Copyright R. Janow – Spring 2012Physics 111 Lecture 13Universal GravitationSJ 8th Ed.: Chap 13.1 – 13.6• Overview - Gravitation• Newton’s Law of Gravitation• Free Fall – Acceleration = Field• Gravitational Potential Energy• Kepler’s Laws of Planetary Motion• Satellite Orbits and Energy13.1 Newton’s Law of Universal Gravitation13.2 Free-Fall Acceleration and the Gravitational Force13.3 Kepler’s Laws and the Motion of Planets13.4 The Gravitational Field13.5 Gravitational Potential Energy13.6 Energy Considerations in Planetary and Satellite MotionCopyright R. Janow – Spring 2012Basic Gravitation Concepts2122112rmmG |F| ====• “Action at a Distance”– No contact needed.– Cannot be screened out, unlike electrical forces.– Always attractive unlike electrical forces (except for “dark energy”, maybe).• Inertial mass (notion used already):– Measures resistance to acceleration, e.g.: F = ma.– Measures response to gravitational acceleration - a field.• Gravitational mass (added for gravitation):– Mass is the source of the gravitational acceleration field - always– Gravitational mass measures strength of a gravitational field produced.• Duality/Equivalence: – Every bit of mass acts as both inertial and gravitational mass with the same value of m in each role. • Very weak compared to electrical forces. – Too small to notice between most human-scale objects and smaller.– Long range gives it great effect on cosmic scale.Every particle in the Universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between themFor a pair of masses:Copyright R. Janow – Spring 2012122122112rˆrmmGF −−−−====rNewton’s Law of Gravitation – Vector Formm1m23rd law pair of forcesForce due tom1on m2Gravitational ConstantG=6.67x10-11 Nm2/kg2Unit vector along r1221121212rˆrrrˆ−−−−====≡≡≡≡rr• Force is radially between pairs of point masses• Symmetric in m1& m2 so F12= - F21• Not screened or affected by other bodies• Easy to miss between masses near athird large mass (e.g. on Earth surface)12rr21Fr12FrDisplacement from m1to m2EarthGravitational “field” - acceleration1221211212212rˆrGmg gmF −−−−====⇒⇒⇒⇒≡≡≡≡rrrInverse square law: as sphere grows field (or force)x area is constantrA=4ππππr2field due to m1at location of m2 If a mass m1 is placed at point 1, is anything now different at the empty point 2?A FIELD transmits the force (no contact, action at a distance)Copyright R. Janow – Spring 2012Superposition: The net force on a point mass when there are many others nearby is the vector sum of the forces taken one pair at a timeAll gravitational effects are between pairs of masses. No known effects depend directly on 3 or more masses.141312111 ,,,i,i onFFF F Frrrrr++++++++========∑∑∑∑≠≠≠≠m1m4m3m221Fr12rr41Fr31Fr14rr13rrFor continuous mass distributions, integrate Fd F dist mass on∫∫∫∫====11rr rGm- a g symmetry by 0 g m F 2i1,igi,1,im at1m on11====≡≡≡≡========1rvExample:m1m4m3m2rrm5rr'rr'rrm2= m3 m4= m5Copyright R. Janow – Spring 2012 13.1. In the sketch, equal masses are placed at the vertices of an equilateral triangle, each of whose sides equals “s”. In which direction would the top-most chunk of mass try to accelerate (ignore the Earth’s gravity) with the bottom two held in place?Superposition for a triangleA) B) C) D) E) a = 0 F Fjij,ij,net∑∑∑∑≠≠≠≠====rr 13.2. Another chunk of mass is placed at the exact center of the triangle in the sketch. In which direction does it tend to accelerate?msss+mmCopyright R. Janow – Spring 2012Shell Theorem: superposition applied to masses with spherical symmetry1. For a test mass OUTSIDE of a uniform spherical shell of mass, the shell’s gravitational force (field) is the same as that of a point mass concentrated at the shell’s mass centerxmrxmrSame for a solid sphere (e.g., Earth, Sun) via nested shellsxrxmrxr+ +2. For a test mass INSIDE of a uniform spherical shell of mass, the shell’s gravitational force (field) is zero• Obvious by symmetry for center• Not obvious elsewhere, need to integrate over sphere xmx3. For a solid sphere, the force on a test mass INSIDE includes only the mass closer to the CM than the test mass.mx• Example: On surface, measure acceleration ga distance rfrom center• Example: Halfway to center, ag= g/2334rVsphereππππ====Copyright R. Janow – Spring 2012What do “g” and “weight = mg” mean?• Object with mass m is at altitude h……above the surface, so r = re+h• Radius of Earth = re• Earth’s mass acts as like a point mass meat the center (by the Shell Theorem)• Weight W = magwith acceleration given byNewtons Law of Gravitation (any altitude) rˆ)hr(mGa altitude anyat eeg2++++−−−−====rmemrehWhen m is on or near the surface: r h r rhee wordsother in or,e≈≈≈≈++++<<<<<<<< m/s 9.8g rmG ag2eeg where ≅≅≅≅≅≅≅≅≡≡≡≡2r kg 10 x 5.98 m 10 x 6.6710 x 6370 x 9.8 Grg m 24e11-3ee============⇒⇒⇒⇒2Example:Use the formula above to find the mass of the Earth, given:•g = 9.8 m/s2 (measure in lab)• G = 6.67x10-11 m3/kg.s2 (lab)• re = 6370 km (average - measure)Copyright R. Janow – Spring 2012Altitude dependence of gWeight decreases withaltitude hThe work needed to increase ∆h declines, sinceweight decreasesCopyright R. Janow – Spring 201213.3 What is the magnitude of the free-fall acceleration at a point that is a distance 2reabove the surface of the Earth, where re is the radius of the Earth?Free fall accelerationa) 4.8 m/s2b) 1.1 m/s2c) 3.3 m/s2d) 2.5 m/s2e) 6.5 m/s22 altitude any ateegm/s 9.8 g )h r(mGa ====++++====2Copyright R. Janow – Spring 2012Gravitational Potential Energy ∆∆∆∆U RGmMUg −−−−====Note: NOT R 2 |RR2R2RgRrGmM rdrGmM dr rGmM rd F dW U∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞====−−−−====−−−−====••••−−−−====−−−−≡≡≡≡∆∆∆∆∫∫∫∫∫∫∫∫∫∫∫∫∫∫∫∫rr The gravitational potential energy between any two particlesvaries as 1/R. The force varies as 1/R2 The potential energy is negative because the force is attractive and we chose the potential energy to be zero at infiniteseparation. Units are