Gravity A fundamental and universal law § 13.1–13.3

Historical Background Earthly and celestial objects thought fundamentally different –Earthly objects seek a lowly position –Celestial objects move in perfect circles

Newton’s Insight The force pulling us to the ground is the same force curving the moon’s path: gravity!

Poll Question Which is greater? A.The pull of gravity from the earth on the moon. B.The pull of gravity from the moon on the earth. C.Both forces are equally strong. D.Cannot tell without more information.

Force of Gravity We have seen F = mg (g = gravitational field) This is true, but: g varies with location both in direction and magnitude

Newton’s Law of Universal Gravitation Gravitational force between two particles: G: universal gravitational constant,  10 –11 Nm 2 /kg 2 m 1, m 2 : masses of the particles r: distance between particles F = –G m1m2m1m2 r2r2 r

F = GF = G m1m2m1m2 r2r2 Properties of Gravity direction is toward the other object magnitude decreases as (distance) 2 increases never becomes zero infinite at zero separation (?!)

Poll Question When two particles are separated by a distance D, the gravitational force between them is F. If they move closer, to 1/2 the distance (D/2), the force becomes A. F/2. B. 2F. C. 4F. D. Cannot tell without more information. D FF D/2

Gravitational Field F = m · field Gives the force acting on an object of mass m M = mass of object creating the field d = distance from the object field is a vector! field = g = GM d2d2

Gravity between Real Objects The actual force on an extended object is the sum of all forces on all its particles from all the particles of the other object!..

A Lucky Break The gravitational field around a sphere is the same as would be around a point particle of the same mass at the center of the sphere.

Poll Question The planet Saturn is about 100 times more massive than Earth, and about 10 times farther from the Sun. How does the gravitational attraction between Saturn and the Sun compare to that between Earth and the Sun? A.Saturn is attracted with 1/100 the force as Earth. B.Saturn is attracted with 1/10 the force as Earth. C.Their attractions to the Sun are about the same. D.Saturn is attracted with 10 times the force as Earth. E.Saturn is attracted with 100 times the force as Earth.

Poll Question The planet Saturn is about 100 times more massive than Earth, and about 10 times farther from the Sun. How does the acceleration of Saturn toward the Sun compare to that of Earth toward the Sun? A.Saturn accelerates 1/100th as much as Earth. B.Saturn accelerates 1/10th as much as Earth. C.Their accelerations are about the same. D.Saturn accelerates 10 times as much as Earth. E.Saturn accelerates 100 times as much as Earth.

Gravity at Earth’s Surface G =  10 –11 Nm 2 /kg 2 Earth’s mass =  kg = M Earth’s radius = m = d so the field g is g = (6.670  10 –11 Nm 2 /kg 2 ) (5.976  kg ) ( m ) 2 = 9.8 N/kg look familiar?

Quick Question At which position(s) is the satellite speeding up? 1.Position A. 2.Position B. 4.Position C. 8.Position D. 0.None of these.

Quick Question At which position is the satellite slowing down? 1.Position A. 2.Position B. 4.Position C. 8.Position D. 0.None of these.

Quick Question At which position is the satellite changing its direction but not its speed? 1.Position A. 2.Position B. 4.Position C. 8.Position D. 0.None of these.

Group Board Work Rank positions A, B, C, and D in descending order of: a.Gravitational force. b.Acceleration. c.Speed. d.Kinetic energy. e.Gravitational potential energy. f.Total mechanical energy. g.Angular momentum.

Integrate Newton’s Formula Potential = –(Integral of force over space) F = –G m1m2m1m2 r2r2 Convention: U g = 0 at r =  Integrate from r =  in to r = R