Newton’s law of Universal Gravitation We will be considering a lot of individual topics.

Specific Topics  The inverse square law  How did Newton figure it out?  Universal law to W = mg  Gravitational Field  General “g” and Latitude  General “g” and Altitude; weightlessness  See the next slide

You mean, there is still more??  Yup  Kepler’s Laws  Discovery of New Planets  Explanation of Tides

Okay don’t get overwhelmed I’ll be doing most of the math just to, hopefully, prove to you that I’m not just making it all up. Mathematically, you need to know the inverse square law, its ramifications, and the gravitational field. The rest is conceptual.

Inverse Square law The force of gravity depends inversely on the square of the distance between the two objects interacting gravitationally. Triple the distance, force decreases by 9 Halve the distance, force increases by 4 A few numerical examples (see the board)

How did Newton know?  Newton assumes that gravity both (1) keeps the moon in orbit and (2) pulls the apple to the ground (gives us “g”).  Calculate a c = a moon = v 2 /r  Calculate a apple = g (measure)  Compare a c /a apple to R apple /R moon  Results agree “pretty nearly”.

General gravitational field Define general gravitational field “g” = G M/R 2 with G = 6.67 × 10 –11 Nm 2 /kg 2 See board for sample calculations. The Force of gravity is F gr = mass × “g”

So why does Weight = mg? We can calculate the gravitational field at surface of earth. Get “g” = 9.83 m/s 2. Why not 9.81 m/s 2 ? Rotation of earth; depends on latitude. Gives measured acceleration due to gravity.

Orbital Motion Only “some” of gravity is used to keep object moving in a circle; the “rest” is used to push an object against a scale. If all of gravity is “used up” keeping the orbiting object in a circular orbit, then there is nothing left to push against a scale: apparent weightlessness.

Kepler’s Laws (1 and 2)  Gravitational force is an inverse square law leads to Kepler’s first law (ellipses)  Gravitational force is along the line between the two bodies leads to Kepler’s second law (equal areas  equal times)

Kepler’s Third law  Gravity provides the centripetal force for orbiting bodies leads to Kepler’s Third law, R 3 / T 2 = constant  The constant depends on the mass of the central body (sun and planets, earth and satellites)

Geosynchronous Orbit Use Kepler’s third law, with Earth at the center, and T = 23 hours 56 minutes to determine where a sattelite should be put. R = 42 million meters, height = 26,000 miles

Discovery of New Planets  Neptune and Pluto (?)  Planets around other stars  Broke down with Mercury (General Relativity)

Explained Tides Different pulls on “close side” of water; earth’s center; and “far side” of water gives two high and two low tides per day. Bay of Fundy in Nova Scotia; tides can change by about 15 meters (50 feet!)

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