
10-6
6x10-39
Range
10-15 m
∞
10-18 m
∞
Weak
Gravity
Universal Gravitation Example
Calculate the force of gravity between two students
with mass 55 kg and 45kg, and they are 1 meter
away from each other
Universal Gravitation Example
Calculate the force of gravity between two students
with mass 55 kg and 45kg, and they are 1 meter
away from each other
Fg = G m1 m2 / d2
Fg = (6.67 x 10-11 N·m2/kg2)(55 kg)(45 kg)/(1 m)2
= 1.65 x 10-7 N
Universal Gravitation Example
Calculate the force of gravity between Earth (mass
= 6.0 x 1024 kg) and the moon (mass = 7.4 x 1022
kg). The Earth-moon distance is 3.8 x 108 m
Universal Gravitation Example
Calculate the force of gravity between Earth (mass
= 6.0 x 1024 kg) and the moon (mass = 7.4 x 1022
kg). The Earth-moon distance is 3.8 x 108 m
Fg = G m1 m2 / d2
Fg = (6.67 x 10-11 N·m2/kg2)(6.0 x 1024 kg)
(7.4 x 1022 kg)/(3.8 x 108 m)2
= 2.1 x 1020 N
Acceleration Due to Gravity
Law of Universal Gravitation:
Fg = G m M /
r2
Weight
Fg = m g
Acceleration due to gravity
g = G M / r2
Fg:
G:
M:
m:
r:
g:
gravitational force / weight
univ. gravitational constant
mass of Earth
mass of the object
radius of Earth
acceleration due to gravity
Universal Gravitation Example
Calculate the acceleration due to gravity of Earth
(mass = 6.0 x 1024 kg, radius = 6.37 × 106 m )
Universal Gravitation Example
Calculate the acceleration due to gravity of Earth
(mass = 6.0 x 1024 kg, radius = 6.37 × 106 m )
g = G M / r2
g = (6.67 x 10-11 N·m2/kg2)(5.98 x 1024 kg)/(6.37 x 106 m)2
= 9.83 m/s2
Universal Gravitation Example
In The Little Prince, the Prince visits a small asteroid
called B612. If asteroid B612 has a radius of only
20.0 m and a mass of 1.00 x 104 kg, what is the
acceleration due to gravity on asteroid B612?
Universal Gravitation Example
In The Little Prince, the Prince visits a small asteroid
called B612. If asteroid B612 has a radius of only
20.0 m and a mass of 1.00 x 104 kg, what is the
acceleration due to gravity on asteroid B612?
g = G M / r2
g = (6.67 x 10-11 N·m2/kg2)(1.00 x 104 kg)/(20.0 m)2
= 1.67 x 10-9 m/s2
Universal Gravitation Example
The planet Saturn has a mass that is 95 times as
massive as Earth and a radius that is 9.4 times
Earth’s radius. If an object is 1000 N on the surface
of Earth, what is the weight of the same object on
the surface of Saturn?
Universal Gravitation Example
The planet Saturn has a mass that is 95 times as
massive as Earth and a radius that is 9.4 times
Earth’s radius. If an object is 1000 N on the surface
of Earth, what is the weight of the same object on
the surface of Saturn?
Fg = G m M / r2
Fg ~ M / r2
Fg = 1000 N x 95 / (9.4)2 = 1075 N
Relative Weight on Each Planet
Isaac Newton’s Influence
Defined the World
People could uncover the workings of the physical
universe
Moons, planets, stars, and galaxies have such a
beautifully simple rule to govern them
Phenomena of the world might also be described by
equally simple and universal laws
Summary
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Isaac Newton
Universal gravitation – Apple and Moon?
Moon’s centripetal acceleration
Gravity’s inverse square law
Law of universal gravitation
Universal gravitational constant – Henry Cavendish
Calculate the mass of Earth
Weak gravitational force
Acceleration due to gravity
Newton’s influence
Gravitational Interaction
Chin-Sung Lin
Gravitational Field
Force Field
A force field exerts a force on objects in its vicinity
Magnetic Field
A magnetic field is a force field that surrounds a magnet
and exerts a magnetic force on magnetic substances
Electric Field
An electric field is a force field surrounding electric
charges
Gravitational Field
Gravitational Field
A gravitational field is a force field
that surrounds massive objects
Earth’s Gravitational Field
Earth’s Gravitational Field
 Earth’s gravitational field is
represented by imaginary field lines
 Where the field lines are closer together, the
gravitational field is stronger
 The direction of the field at any point is along the
line the point lies on
 Arrows show the field direction
 Any mass in the vicinity of Earth will be accelerated
in the direction of the field line at that location
Strength of Gravitational Field
Strength of the gravitational field is the force per unit mass
exerted by Earth on any object
Gravitational Field
g = Fg / m = (G m M / r2) / m = G M / r2
F:
G:
m:
M:
r:
weight of the object
universal gravitational constant (6.67 x 10-11 N·m2/kg2)
mass of the object
mass of Earth (5.98 x 1024 kg)
Earth’s radius (6.37 x 106 m)
Gravitational Field Inside a Planet
Gravitational Field Inside a Planet
P
r
C
Gravitational Field Inside a Planet
Cancellation of gravitational force
P
If Earth were of uniform density, the
gravity of the entire surrounding
shell of inner radius equal to your
radial distance from the center will
completely cancel out
r
C
Gravitational Field Inside a Planet
Cancellation of gravitational force
The gravity of area A and area B
on P completely cancel out
A
P
B
Gravitational Field Inside a Planet
Cancellation of gravitational force
P
You are pulled only by the mass
within this shell – below you
r
C
At Earth’s center, the whole Earth
is the shell and complete
cancellation occurs
Gravitational Field Inside a Planet
Strength of the gravitational field is proportional to M / r2
g = GM/r2 = GDV/r2 = GD(4/3)πr3/r2 = (4/3)GDπr
g ~ r
G:
M:
D:
V:
r:
universal gravitational constant (6.67 x 10-11 N·m2/kg2)
mass of Earth (5.98 x 1024 kg)
density of Earth
volume of Earth
distance to the Earth’s center
Gravitational Field Inside a Planet
a=g
a = g/2
a=0
a = g/2
a=g
Gravitational Field Inside a Planet
Without air drag, the trip would
take nearly 45 minutes. The
gravitational field strength is
steadily decreasing as you
continue toward the center
At the center of Earth, you are
pulled in every direction equally,
so that the net force is zero and
the gravitational field is zero