Universal Gravitation

Recap•You’ve been exposed to basic Fg •Standing on Earth, you experience gravitational

force.• Jumping, you still experience it. It slows you down

until you reach the peak of your motion, then it pulls you down faster and faster.

•Gravity provides an acceleration on your body while in the air. (g- the ACCELERATION due to gravity)

•The value for g on Earth is 9.8 m/s2, regardless of mass. Regardless of mass. Regardless of mass. Regardless of mass? Yes, regardless of mass.

Kepler’s 3 Laws• Johannes Kepler: German mathematician and

astronomer.•Developed 3 laws to explain the planetary orbits

about the Sun.•Developed from years of data collection by his

Danish teacher, Tycho Brahe.

Kepler’s 3 Laws

Kepler (German) Brahe (Danish)

•Law of Ellipses: the paths of the planets are elliptical in shape, with the center of the sun being located at one focus.

•Law of Equal Areas: an imaginary line drawn from the center of the sun to the center of the planet will sweep out equal areas in equal time intervals.

•Law of Harmonies: the ratio of the squares of the periods of any 2 planets is equal to the ratio of the cubes of their avg distances from the sun.

•They provided a what, but not a why•Kepler believed the planets were “magnetically”

driven by the sun to orbit elliptically.• Isaac Newton hates not having explanations for

things.

•Newton wanted to know why the planets orbit the sun elliptically, but the moon has a circular orbit around the Earth.

•And, both circular and elliptical paths were departures from inertial straight line paths.

•What force is causing them to change direction (accelerate)?

•According to legend, it all came together for him in an apple orchard at age 24.

•Newton was able to relate the cause for heavenly motion (the orbit of the moon around the Earth) to the cause for Earthly motion (why things fall).

Newton’s Mountain

•His dilemma now was to explain why the moon doesn’t come crashing in.

•He needed to show that gravity became weaker over a distance.

•Newton knew things on Earth accelerated at 9.8 m/s2.• It had also been measured that the moon accelerates

towards Earth at 0.00272 m/s2.•Why does the moon accelerate at a MUCH slower

rate?

•Solved the riddle by looking at the distance from the center of the Earth.

•The moon is 60x further from the center of the Earth. It also experiences 3600x less gravitational acceleration.

• (602 = 3600).•Gravity must follow an inverse square law.

•The force of gravity is inversely related to the square of the distance.

• I.e., the further away you get, the less gravitational force.

Law of Universal Gravitation•Newton goes into the HOF, not for discovering

gravity, but for realizing it’s UNIVERSAL.•All objects have a gravitational force of attraction

between them.

•Lord Henry Cavendish discovered the value of the constant G (universal gravitation constant) in 1798, nearly 100 years after Newton.

•G = 6.673 x 10-11 Nm2/kg2

Fg = Gm1m2

d2

Example 1•Determine the force of gravitational attraction

between the earth (m = 5.98 x 1024 kg) and a 70-kg physics student if the student is standing at sea level, a distance of 6.38 x 106 m from earth's center.

•Determine the force of gravitational attraction between the earth (m = 5.98 x 1024 kg) and a 70-kg physics student if the student is in an airplane at 40000 feet above earth's surface. This would place the student a distance of 6.39 x 106 m from earth's center.

•We know that Fg = mg.•We know that Fg = Gm1m2

d2

So, m1g = Gm1m2

d2

g = Gmplanet

R2

Where R is the radius of the planet