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Rutgers University Department of Physics & Astronomy 01:750:271 Rutgers University Home Page Department of Physics & Astronomy Title Page 01:750:271 Honors Physics I JJ II Fall 2015 J I Lecture 20 Page1 of 32 Go Back Full Screen Close Quit 13. Gravitation Home Page Newton's law of gravitation Title Page Every point particle attracts every other particle with JJ II a gravitational force J I m m Page2 of 32 F = G 1 2 r2 Go Back Full Screen G = 6:67 × 10−11 N · m2=kg2 = 6:67 × 10−11 m3=kg · s2 Close Quit Home Page Title Page JJ II J I Page3 of 32 Gravitational force on particle 1 { vector form: Go Back m1m2 Full Screen F~2 on 1 = G r 2 b r Close Quit br is the radial unit vector: j~rj = 1. • Principle of Superposition Home Page Given n interacting particles: Title Page JJ II F~1;net = F~12 + F~13 + ··· + F~1n J I Page4 of 32 F~1;net = net gravitational force acting on particle 1 Go Back ~ gravitational force on particle from particle F1;i = 1 i Full Screen Close Quit Example Home Page Title Page JJ II J I Page5 of 32 • Isolated system of particles far from other massive Go Back objects. Full Screen • What is the magnitude of the gravitational force on particle 1 ? Close ~ ~ ~ F1;net = F12 + F13 Quit m m Home Page F~ = G 1 2j 12 2 b a Title Page ~ m1m3 JJ II F13 = −G bi 4a2 J I Page6 of 32 Go Back Gm1 m3 F~1 net = m j − i ; 2 2b b a 4 Full Screen s 2 Close Gm1 2 m3 jF~1 netj = m + ; 2 2 a 16 Quit • Gravitational force on a particle from an extended object: Home Page Title Page JJ II J I Page7 of 32 Go Back Full Screen Z Z mdM Close F~ = dF~ = −G r 2 b r Quit Shell theorem Home Page A uniform spherical shell of matter attracts a particle that is outside the shell as if all the shell's mass were Title Page concentrated at its center. mM JJ II F = G 2 r J I • m mass of particle F Page8 of 32 • M mass of shell r Go Back • r distance from particle to center of spherical shell Full Screen (http://hyperphysics.phy-astr.gsu.edu/hbase/ mechan- Close ics/sphshell.html) Quit Home Page Title Page Consequence: The gravitational force be- JJ II tween two uniform spheri- J I cal distributions of mass is the same as if all the mass Page9 of 32 of each sphere were con- Go Back centrated at its center Full Screen Close Quit • Gravitation near Earth's surface Home Page • Assume the Earth is ex- actly spherical. Title Page • Assume uniform distri- JJ II bution of mass throughout the Earth J I • Neglect rotation effects. Page 10 of 32 Gravitational force on a particle near Earth's surface: Go Back mM GM F~g = −G r ) ag = Full Screen r2 b r2 Close r = distance between particle and center of the Earth Quit Note: the free fall acceleration decreases with r Home Page Title Page GM GM JJ II ag = = r2 (R + h)2 J I h = altitude = dis- tance from particle Page 11 of 32 to the surface of the Go Back Earth Full Screen Close Quit • Rotation effects Home Page Title Page JJ II J I Page 12 of 32 Go Back m~ac = m~ag + F~N Full Screen 2 FN = mag − m! R Close 2 Quit g = ag − ! R i-Clicker Home Page The gravitational acceleration on the surface of the Earth is g (neglecting rotation.) What will it be on Title Page the surface of a planet that has half the mass of the JJ II Earth and half its radius? J I A) g=4 Page 13 of 32 B) g=2 Go Back C) g Full Screen D) 2g Close Quit i-Clicker Home Page The gravitational acceleration on the surface of the Earth is g (neglecting rotation.) What will it be on Title Page the surface of a planet that has half the mass of the Earth and half its radius? JJ II J I A) g=4 B) g=2 Page 14 of 32 C) g Go Back M 0 0 2 0 2 Full Screen D) 2gg = GR2 ) g =g = M R =M(R ) = 4=2 = 2 Close Quit • Gravitation inside the Earth Home Page A uniform shell of matter exerts no net gravitational force on a particle located inside it. Title Page JJ II 2 F r1 r2 A1 r1 2 J I = = 2 d1 d2 A2 r2 Page 15 of 32 F1 2 F1 A1=1 Go Back = 2 = 1 F2 A2=d2 Full Screen Close F~1 + F~2 = 0 Quit Suppose a very narrow tun- nel is dug through the Earth Home Page along the North-South axis. Title Page What is the gravitational force on a particle of mass JJ II m at distance r < R from J I the center? Page 16 of 32 Go Back Any thin spherical shell of matter of radius rshell > r does not yield any net force. Full Screen Only thin shells of radius rshell < r give a nonzero Close force. Quit The gravitational force on the particle is the same as the force due to a sphere of Home Page radius r. Title Page Assume spherical shape and uniform mass distribution. JJ II Mass density J I M 3M ρ = = Page 17 of 32 4πR3=3 4πR3 Gm Go Back F = × Mass inside sphere of radius r 2 r Full Screen Gm 4πr3ρ 4πGmρ GmM = × = r = r r2 3 3 R3 Close Quit i-Clicker • All spher- ical shells are uniform Home Page and have the same mass Title Page M. JJ II • Rank the J I situations according to Page 18 of 32 A) F > F > F a b c the magni- Go Back B) Fa = Fb > Fc tude of the gravitational Full Screen C) Fa < Fb < Fc force on the D) Fa < Fb = Fc particle. Close Quit i-Clicker • All spher- ical shells are uniform Home Page and have the Title Page same mass M. JJ II • Rank the J I situations Page 19 of 32 A) Fa > Fb > Fc according to B) Fa = Fb > Fc the magni- Go Back tude of the C) F < F < F Full Screen a b c gravitational D) Fa < Fb = Fc force on the Close 2 particle. Fa = 0, Fb = Fc = GmM=r Quit • Gravitational potential energy Home Page Title Page Work done by gravitational force: suppose a baseball JJ II moves upward from a dis- J I tance r1 to a distance r2 > r1 in the gravitational field Page 20 of 32 of the Earth. Go Back What is the work done by gravitational force? Full Screen Close Quit r Z 2 Home Page ~ WFg = Fg · d~r r1 Title Page Z r2 dr = −GmM 2 JJ II r1 r 1 1 = GmM − J I r2 r1 Page 21 of 32 Does it depend on the path? Go Back Full Screen Close Quit WFg does not depend on the path. Home Page Contribution of circular arcs is 0: Title Page F~g · d~r = 0 JJ II Contribution of radial arcs: J I GmM Page 22 of 32 F~g · d~r = − dr r2 Go Back Full Screen WFg;curved path = WFg;radial path Close Quit • The gravitational force is conservative Home Page • Gravitational potential Title Page energy: JJ II U(1) − U(r) = −WFg;r!1 GmM J I = r Page 23 of 32 Go Back GmM U(r) = − r Full Screen Close Quit Gravitational potential energy: system of particles • For any pair of particles Home Page (i; j) Title Page Gmimj Uij = − JJ II rij J I X Gmimj Utotal = − Page 24 of 32 r i<j ij Go Back Gm1m2 Gm2m3 Gm1m3 Utotal = − + + Full Screen r12 r23 r13 Close Quit How fast • Escape speed should a projectile Home Page be launched such that Title Page it escapes Earth's grav- JJ II itational J I field? Page 25 of 32 How fast should it Go Back be launched Full Screen such that it does not Close fall back on Earth? Quit i-Clicker Home Page Find the initial speed of the rocket such that it escapes the Earth's gravitational Title Page field, moving infinitely far away. JJ II A) pMG=R J I B) p2MG=R Page 26 of 32 C) pMG=2R Go Back D) It is impossible for the rocket to es- cape. Full Screen Close Quit i-Clicker Home Page Find the initial speed of the rocket such that it escapes the Earth's gravitational Title Page field, moving infinitely far away. JJ II A) pMG=R J I B) p2MG=R Page 27 of 32 C) pMG=2R Go Back D) It is impossible for the rocket to es- cape. Full Screen Close Quit • Energy conservation Initially: object on the surface of the Home Page Earth; kinetic and potential energy Title Page Emec = K0 + Ugrav JJ II Finally: object at 1; kinetic energy J I Emec = K ≥ 0 Page 28 of 32 Go Back r 1 2 GmM 1 2 2GM mv − = mv ≥ 0 ) v0 ≥ Full Screen 2 0 R 2 R Close Quit Home Page r2GM vesc = Title Page R JJ II Escape speed J I Page 29 of 32 Go Back Full Screen Close Quit A projectile is launched • i-Clicker horizontally from altitude h above the Earth's sur- Home Page face.
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