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Transcript
Welcome back to Physics 215 Today’s agenda: • Newtonian gravity • Planetary orbits • Gravitational Potential Energy Physics 215 – Fall 2014 Lecture 13-1 1 Current homework assignment • HW10: – Knight Textbook Ch.14: 32, 52, 56, 74, 76, 80 – Due Wednesday, Nov. 19th in recitation Physics 215 – Fall 2014 Lecture 13-1 2 Gravity • Before 1687, large amount of data collected on motion of planets and Moon (Copernicus, Galileo, Brahe, Kepler) • Newton showed that this could all be understood with a new Law of Universal Gravitation Physics 215 – Fall 2014 Lecture 13-1 3 Universal Gravity • Mathematical Principles of Natural Philosophy: Every particle in the Universe attracts every other with a force that is directly proportional to their masses and inversely proportional to the square of the distance between them. Physics 215 – Fall 2014 Lecture 13-1 4 Inverse square law m1 F12 F = Gm1m2/r2 r m2 Physics 215 – Fall 2014 Lecture 13-1 5 Interpretation • F acts along line between bodies • F12 = -F21 in accord with Newton’s Third Law • Acts at a distance (even through a vacuum) … • G is a universal constant = 6.7 x 10-11 N.m2/kg2 Physics 215 – Fall 2014 Lecture 13-1 6 The Earth exerts a gravitational force of 800 N on a physics professor. What is the magnitude of the gravitational force (in Newtons) exerted by the professor on the Earth ? • • • • 800 divided by mass of Earth 800 zero depends on how fast the Earth is spinning Physics 215 – Fall 2014 Lecture 13-1 7 Motivations for law of gravity • Newton reasoned that Moon was accelerating – so a force must act • Assumed that force was same as that which caused ‘apple to fall’ • Assume this varies like r-p • Compare acceleration with known acceleration of Moon find p Physics 215 – Fall 2014 Lecture 13-1 8 Apple and Moon calculation aM = krM-p aapple = krE-p aM/aapple = (rM/rE)-p But: aM = (2prM/T)2/rM = 4p2rM/T2 = 2.7 x 10-3 m/s2 aM/aapple = 2.7 x 10-4 rM/rE = 3.8 x108/6.4 x 106 = 59.0 p = 2! Physics 215 – Fall 2014 Lecture 13-1 9 What is g ? • Force on body close to rE = GMEm/rE2 = mg g = GME/rE2 = 9.81 m/s2 • Constant for bodies near surface • Assumed gravitational effect of Earth can be thought of as acting at center (ultimately justified for p = 2) Physics 215 – Fall 2014 Lecture 13-1 10 Kepler’s Laws experimental observations 1. Planets move on ellipses with the sun at one focus of the ellipse (actually, CM of sun + planet at focus). Physics 215 – Fall 2014 Lecture 13-1 11 Kepler’s Laws experimental observations 2. A line from the sun to a given planet sweeps out equal areas in equal times. *Conservation of angular momentum Physics 215 – Fall 2014 Lecture 13-1 12 Kepler’s Laws experimental observations 3. Square of orbital period is proportional to cube of semimajor axis. • Deduce ( for circular orbit) from gravitational law • assume gravity responsible for acceleration in orbit GMSM/r2 = M(2pr/T)2/r T2 = (4p2/GMS)r3 !! Physics 215 – Fall 2014 Lecture 13-1 13 Orbits of Satellites • Following similar reasoning to Kepler’s 3rd law GMEMsat/r2 = Msatv2/r v = (GME/r)1/2 Physics 215 – Fall 2014 Lecture 13-1 14 Gravitational Field • Newton never believed in action at a distance • Physicists circumvented this problem by using new approach – imagine that every mass creates a gravitational field G at every point in space around it • Field tells the magnitude (and direction) of the gravitational force on some test mass placed at that position F = mtestG Physics 215 – Fall 2014 Lecture 13-1 15 Gravitational Potential Energy Work done moving small mass along path P W = SF.Dx But F acts along line of action! m Dx P M Therefore, only component of F to do work is along r W = - F(r)dr Independent of P! Physics 215 – Fall 2014 Lecture 13-1 16 Gravitational Potential Energy Define the gravitational potential energy U(r) of some mass m in the field of another M as the work done moving the mass m in from infinity to r Physics 215 – Fall 2014 U = - F(r)dr = -GMm/r Lecture 13-1 17 A football is dropped from a height of 2 m. Does the football’s gravitational potential energy increase or decrease ? 1. 2. 3. 4. decreases increases stays the same depends on the mass of football Physics 215 – Fall 2014 Lecture 13-1 18 Gravitational Potential Energy near Earth’s surface U = -GMEm/(RE+h) = -(GMEm/RE) 1/(1+h/ RE) For small h/RE (GMEm/RE2)h = mgh!! as we expect Physics 215 – Fall 2014 Lecture 13-1 19 Energy conservation • Consider mass moving in gravitational field of much larger mass M • Since W = -DU = DK we have: DE = 0 where E = K+U = 1/2mv2 - GmM/r • Notice E < 0 if object bound Physics 215 – Fall 2014 Lecture 13-1 20 Escape speed • Object can just escape to infinite r if E=0 (1/2)mvesc2 = GMEm/RE vesc2 = 2GME/RE • Magnitude ? 1.1x104 m/s on Earth • What about on the moon ? Sun ? Physics 215 – Fall 2014 Lecture 13-1 21 Consequences for planets • Planets with large escape velocities can retain light gas molecules, e.g. Earth has an atmosphere of oxygen, nitrogen • Moon does not • Conversely Jupiter, Sun manage to retain hydrogen Physics 215 – Fall 2014 Lecture 13-1 22 Black Holes • Suppose light travels at speed c • Turn argument about – is there a value of M/R for some star which will not allow light photons to escape ? • Need M/R = c2/2G density = 1027 kg/m3 for object with R = 1m approx • Need very high densities – possible for collapsed stars Physics 215 – Fall 2014 Lecture 13-1 23 Precession • Simple model of solar system based on assuming only important force due to Sun -- ellipses • Not true. Other planets exert mutual gravitational forces also – most important due to Jupiter • Ellipses rotate in space - precession Physics 215 – Fall 2014 Lecture 13-1 24 Reading assignment • Prepare for Exam 3 ! • Chapter 15 in textbook (fluids) Physics 215 – Fall 2014 Lecture 13-1 25