Universal Gravitation Chin-Sung Lin. Isaac Newton 1643 - 1727

Universal Gravitation

Chin-Sung Lin

Isaac Newton

1643 - 1727

Newton & Physics

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

Newton’s question:

Can gravity be the force keeping the Moon in its orbit?

Newton’s approximation: Moon is on a circular orbit

Even if its orbit were perfectly circular, the Moon would still be accelerated

v

v

v

v


The Moon’s Orbital Speed

radius of orbit: r = 3.8 x 108 m

Circumference: 2r = ???? m

orbital period: T = 27.3 days = ???? sec

orbital speed: v = (2r)/T = ??? m/sec

The Moon’s Orbital Speed

radius of orbit: r = 3.8 x 108 m

Circumference: 2r = 2.4 x 109 m

orbital period: T = 27.3 days = 2.4 x 106 sec

orbital speed: v = (2r)/T = 103 m/sec = 1 km/s

The Moon’s 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


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

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?


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


r 2r 3r 4r 5r 6r 60r

g g g g g g g1 4 9 16 25 36 3600

Bottom LineThe Moon’s Centripetal Acceleration



Bottom LineGravity’s Inverse Square Law


r 2r 3r 4r 5r 6r 60r

Fg Fg Fg Fg Fg Fg Fg

1 4 9 16 25 36 3600


Fg ~ 1/r2



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%


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

LocationDistance fromEarth'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


Bottom LineLaw 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


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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m1m2

d

Fg Fg

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Fg = Gm1m2

d 2

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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 LineLaw 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

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


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


Calculate the Mass of Earth

Calculate the Mass of Earth

Universal Gravitational Force



Force StrongElectro-

magneticWeak Gravity

Strength 1 1/137 10-6 6x10-39

Range 10-15 m ∞ 10-18 m ∞

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


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


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


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: 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


Calculate the acceleration due to gravity of Earth (mass = 6.0 x 1024 kg, radius = 6.37 × 106 m )


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


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?


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


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?


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

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

• 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

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

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

Earth’s 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: weight of the objectG: universal gravitational constant (6.67 x 10-11 N·m2/kg2) m: mass of the objectM: mass of Earth (5.98 x 1024 kg)r: Earth’s radius (6.37 x 106 m)

Strength of Gravitational Field

Gravitational Field Inside a Planet


P

C

r


Cancellation of gravitational force

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

P

Cr


A BP


The gravity of area A and area B on P completely cancel out



You are pulled only by the mass within this shell – below you

At Earth’s center, the whole Earth is the shell and complete cancellation occurs

P

Cr


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: universal gravitational constant (6.67 x 10-11 N·m2/kg2) M: mass of Earth (5.98 x 1024 kg)D: density of EarthV: volume of Earthr: distance to the Earth’s center


a = g

a = g/2

a = g

a = g/2

a = 0


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

Documents

Universal Gravitation Chin-Sung Lin. Isaac Newton 1643 - 1727