Universal Gravitation

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Universal Gravitation

• Chin-Sung Lin

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Isaac Newton
1643 - 1727
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Newton Physics
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Universal Gravitation

• Newtons 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

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Universal Gravitation - Apple
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Universal Gravitation - Moon
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Universal Gravitation - Moon
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Universal Gravitation

• Newtons question
• Can gravity be the force keeping the Moon in its
  orbit?
• Newtons approximation Moon is on a circular
  orbit
• Even if its orbit were perfectly circular, the
  Moon would still be accelerated

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• The Moons 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

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• The Moons 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

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The Moons Centripetal Acceleration

• The centripetal acceleration of the moon
• orbital speed v 103 m/s
• orbital radius r 3.8 x 108 m
• centripetal acceleration Ac v2 / r ???? m/s2

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The Moons Centripetal Acceleration

• The centripetal acceleration of the moon
• orbital speed v 103 m/s
• orbital radius 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

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The Moons 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
• Whats relation between them?

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The Moons 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

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Bottom Line
The Moons Centripetal Acceleration
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Bottom Line
The Moons Centripetal Acceleration

• 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)

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Bottom Line
The Moons Centripetal Acceleration

• 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 !!!

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Bottom Line
Gravitys Inverse Square Law

• The acceleration due to gravity is inverse
  proportional to the square of the distance
• Ac 1/r2
• The gravity is inverse proportional to the
  square of the distance
• Fg Fc m Ac Fg Ac Fg 1/r2

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Bottom Line
Gravitys Inverse Square Law

• Gravity is reduced as the inverse square of
  its distance from its source increased
• Fg 1/r2

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Bottom Line
Gravitys Inverse Square Law
Fg 1/r2
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Bottom Line
Gravitys Inverse Square Law
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Bottom Line
Gravitys Inverse Square Law
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
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Bottom Line
Gravitys Inverse Square Law
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
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Bottom Line
Gravitys Inverse Square Law
Location Distance from Earth's center (m) Value of g (m/s2)
Earth's surface 6.38 x 106 m 9.8
1000 km above 7.38 x 106 m 7.33
2000 km above 8.38 x 106 m 5.68
3000 km above 9.38 x 106 m 4.53
4000 km above 1.04 x 107 m 3.70
5000 km above 1.14 x 107 m 3.08
6000 km above 1.24 x 107 m 2.60
7000 km above 1.34 x 107 m 2.23
8000 km above 1.44 x 107 m 1.93
9000 km above 1.54 x 107 m 1.69
10000 km above 1.64 x 107 m 1.49
50000 km above 5.64 x 107 m 0.13
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Bottom Line
Law of Universal Gravitation
Newtons discovery Newton didnt 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
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Bottom Line
Law of Universal Gravitation
Universal gravitation formula Fg G m1 m2 /
d2 Fg gravitational force between
objects G universal gravitational
constant m1 mass of one object m2 mass of the
other object d distance between their centers
of mass
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Bottom Line
Law of Universal Gravitation
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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
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Bottom Line
Law of Universal Gravitation Example
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
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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
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Universal Gravitational Constant
The Universal Gravitational Constant (G) was
first measured by Henry Cavendish 150 years after
Newtons discovery of universal gravitation
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Henry Cavendish
1731 - 1810
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Universal Gravitational Constant

• Cavendishs 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

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Cavendishs Experiment
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Universal Gravitational Constant
G Fg d2 / m1 m2 6.67 x 10-11 Nm2/kg2 Fg
G m1 m2 / d2
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Calculate the Mass of Earth
G 6.67 x 10-11 Nm2/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
Earths radius (r), 6.4 x 106 m
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Calculate the Mass of Earth
G 6.67 x 10-11 Nm2/kg2 Fg G M m / r2 9.8
N 6.67 x 10-11 Nm2/kg2 x 1 kg x M / (6.4 x 106
m)2 where M is the mass of Earth M 6 x 1024
kg
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Universal Gravitational Force
G 6.67 x 10-11 Nm2/kg2 Fg G m1 m2 /
d2 Gravitational force is a VERY WEAK FORCE
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Universal Gravitational Force
G 6.67 x 10-11 Nm2/kg2 Gravity is is the
weakest of the presently known four fundamental
forces
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Universal Gravitational Force
Force Strong Electro-magnetic Weak Gravity
Strength 1 1/137 10-6 6x10-39
Range 10-15 m 8 10-18 m 8
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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
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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 Nm2/kg2)(55 kg)(45
kg)/(1 m)2 1.65 x 10-7 N
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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
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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
Nm2/kg2)(6.0 x 1024 kg) (7.4 x 1022 kg)/(3.8 x
108 m)2 2.1 x 1020 N
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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 gravitational force / weight G univ.
gravitational constant M mass of Earth m
mass of the object r radius of Earth g
acceleration due to gravity
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Universal Gravitation Example
Calculate the acceleration due to gravity of
Earth (mass 6.0 x 1024 kg, radius 6.37 106
m )
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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
Nm2/kg2)(5.98 x 1024 kg)/(6.37 x 106 m)2
9.83 m/s2
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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?
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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 Nm2/kg2)(1.00 x 104 kg)/(20.0 m)2
1.67 x 10-9 m/s2
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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
Earths 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?
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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
Earths 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
2.1 x 1020 N
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Relative Weight on Each Planet
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Isaac Newtons 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
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Summary

• Isaac Newton
• Universal gravitation Apple and Moon?
• Moons centripetal acceleration
• Gravitys inverse square law
• Law of universal gravitation
• Universal gravitational constant Henry
  Cavendish
• Calculate the mass of Earth
• Weak gravitational force
• Acceleration due to gravity
• Newtons influence