Physics 106 Lecture 9 Newton’s Law Of Gravitation

Physics 106 Lecture 9Newton’s Law of GravitationSJ 7th Ed.: Chap 13.1 to 2, 13.4 to 5 Historical overview Newton’sN’ inverse-squareillawoff gravitationii Force Gravitational acceleration “g” SuperpositionGravitation near the Earth’s surfaceGravitation inside the Earth (concentric shells)Gravitational potential energy Related to the force by integration A conservative force means it is path independent Escape velocityGravitation – IntroductionWhy do things fall?Why doesn’t everything fall to the center of the Earth?What holds the Earth (and the rest of the Universe) together?Why are there stars, planets and galaxies, not just dilute gas? Aristotle – Earthly physics is different from celestial physics Kepler – 3 laws of planetary motionmotion, Sun at the centercenter. Newton – English, 1665 (age 23) Numerical fit/no theoryPhysical Laws are the same everywhere in the universe (same laws for legendary fallingapple and planets in solar orbit, etc).Invented differential and integral calculus (so did Liebnitz)Proposed the law of “universal gravitation”Deduced Kepler’s laws of planetary motionRevolutionized “Enlightenment” thought for 250 years Reason ÅÆ prediction and control, versus faith and speculation Revolutionary view of clockwork, deterministic universe (now dated) Einstein - Newton 250 years (1915, age 35)General Relativity – mass is a form of concentrated energy (E mc2), gravitation is adistortion of space-time that bends light and permits black holes (gravitational collapse). Planck, Bohr, Heisenberg, et al – Quantum mechanics (1900–27)Energy & angular momentum come in fixed bundles (quanta): atomic orbits, spin,photons, etc.Particle-wave duality: determinism breaks down.There should be a “graviton” (quantum gravity particle). No progress yet. Current Issues Dark Matter – Luminous mass of galaxies is too small to explain stars’ orbits.Dark Energy and inflation – Possible anti-gravity at long range fuels acceleratingexpansion of the universe, and also early Big Bang.1

Gravitation – Basic ConceptsEvery particle in the Universe attracts every other particle with a force thatis directly proportional to the product of their masses and inverselyproportional to the square of the distance between themFor a pair of masses: Inertial mass: 2r12Measures resistance to acceleration, e.g.: F ma.Measures response to gravitational acceleration - a field.Mass is the source of the gravitational acceleration field - alwaysGravitational mass measures strength of a gravitational field produced.Duality/Equivalence: G m1m2Gravitational mass: F12 Every bit of mass acts as both inertial and gravitational mass with the same valueof m in each role.Gravitational Force No contact needed: “action at a distance”.Cannot be screened out, unlike electrical forces.Always attractive unlike electrical forces (except for “dark energy”, maybe).Very weak compared to electrical forces. Too small to notice between most humanscale objects and smaller (e.g., p and e- ).Gravitation is long range, has cosmological effects over long times.But it is a weak force on the human scale.Newton’s Law of Universal Gravitation3rd law pair ofGGravitational ConstantForce onforcesF21mG 6.67x10-11 Nm2/kg2m2 due to m1GGF12Gr12G m1m21Displacementfrom m1 to m2 m2F12 Force is between pairs of point massesSymmetric in m1 & m2 so F12 - F21Not screened or affected by other bodiesEasy to miss between masses near athird large mass (e.g. on Earth surface)2r12r̂12Unit vectorr12alongGr12r̂12 G r̂21r12IF m2 IS REMOVED, IS ANYTHING AT POINT 2 DIFFERENT BECAUSE m1 IS STILL AT POINT 1?FIELD transmits the force (no contact, action at a distance)Acceleration fieldEarthInverse square law:as sphere grows field(or force)x area is constantGGGGmF12 m2g12 g12 2 1 r̂12r12A 4πr2rfield at locationof m2 due to m12

Finding the Value of G Henry Cavendish firstmeasured G directly (1798)Two masses m are fixed at theends of a light horizontal rod(torsion pendulum)Two large masses M wereplaced near the small onesThe angle of rotation wasmeasuredResults were fitted intoNewton’s LawG 6.67x10-11 N.m2/kg2G versus g: G is the universal gravitational constant, the same everywhere g ag is the acceleration due to gravity. It varies by location. g 9.80 m/s2 at the surface of the EarthWhy was the Law of Gravitation not obvious (except to Newton).How big are gravitational forces between ordinary objects? F12 G m1m22r121 Newton is about the force needed tosupport 100 grams of mass on the Earthm1m2r12F121 kga liter of soda1 kg sandwich1 meter6.67x10-11 N.100 kga person100 kganother person1 meter6.67x10-7 N.106 kga ship106 kganother ship100 meters0.67 N.still hard to detectConclusion:Cl i G is very small, so need huge masses to get perceptible forcesDoes gravitation play a role in atomic physics & chemistry?9.1x10-31 kgelectron1.7x10-27 kgproton5x10-11 meterorbit radius4x10-47 N.3

Superposition:The net force on a point mass when there are many others nearby is thevector sum of the forces taken one pair at a timeGFon 1 G Fi,1i 1GGG F2,1 F3,1 F4,1GF21All gravitational effects are betweenpairs of masses. No known effectsdepend directly on 3 or more masses.m4Example:m2m1m1GF41Gr13GF31Grm4 m5m3KFonm1G m1 gatGr'gi,1m3Gr14m4m2 m3Gr'Grm2Gr12 0 by symmetryGm a gi,1 - 2 ir1,im1m5For continuous massdistributions, integrateGFon 1 GdF 1mass distNumerical Example:yFind net force on m1 due to m2 and m3 Use superposition Basic forces at right anglesGFnet , j G Fi, ji jm2am3F122aF13m1x4

Superposition for a triangle 9.1. In the sketch, equal masses are placed at the vertices of anequilateral triangle, each of whose sides equals “s”. In whichdirection would the top-most chunk of mass try to accelerate(ignore the Earth’s gravity) with the bottom two held in place?mA)B)C)D)ssE) a 0 smm 99.2.2 Another chunk of mass is placed at the exact center of thetriangle in the sketch. In which direction does it tend to accelerate?GGFnet , j Shell Theorem: Fi, ji jsuperposition for masses with spherical symmetry1. For a test mass OUTSIDE of a uniform spherical shell of mass, theshell’s gravitational force (field) is the same as that of a point massconcentrated at the shell’s mass centermrmxrxSa e forSamefo a solidolid spherehe e (e.g.,(eEarth,Ea th Sun)S ) viaia nestede ted shellshellmrrxx rx 2. For a test mass INSIDE of a uniform spherical shell of mass, theshell’s gravitational force (field) is zero Obvious by symmetry for center Elsewhere, need to integrate over spheremx x3. For a solid sphere, the force on a test mass INSIDE includes onlythe mass closer to the CM than the test mass.mx Example: On surface, measure accelerationdistancerfrom center Example: Halfway to center,ag g/2ga43Vsphere πr 35

Gravitation near the surface of the Earth:What do “g” and “weight mg” mean?mhreme Earth’s mass acts as like a point mass meat the center (by the Shell Theorem) Radius of Earth re Object with mass m is at altitude h above the surface, so r re h Weightg W mag with acceleration ggiven byyNewtons Law of Gravitation (any altitude)G meGag r̂(re h) 2When m is “on ornear the surface:Example:at any altitudeh re or, in other words re h reG meG2g ag where g 9.8 m/sre2Use the above to find the mass of the Earth, given: g 9.8 m/s2 (measure in lab)6.67x10-11 m3/kg.s2 G (lab) re 6370 km (average - measure) me g re2G 9.8 x 6370 x 1036.67 x 10-11me 5.98 x 1024 kgAltitude dependence of g Weight decreases withaltitude h The work needed toincrease Δh declines, sinceweight decreases6

Free fall acceleration9.3 What is the magnitude of the free-fall acceleration at a point that isa distance 2re above the surface of the Earth, where re is the radiusof the 2ag G me(re h) 2at any altitudeg 9.8 m/s 2Gravitational “field” transmits the force A piece of mass m1 placed somewhere creates a “gravitational field” thathas values described by some function g1(r) everywhere in space. Another piece of mass m2 feels a force proportional to g1(r) and in the samedirection, also proportional to m2.GG m1m2GF12 m2g1 (r ) r̂122Conceptsp for g-fields:gr12 No contact needed: “action at a distance”. Acceleration Field created by gravitational mass transmits the force as adistortion of space that another (inertial) mass responds to. Gravitational field is “Conservative” (i.e. can have a potential energy function). g cannot be screened out, unlike electrical fields. g is always attractive (except cosmologically, maybe), unlike electrical fields. The field g1(r) is the gravitational force per unit masscreated by mass m1, present at all points whether or notthere is a test mass m2 located there The gravitational field vectors point in the direction ofthe acceleration a particle would experience if placed inthe field at each point.GG FgGMg 2 rˆmrField lines help to visualize strength and direction. close together Æ strong field, direction Æ force on test mass7

Gravitational Potential Energy ΔUΔU mgΔh fails unless ag is constant – force depends on rGGWork done byDefinition:potentialdW Fg d r dUggravity on a testenergymass m movedthrough drforce variesalong pathdisplacementchangeChoose: gravitational potential zero at R where the force zeroi.e.: U(R) Æ 0 as R Æ .Mass M creates the g-field. Integrate along a radial path from R to infinityG G GmMΔU dW Fg d r drRRR r2 drGmM GmM 2 RRrrUg GmMRNote: The gravitational potential energy between any two particlesNOT R 2varies as 1/R. The force varies as 1/R2 The potential energy is negative because the force isattractive and we chose the potential energy to be zero at infiniteseparation. Another form of energy (external work or kinetic energy) is convertedwhen the potential energy and separation between masses increase.Gravitational Potential EnergyMutual potential energy of a system of many particles Uij (rij )Utotal all pairsshared, sum overall possible pairingsThe total gravitational potential energy of the system is thesum over all pairs of particles.Gravitational potential energy obeys the superposition principleExamplem13 possible pairsm2r12r23r13m3 Gm mGm1m3 Gm2m3 1 2 Utotal U12 U13 U23 r13r23 r12The slope of theenergy curve is related to the force.G potentialGRecall: dW Fg d r dUgforce due togravitationGdUgFg drminusSlope of potentialenergy function(derivative,gradient)8

Gravitational Potential Energy, cont As a particle moves from A to B, itsgravitational potential energy changes by ΔU But the mechanical energy remains constant,i dindependentd t off path,th so longlas no otherthforce is actingEmech K U(r ) Graph of the gravitationalpotential energy U versus r foran object above the Earth’ssurfaceThe potential energy goes tozero as r approaches infinityThe mechanical energy may bepositive, negative, or zeroEmech2Emech1Conservation of mechanical energy with gravitation Emech determines whether motion is bound, free, or at escape thresholdEmech is constantEmech K Ug (r )always negativeFor Emech 0, particle is bound andcannot escape. It cannot move beyonda turning point (e.g., r2)For Emech 0, particle is free. It canreach r infinity and still have some KEleftrr2EmechTurningpoint r2KE2 0U(r1)BOUNDEmech 0 iis theh escape condition.di iAparticle at any location r would need atleast KE -Ug(r) to move off to the rightand never return.How much energy does it cost per kilogram toescape completely from the surface of the Earth?Ug -r1KE1GmemrUg 0EUg (r ) FREEUg GmerGdUgFg drThe tangent to the potentialenergy graph measures thegravitational forceGme x1 17.4 KWH/kgre9