1. Universal Gravitation 1546-1601 Tycho Brahe 1571-1630 Johannes Kepler 1. Law of Ellipses : –If e = 0, then the two foci are at the same location and it is a circle 2. 3. Kepler provided a framework for describing the motion and paths of the planets about the sun but not the WHY!!!

2. Con’t Newton wanted to know the cause of the elliptical motion of planets and the circular motion of the moon: –1. –2. –3. –4. –Universal Gravitation

3. Moon as a Projectile Orbiting moon ~ cannonball ~ apple Laws of mechanics apply universally

4. Newton’s Evidence for Gravity: Inverse Square Law INVERSE SQUARE LAW: F grav ~1/d 2

5. Newton’s Law of Universal Gravitation Distance is not the only variable affecting the magnitude of a gravitational force Fnet=ma Force causing the apple to fall must be dependent (directly proportional) on the mass of the apple Same force causes Earth’s upward acceleration (Newton’s 3 rd law) Force is also dependent (directly proportional) on mass of Earth Force is inversely proportional to square the distance UNIVERSAL not just here on Earth

6. Effect of Mass on Fgravity F gravity =Gm 1 m 2 /d 2 G=6.67x10 -11 Nm 2 /kg 2

7. G constant F gravity =Gm 1 m 2 /d 2 G=6.67x10 -11 Nm 2 /kg 2

8. Universality ALL objects attract in proportion to their mass Ex. Jupiter approaches Saturn in its orbit, it deviates from its smooth path (perturbation) Uranus had an orbit that Newton’s law of gravity did not correctly predict Proposed that another planet was attracting Uranus in addition to the sun (Giant: Neptune)

9. Value of g F gravity =Gm 1 m 2 /d 2 F gravity =mg mg=Gm 1 m 2 /d 2 M Earth : 5.98x10 24 kg g=GM Earth /d 2 d Earth : 6.38x10 6 kg

10. Inverse Law Value of g varies inversely with the distance from the center of the earth g=G M planet /R planet 2

11. Equations F gravity =Gm 1 m 2 /d 2 F gravity =mg F=ma (F is gravitational Force) (a is centripetal acceleration) a c =(4pi 2 )/(T 2 ) F c =m p (4pi 2 )/(T 2 ) mg=Gm 1 m 2 /d 2 (Ta/Tb) 2 =(Ra/Rb) 3 F c = (M p v 2 )/R (M p v 2 )/R=Gm p m s /R 2 v=(2piR)/T T 2 /R 3 =(4pi 2 )/(GM s ) T 2 =(4pi 2 /GM s ) R 3 g=GM Earth /d 2