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
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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
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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
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Page3 of 32 Gravitational force on particle 1 – vector form:
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m1m2 Full Screen F~2 on 1 = G r 2 b r Close
Quit br is the radial unit vector: |~r| = 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
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F~1,net = net gravitational force acting on particle 1 Go Back ~ gravitational force on particle from particle F1,i = 1 i Full Screen
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• 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
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Go Back Gm1 m3 F~1 net = m j − i , 2 2b b a 4 Full Screen s 2 Close Gm1 2 m3 |F~1 net| = m + , 2 2 a 16 Quit • Gravitational force on a particle from an extended object: Home Page
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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
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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
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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
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Quit • Rotation effects
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m~ac = m~ag + F~N Full Screen
2 FN = mag − mω R Close
2 Quit g = ag − ω 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
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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
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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
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Close F~1 + F~2 = 0
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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
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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
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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
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WFg,curved path = WFg,radial path Close
Quit • The gravitational force is conservative Home Page • Gravitational potential Title Page energy:
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U(∞) − U(r) = −WFg,r→∞ GmM J I = r Page 23 of 32
Go Back GmM U(r) = − r Full Screen
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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 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 ∞; 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. How fast should a projectile be launched Title Page such that it moves on a circular trajectory around JJ II the Earth? J I p A) MG/(R + h) Page 30 of 32 p B) 2MG/(R + h) Go Back C) pMG/2(R + h) Full Screen D) It is impossible for the projectile to circle the Close Earth. Quit A projectile is launched • i-Clicker horizontally from altitude h above the Earth’s sur- Home Page face. How fast should a projectile be launched Title Page such that it moves on a circular trajectory around JJ II the Earth? J I p A) MG/(R + h) Page 31 of 32 p B) 2MG/(R + h) Go Back C) pMG/2(R + h) Full Screen D) It is impossible for the projectile to circle the Close Earth. Quit Home Page m~ac = F~g Title Page mv2 GmM = R + h (R + h)2 JJ II s GM J I v = R + h Page 32 of 32 Orbital velocity Go Back Full Screen Close Quit