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

                                                                                                                

Launch Speed less than 8000 m/s Projectile falls to

Earth

                                                                                                                

Launch Speed equal to 8000 m/s Projectile orbits Earth - Circular

Path

                                                                                                                

Launch Speed greater than 8000 m/s

Projectile orbits Earth - Elliptical Path

                                                                                                                

A. B.

C. D.

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 3600Distance 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 linkedto 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 = 1___ gearth = 9.8 m/s/s 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 demon-strates the inverse relation-ship of distance and g (accel-eration due to gravity)

Planetary gPlanet 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

http://www.ioncmaste.ca/homepage/resources/web_resources/CSA_Astro9/files/multimedia/unit4/weight_planets/weight_planets.html

And Now ….

• BACK TO KEPLER’s LAWS

Elliptical orbits cause planetary motion to vary orbital speed.http://www.ioncmaste.ca/homepage/resources/web_resources/CSA_Astro9/files/multimedia/unit4/planetary_orbits/planetary_obits.html

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