1 Lecture 27 Newton?s Law of Gravity Newton and the Moon What is the centripetal acceleration of the Moon for its rotation around the Earth? We know (Newton also knew): T = 27.3 days = 2.36 × 10 6 s r moon-Earth = 3.84 × 10 8 m R Earth = 6.35 × 10 6 m (see Appendix) r moon-Earth R Earth Newton noticed that 2 2 0.000273 E R r = Angular frequency of the moon: 2 0.00272 m/s 0.000278 c ar g?== = 2 pointing toward the Earth This inspired him to suggest that F gravity ? 1/r 2 Yes!! = 0.000278 c a g ? ===× -6 -1 1rotation ... 2.66 10 s 27.3 day = 2 E constant g R = 2 constant a r 2 Newton?s Law of Gravitation = 12 2 mm FG r Two point-like bodies of mass m 1 and m 2 that are separated by a distance r attract each other with a force with magnitude ? =× 2 11 2 Nm 6.67 10 kg G and direction along the line between both bodies. Gravitational constant m 1 r F F m 2 Spherical shell theorem An important result that we will not prove until Gauss?s law in 222. The gravitational force exerted by a spherically symmetric object of radius R and mass M is the same as the force exerted by a particle of mass M located at the center of the sphere, for distances r ? R. m 2 m 1 r m 2 m 1 r Same force in both cases Weight Force by the Earth on an object of mass m at distance h from the ground = + E 2 E () g M F m G Rh E Rh E 2 E mM G R = mg ? E 2 E M gG R Example of application of the spherical shell theorem: Object near the surface of the Earth ACT: Earth model Paul?s weight on Earth is 1000 N. What will be his weight if he stands on the surface of a scale model of the Earth, made of the same type of material but scaled to half the size? A. 250 N B. 500 N C. 4000 N VIDEO: On the Moon E 3 X X E22 X E 1 2 2 2 M M g GG g R R == = ?? ?? ?? Mass scales like volume 3 Gravity is a very weak force Example: Find the magnitude of the gravitational attraction between two 3-kg books sitting 1 cm apart. Completely negligible. ?? ? =× =× 22 11 6 222 (3 kg)Nm (6.67 10 ) 6.0 10 N !!! kg (10 m) F Gravitational force matters when at least one of the objects is very massive, like stars, planets or galaxies. In fact, it is the dominant force in the structures at astronomical scale. Outer 12 1 Astronomical Unit (AU) = 1.5×10 11 m (Average distance Earth-Sun) Outer 13 ~10 20 mfor our galaxy 4 Gravitational potential A B M m r A r B Find the work done by gravity on an object of mass m that moves from point A to point B in the vicinity of a planet of mass M. Gravitational forces are conservative. A B M m r A r B ? = ? ? G G AB WFdl ?? =? ? ?? ?? 1 B A r r GMm r ?? =? ? + ?? ??BA GMm GMm rr =? ? 2 B rA r Mm Gdr r For any path, the dot product selects the radial part of dl =? +( ) constant g GMm Ur r Gravitational potential energy ( )=? ? BA UU Example: How high? A projectile of mass m is launched straight up from the surface of the Earth with initial speed v 0 . What is the maximum distance from the center of the Earth it reaches before falling back down? R MAX ?+ =? 2 0 11 1 2 MAXEE v RGMR = ? 2 0 1 2 E MAX E E R R vR GM M E m v 0 2 000 1 2 E E Mm UG KE mv R =? = R MAX 0 E ff MAX Mm UG KE R = ?= 2 0 1 2 EE MAXE Mm Mm GmvG RR ?+=? Conservation of mechanical energy Note: Independent of m ! 5 =????= ? 2 esc 2 esc 10 2 1 2 EE MAX E E E RvR R vR GM GM What is the minimum initial speed of a projectile of mass m to escape from Earth? Escape = reach infinity (where the attraction of Earth is zero) ()= esc 2= 11.2 km/s Es cape speed E vgR == 2 esc 2 2 E E E GM vgR R Example: Escape speed A. 9.8 m/s B. 9.8 km/s C. 11.2 km/s D. 46.4 km/s E. 98 km/s Also independent of m, of course! Types of orbits Special ellipse: Circular orbit = circular M vG r <= escape 2 M vv G r ? Closed orbits (ellipses): ? escape vv? Open orbits (parabolas and hyperbolas): For the Earth, = = escape circular 11.2 km/s 7.9 km/s v v < escape Elliptical orbits: vv = escape Parabolic orbit: vv > escape Hyperbolic orbit: vv For g ~ constant, approximately a parabola = circular Circular orbit: vv Kepler?s Laws Originally formulated for the planets in the Solar system, but they apply to any closed orbit (changing ?Sun? by whatever applies). 1. Each planet moves in an elliptical orbit, with the Sun at one focus of the ellipse. 2. A line from the Sun to the given planet sweeps out equal areas in equal times ? Angular momentum about the Sun is conserved. 3. The periods of the planets are proportional to the 3/2 power of the semi-major axis a. F F? P PF + PF? = constant ? Faster near the Sun a 6 Appendix: Measuring the radius of the Earth in 200 BC Eratosthenes of Cyrene measured the shadow cast by a vertical stick at noon in Alexandria and Syene Alexandria Syene ? ? (no shadow) length of the shadow ? ~ length of the stick distance between Alexandria and Syene ? = radius of the Earth He got R E ~ 6400 km (The actual value!!!) Paula Microsoft PowerPoint - Lecture 27 - Gravitation.
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