Universal Gravitation.pptx

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And Kepler’s Laws

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ISAAC NEWTON (1642-1727)

• Born on Christmas Day, in same year Galileo died

• _{Father had recently died, Isaac was premature, and not expected to}

live

• _{Isaac’s mother remarried when he was 3, and he was left with his}

grandmother; resentment toward his mom gave him anxiety and insecurity issues for his whole life.

• _{Trained as an apothecary as a young man, but returned home when}

his mother’s husband died and she needed him to run the family farm. Did not succeed as a farmer.

• _{Mother convinced by local clergyman that Isaac should go to college.}

Entered Trinity College in 1661.

• _{In the summer of 1665, Trinity was closed due to the Black Plague.}

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NEWTON, CONTINUED…

• _{During the plague years, Newton concentrated on physics and math,}

and came up with his theories of Universal Gravitation, Optics, and Integral and Differential math, now known as calculus.

• _Published_Principia_{in 1687, stating his three famous laws of motion.} • _Published_Opticks_{in 1707. Proponent of a particle theory of light.} • _{Continued studies of alchemy, and was extremely religious his whole}

life

• _{Eccentric behavior may be explained by excessive mercury exposure} • _{Many honors given to Isaac Newton, including the fact that he was}

knighted by Queen Anne in 1705.

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SATELLITE MOTION

• Objects such as apples fall to the earth with a constant acceleration. Why didn’t the moon fall toward the earth in the same way?

• _{Newton’ cannon: A cannon shot from a top of}

a tall mountain accelerates downward and

continues moving at a constant speed. Path is a PARABOLA… and the faster it is launched from the cannon the farther it will go horizontally before landing.

• If the ground isn’t flat, however, the cannon will keep falling until it hits the curved surface of the ground.

• _{It is possible to shoot the cannon so fast that it}

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A SATELLITE IS HELD IN ORBIT BY GRAVITY

•

Isaac Newton hypothesized that the moon was attracted to the

earth in the same way that an apple was attracted to the ground.

However, they didn’t behave the same way.

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•

To follow his line of thinking, first calculate the centripetal

acceleration of the moon. The distance from the moon to the

earth is 3.84 x 10

8

m. The period of the moon is 27.3 days

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CENTRIPETAL ACCELERATION OF THE MOON…

v

=

2

p

r

T

=

2

*

p

*3.840

*

10

8

2358720

s

=

1023m/s

a

_c

=

v

2

r

=

(1023m/s)

2

3.840

*

10

8

m

=

2.725*10

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COMPARE TO ACCELERATION OF AN OBJECT

NEAR EARTH’S SURFACE, LIKE AN APPLE…

•

a of moon = 2.725 * 10

-3

m/s

2

•

a of apple near earth’s surface = 9.8 m/s

2

•

a

_moon

/a

_apple

=2.781 * 10

-4

which is about 1/3600

•

The moon’s distance from the center of the earth is

about 60 times the radius of the earth.

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THE LAW OF UNIVERSAL GRAVITATION

Where G is the universal gravitation constant –

G = 6.67 *10

-11

Nm

2

/kg

2

, m

1

and m

2

are two objects

exerting gravitational forces on each other, and r is the

distance between them.

F=

Gm

1

m

2

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CALCULATE:

• Very small! Gravity is a very weak force… you need a lot of mass

to make the force significant.

The gravitational force between you and the person sitting next to you

• Compare this force to your weight, finding it with Fg = mg

The gravitational force between you and the earth The gravitational force between you and the earth

• Fg=mg AND Fg=Gm1m2/r2

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General formula for the acceleration due to gravity for

any planet, any distance from the

center

CALCULATING THE ACCELERATION DUE TO GRAVITY

If we plug in the earth’s mass and radius, we get:

(6.67*10-11_Nm2_/kg2_)(5.98*1024_kg)/

(6.38*108_m)2₌_{9.8 m/s}2_!

F

_g

=

G

m

p

m

e

r

2

=

m

p

g

G

m

_e

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CALCULATING ORBITAL SPEED

Gravity provides the force to keep an object moving in a circle:

F

_net

=m

₁

a = F

_c

v=

Gm

2

r

where m

2

is the mass of the planet that is being orbited

Gm

₂

r

=v

2

Gm

₁

m

₂

r

2

=

m

₁

v

2

r

F

_g

=

m

₁

a

_c

=

m

1

v

2

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BONUS QUESTION - SATELLITES

Geostationary satellites are satellites which are orbiting the Earth above the equator and

make one complete orbit every 24 hours. Because their orbital period is synchronized with the Earth's rotational period, a geostationary satellite can always be found in the same position in the sky relative to an observer on Earth. (GIVEN: M_Earth = 5.98 x 1024_kg)

a. Determine the orbital radius of a geostationary satellite and its altitude.

b. Determine the orbital speed of a geostationary satellite.

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JOHANNES KEPLER (1571-1630)

• German astronomer and natural philosopher.

• Chair of astronomy and mathematics at

University of Graz from 1594 to 1600, when he became assistant to Tycho Brahe.

• Stole Brahe’s data, after he died, to formulate his three laws of planetary motion.

• A Lutheran, caught up in religious trouble of his time period. Persecuted by Catholics and

forced him to move several times.

• His mother was raised by an aunt who was burned at the stake for witchcraft; she narrowly escaped the same fate.

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KEPLER’S LAWS

• #1 - Law of Ellipses: The path of a planet’s orbit is an ellipse, and the sun is at one of the focal points.

• #2 – Law of Equal Areas: In their orbits about the sun, the planets sweep out equal areas in equal times

Gravity Lab