1. Newton’s Law of Gravitation The “4 th Law”

2. Quick Review NET FORCE IS THE SUM OF FORCES… IT IS NOT ACTUALLY A FORCE ON ITS OWN!

3. Sir Isaac Newton  Remember him?  Well… he’s very interesting.interesting  Remember him?  Well… he’s very interesting.interesting

4. Newton’s Universal Law of Gravitation  Every object in the universe attracts every other object with a force that is directly proportional to the masses of the bodies and inversely proportional to the square of the distance between the object.

5. Newton’s Universal Law of Gravitation

7. Example  Determine the gravitational attraction between the sun and the Earth at its mean radius of 1.5 x 10 8 m.  m Earth = 5.98 x 10 24 kg  m sun = 1.99 x 10 30 kg  Determine the gravitational attraction between the sun and the Earth at its mean radius of 1.5 x 10 8 m.  m Earth = 5.98 x 10 24 kg  m sun = 1.99 x 10 30 kg

9. Example  Determine the gravitational attraction between Jane (m = 100 kg) and the Earth if Jane is on the surface.  m Earth = 5.98 x 10 24 kg  r Earth = 6.38 x 10 6 m  Determine the gravitational attraction between Jane (m = 100 kg) and the Earth if Jane is on the surface.  m Earth = 5.98 x 10 24 kg  r Earth = 6.38 x 10 6 m

11. Example  If Tarzan (m = 60 kg) is 3 m from Jane (m = 100 kg), determine the gravitational force of attraction between Tarzan and Jane.

13. Finding ‘g’:  How is g determined?  ‘g’ is the ACCELERATION DUE TO GRAVITY.  Two equations for F g :  How is g determined?  ‘g’ is the ACCELERATION DUE TO GRAVITY.  Two equations for F g :

15. Example  Use Newton's law of gravitation to determine the acceleration of an 85-kg astronaut on the International Space Station (ISS) when the ISS is at a height of 350 km above Earth's surface. The radius of the Earth is 6.37 x 10 6 m. (GIVEN: M Earth = 5.98 x 10 24 kg)

17. Example  Determine the acceleration of the Earth about the sun. (GIVEN: M sun = 1.99 x 10 30 kg and Earth-sun distance = 1.50 x 10 11 m)

19. Gravitational Motion:

20. Example  Determine the orbital speed of the International Space Station - orbiting at 350 km above the surface of the Earth. The radius of the Earth is 6.37 x 10 6 m. (GIVEN: M Earth = 5.98 x 10 24 kg)

22. Example  Hercules is hoping to put a baseball in orbit by throwing it horizontally (tangent to the Earth) from the top of Mount Newton - 97 km above Earth's surface. With what speed must he throw the ball in order to put it into orbit? (GIVEN: M Earth = 5.98 x 10 24 kg; R Earth = 6.37 x 10 6 m)

24. Kepler’s Laws Mathematics of Planetary Motion

25. Newton and Kepler  Newton uses gravity to explain Kepler’s mathematics.  Satellites go around in circular orbits due to gravity  Newton uses gravity to explain Kepler’s mathematics.  Satellites go around in circular orbits due to gravity

26. Try These:

27. Review of the Ellipse  An ellipse is the set of all points on a plane whose distance from two fixed points F and G add up to a constant.  F and G are focal points (foci)  An ellipse is the set of all points on a plane whose distance from two fixed points F and G add up to a constant.  F and G are focal points (foci)

28. Review of the Ellipse  Major Axis – longest diameter  Minor Axis – shortest diameter  Semi-Major Axis – half of the major axes (this will be important later)  Major Axis – longest diameter  Minor Axis – shortest diameter  Semi-Major Axis – half of the major axes (this will be important later)

29. Kepler’s Laws  3 Mathematical Laws  Took him 30 years to complete his research after Brahe dies.  Mathematical laws and described the motion.  He dies not knowing how any of it worked.  3 Mathematical Laws  Took him 30 years to complete his research after Brahe dies.  Mathematical laws and described the motion.  He dies not knowing how any of it worked.

30. Kepler’s First Law  The orbits of the planets are elliptical, with the sun at one focus of the ellipse.  Perihelion (Perigee) - point of closest approach distance  Aphelion (Apogee) - point of furthest approach distance  The orbits of the planets are elliptical, with the sun at one focus of the ellipse.  Perihelion (Perigee) - point of closest approach distance  Aphelion (Apogee) - point of furthest approach distance

31. Eccentricity  A measure of elongation of ellipse  r a = radius of aphelion  r p = radius of perihelion  if e = 0, then ellipse is a circle  if e = 1, then ellipse is a parabola  A measure of elongation of ellipse  r a = radius of aphelion  r p = radius of perihelion  if e = 0, then ellipse is a circle  if e = 1, then ellipse is a parabola

32. Example  The Earth has an eccentricity of 1.67%. If the aphelion distance is 1.52 x 10 8 km, what is the perihelion distance?

34. Example  If the aphelion distance is 5r and the perihelion distance is 3r, determine the eccentricity.