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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!!! 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 Moon as a Projectile • Orbiting moon ~ cannonball ~ apple • Laws of mechanics apply universally Newton’s Evidence for Gravity: Inverse Square Law • INVERSE SQUARE LAW: Fgrav~1/d2 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 3rd 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 Effect of Mass on Fgravity • Fgravity=Gm1m2/d2 G=6.67x10-11 Nm2/kg2 G constant • Fgravity=Gm1m2/d2 • G=6.67x10-11 Nm2/kg2 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) Value of g • Fgravity=Gm1m2/d2 • mg=Gm1m2/d2 • g=GMEarth/d2 Fgravity=mg MEarth: 5.98x1024 kg dEarth: 6.38x106 kg Inverse Law • Value of g varies inversely with the distance from the center of the earth • g=G Mplanet/Rplanet2 Equations • Fgravity=Gm1m2/d2 Fgravity=mg • F=ma (F is gravitational Force) (a is centripetal acceleration) • ac=(4pi2)/(T2) Fc=mp(4pi2)/(T2) • mg=Gm1m2/d2 (Ta/Tb)2=(Ra/Rb)3 • Fc= (Mpv2)/R (Mpv2)/R=Gmpms/R2 • v=(2piR)/T T2/R3=(4pi2)/(GMs) • T2=(4pi2/GMs) R3 • g=GMEarth/d2
