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GRAVITATION

GRAVITATION. Early Astronomy. As far as we know, humans have always been interested in the motions of objects in the sky. They used the sky to navigate and to predict the changing of the seasons.

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GRAVITATION

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  1. GRAVITATION

  2. Early Astronomy • As far as we know, humans have always been interested in the motions of objects in the sky. They used the sky to navigate and to predict the changing of the seasons. • There were many early attempts both to describe and explain the motions of stars and planets in the sky. • All were unsatisfactory, for one reason or another.

  3. Geocentric Solar System • Ptolemy came up with a geocentric (Earth-centered) solar system. • Ptolemy’s solar system could be made to fit the observational data pretty well, but only by becoming very complicated.

  4. Heliocentric Solar System • The Polish cleric Copernicus proposed a heliocentric (Sun centered) solar system in the 1500’s. • Objections: • How could Earth be moving at enormous speeds when we don’t feel it? (Copernicus didn’t know about inertia. • Copernicus’ model didn’t fit the observational data very well. • Galileo supported Copernicus’s idea – and was excommunicated from the Catholic church and placed under house arrest

  5. Tycho and Kepler • In the late 1500’s, a Danish nobleman namedTycho Braheset out to make the most accurate measurements of planetary motions to date, in order to validate his own ideas of planetary motion. • Tycho’s data was successfully interpreted by the German mathematician and scientist Johannes Keplerin the early 1600’s. • Kepler’s Description of planetary motion is known as Kepler’s Laws

  6. Planet Sun Kepler’sLaws : Kepler’s First Law • Kepler determined that the orbits of the planets were not perfect circles, but ellipses, with the Sun at one focus.

  7. Planet Sun Kepler’s Laws: Kepler’sSecond Law • Kepler determined that a planet moves faster when near the Sun, and slower when far from the Sun. The radius vector sweeps out equal areas in equal time intervals Slower Faster

  8. Kepler’s Laws: Kepler’s Third Law • Kepler found that the orbital speed of a planet is proportional to the cube of the semi-major axis of the elliptical orbit. • Usually you will see this law expressed in terms of the period of the orbit (T): • The orbital speed of a planet is 2πr/T

  9. WHY? • Kepler’s Laws provided a complete kinematicdescription of planetary motion (including the motion of planetary satellites, like the Moon) - but why did the planets move like that?

  10. The Apple & the Moon • Isaac Newton realized that the motion of a falling apple and the motion of the Moon were both actually the same motion, caused by the same force- the gravitationalforce.

  11. Universal Gravitation • Newton’s idea was that gravity was a universal force acting between any two objects.

  12. Law of Universal Gravitation • Newton found that the gravitational force is: • Directly proportionalto themassesof bothobjects • Inversely proportionalto the distance between the objects • Where G is a constant of proportionality • G = 6.67 x 10-11 N m2/kg2

  13. Law of Universal Gravitation: An Inverse-Square Force

  14. Deriving Kepler’s Third Law • Derive Kepler’s Third Law using the Law of Universal Gravitation and your knowledge of centripetal force • Remember that Kepler’s Third Law is:

  15. Practice Problem: • Three .30 kg billiard balls are on a table making the corners of a right triangle. Find the gravitational force on the cue ball (m1) as a result of the other two balls. (G = 6.67 x 10-11 N m2/kg2) m2 F21 = 3.75 X 10-11jN F31 = 6.67 X 10-11iN .40 m Ftot= 7.65 x 10-11 N at 29.3° .30 m m1 m3

  16. The Gravitational Field • In Newton’s time, there was a lot of discussion about HOW gravity worked. • How does the Sun, for instance, reach across empty space, with no actual contact at all, to exert a force on the Earth? • This spooky notion was called “action at a distance.” • During the 19th century, the notion of the “field” entered physics (via Michael Faraday). • Objects with mass create an invisible disturbance in the space around themthat is felt by other massive objects - this is a gravitational field.

  17. Gravitational Field Strength • To measure the strength of the gravitational field at any point, measure the gravitational force, F, exerted on any “test mass”, m • Gravitational Field Strength,g = F/m • Near the surface of the Earth, g = F/m = 9.8 N/kg = 9.8 m/s2 • In general, g = GM/r2, where M is the mass of the object creating the field, r is the distance from the object’s center, and G = 6.67 x10-11 Nm2/kg2

  18. Gravitational Force • If g is the strength of the gravitational field at some point, then the gravitational force on an object of mass m at that point is Fgrav= mg • If g is the gravitational field strength at some point (in N/kg), then the free fall acceleration at that point is also g (in m/s2)

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