2 Sec. 8.1 Motion in the Heavens and on Earth ObjectivesRelate Kepler’s laws of planetary motion to Newton’s law of universal gravitation.Calculate the periods and speeds of orbiting objects.Describe the method Cavendish used to measure G and the results of knowing G.
3 Kepler Johannes Kepler (1571-1630) Was convinced that geometry and mathematics could be used to explain the motion of the planets.
4 Kepler’s laws1. The orbits of the planets are ellipses, with the sun at one focus.(Law of Ellipses)The closer the planetsare to one another,the more circular theorbit.
5 Kepler’s Laws2. 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.(Law of Equal Areas)Deals with speed…faster when closerto the sun.
6 Kepler’s Laws3. 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.(Law of Harmonies)(TA/TB)2 = (rA/rB)3
7 Universal Gravitation Isaac Newton24yrs old…Watching an apple fall to the ground made him wonder if gravity extended beyond EarthCame up with universal gravitationAttractive force between two objectsThe apple was also attracting the EarthProposed Law of Universal Gravitation
8 Law of Universal Gravitation The force of attraction between any two masses is constant throughout the universeF = G(mAmB/d2)G is a universal gravitational constant between two masses6.67 x N·m2/kg2
9 Sec. 8.2 Using the Law of Universal Gravitation ObjectivesSolve problems involving orbital speed and periodRelate weightlessness to objects in free fallDistinguish between inertia mass and gravitational massContrast Newton’s and Einstein’s views about gravitation
10 Satellite MotionIf a projectile moves fast enough, it falls at the same rate that the Earth curves
11 How fast are satellites moving? F = ma or F = mv2/r(ac = v2/r)F = G(mAmB/d2)Solve for velocity? Set them equal to each otherG(mAmB/d2) = mv2/rwhich gives you….
12 Period of a Satellite Circling Earth T = 2π√r3/GmEor if we know the velocity…T = 2πr/v
13 Weightlessness What is gravity in outer space? Where space shuttle orbits…g = 8.7m/s2How come astronautsare “floating” then?g = F/m