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Lecture 25 Jupiter and 4 of its moons under the influence of gravity Goal: To use Newton’s theory of gravity to analyze the motion of satellites and planets. Physics 201: Lecture 25, Pg 1 Exam results Mean/Median 59/100 Nominal exam curve A 81-100 AB 71-80 B 61-70 BC 51-60 C 31-50 D 25-30 F below 25 Physics 201: Lecture 25, Pg 2 Newton’s Universal “Law” of Gravity Newton proposed that every object in the universe attracts every other object. The Law: Any pair of objects in the Universe attract each other with a force that is proportional to the products of their masses and inversely proportional to the square of their distance Physics 201: Lecture 25, Pg 3 Newton’s Universal “Law” of Gravity m1m2 Fon 2 by 1 G 2 r̂12 Fon1by 2 r “Big” G is the Universal Gravitational Constant G = 6.673 x 10-11 Nm2 / kg2 Two 1 Kg particles, 1 meter away Fg= 6.67 x 10-11 N (About 39 orders of magnitude weaker than the electromagnetic force.) The force points along the line connecting the two objects. Physics 201: Lecture 25, Pg 4 Little g and Big G Suppose an object of mass m is on the surface of a planet of mass M and radius R. The local gravitational force may be written as FG mg surface where we have used a local constant acceleration: mGM Fon m by M r̂M ,m 2 R GM On Earth, near sea level, it can g surface 2 be shown that gsurface ≈ 9.8 m/s2. R Physics 201: Lecture 25, Pg 5 g varies with altitude With respect to the Earth’s surface g (h) GM E /( RE h) 2 Physics 201: Lecture 25, Pg 6 At what distance are the Earth’s and moon’s gravitational forces equal Earth to moon distance = 3.8 x 108 m Earth’s mass = 6.0 x 1024 kg Moon’s mass = 7.4 x 1022 kg GM E / r GM M / r 2 E 2 M r / r MM / ME 2 M 2 E rM / rE M M / M E 0.11 rM rE 3.8 10 m 8 rE 3.8 10 m/1.11 3.4 10 m 8 8 Physics 201: Lecture 25, Pg 7 The gravitational “field” To quantify this invisible force (action at a distance) even when there is no second mass we introduce a construct called a gravitational “force” field. The presence of a Gravitational Field is indicated by a field vector representation with both a magnitude and a direction. M GM g 2 r̂ r Physics 201: Lecture 25, Pg 8 Compare and contrast Old representation New representation g gr̂ GM g 2 r̂ r GM g r̂ 2 ( RE h) M GM 1 1 g ( h) 2 g (0) 2 RE (1 h / RE ) (1 x) 2 2 h 3 h g (h) g (0) 1 2 ... RE 2 RE Physics 201: Lecture 25, Pg 9 Gravitational Potential Energy Recall for 1D: F ( x) dU ( x) / dx W xx F ( x) dx f i W rf ri Gm1m2 Gm1m2 1 FG dr ( r ) rf ri 2 When two isolated masses m1 and m2 interact over large distances, they have a gravitational potential energy of U (r ) Gm1m2 / r The “zero” of potential energy occurs at r = ∞, where the force goes to zero. Note that this equation gives the potential energy of masses m1 and m2 when their centers are separated by a distance r. Physics 201: Lecture 25, Pg 10 A plot of the gravitational potential energy U (r ) Gm1m2 / r Ug= 0 at r = ∞ We use infinity as reference point for Ug(∞)= 0 This very different than U(r) = mg(r-r0) Referencing infinity has some clear advantages All r allow for stable orbits but if the total mechanical energy, K+ U > 0, then the object will “escape”. Physics 201: Lecture 25, Pg 11 It is the same physics Shifting the reference point to RE. GmM E U ( RE h) RE h GmM E GmM E U ( RE h) U ( RE ) RE h RE RE ( RE h) Gm1m2 R ( R h ) R ( R h ) E E E E GmM E h GmM E 2 h 2 RE RE RE h GM E m 2 h mgh RE Physics 201: Lecture 25, Pg 12 Gravitational Potential Energy with many masses If there are multiple particles U Total U12 U13 U 23 m1m2 m1m3 m2 m3 U (r ) G | r1 r2 | | r1 r3 | | r2 r3 | Neglects the potential energy of the many mass elements that make up m1, m2 & m3 (aka the “self energy” Physics 201: Lecture 25, Pg 13 Dynamics of satellites in circular orbits Circular orbit of mass m and radius r Force: FG= G Mm/r2 =mv2/r Speed: v2 = G M/r (independent of m) Kinetic energy: K = ½ mv2 = ½ GMm/r Potential energy UG = - GMm/r UG = -2 K This is a general result Total Mechanical Energy: E = K + UG = ½ GMm/r -GMm/r E = - ½ GMm/r Physics 201: Lecture 25, Pg 14 Changing orbits With man-made objects we need to change the orbit Examples: 1. Low Earth orbit to geosynchronous orbit 2. Achieve escape velocity 1. We must increase the potential energy but one can decrease the kinetic energy consistent with U G (r ) GMm r GMm K (r ) U G (r ) 2r 1 2 GMm E (r ) U G (r ) K (r ) 2r Physics 201: Lecture 25, Pg 15 Changing orbits Low Earth orbit to geosynchronous orbit GMm E (r ) U G (r ) K (r ) 2r A 470 kg satellite, initially at a low Earth orbit of hi = 280 km is to be boosted to geosynchronous orbit hf = 35800 km What is the minimum energy required to do so? (rE = 6400 km, M = 6 x 1024 kg 1 1 E E (rf ) E (ri ) GMm 2 r 2 r i f 1 1 11 24 E 6.67 10 (6 10 )470 /2 7 6 4.2 10 6.7 10 10 E 1.2 10 J Physics 201: Lecture 25, Pg 16 Escaping Earth orbit Exercise: suppose an object of mass m is projected vertically upwards from the Earth’s surface at an initial speed v, how high can it go ? (Ignore Earth’s rotation) Ei U G ( RE ) mv 1 2 2 i E f U G ( RE h) 0 E f Ei U G ( RE h) 0 GmM /( RE h) GmM / RE mv 2 1 1 /( RE h) 1/ RE 2 mvi /(GmM ) 1 2 2 i Physics 201: Lecture 25, Pg 17 Escaping Earth orbit Exercise: suppose an object of mass m is projected vertically upwards from the Earth’s surface at an initial speed v, how high can it go ? (Ignore Earth’s rotation) 1 /( RE h) 1 / RE 12 mvi2 /(GmM ) h ( R hRE )v /(2GM ) 2 E 2 i h(1 R v /(2GM )) R v /(2GM ) 2 E i 2 2 E i 2 2 E i Rv h 2 (2GM RE vi ) Physics 201: Lecture 25, Pg 18 Escaping Earth orbit Exercise: suppose an object of mass m is projected vertically upwards from the Earth’s surface at an initial speed v, how high can it go ? (Ignore Earth’s rotation) 2 2 E i Rv h 2 (2GM RE vi ) 2GM RE vi2 0 vEscape implies infinite height 2GM 11.2 km/s RE Physics 201: Lecture 25, Pg 19 Some interesting numbers First Astronautical Speed Low Earth orbit: v = 7.9 km/s Second Astronautical Speed Escaping the Earth: v = 11 km/s Third Astronautical Speed Escaping Solar system: v= 42 km/s Physics 201: Lecture 25, Pg 20 Next time Chapter 14 sections 1 to 3, Fluids Physics 201: Lecture 25, Pg 21