1. Physics 215 – Fall 2014Lecture 13-11 Welcome back to Physics 215 Today’s agenda: Newtonian gravity Planetary orbits Gravitational Potential Energy

2. Physics 215 – Fall 2014Lecture 13-12 Current homework assignment HW10: –Knight Textbook Ch.14: 32, 52, 56, 74, 76, 80 –Due Wednesday, Nov. 19 th in recitation

3. Physics 215 – Fall 2014Lecture 13-13 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

4. Physics 215 – Fall 2014Lecture 13-14 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.

5. Physics 215 – Fall 2014Lecture 13-15 Inverse square law F = Gm 1 m 2 /r 2 m1m1 m2m2 r F 12

6. Physics 215 – Fall 2014Lecture 13-16 Interpretation F acts along line between bodies F 12 = -F 21 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. m 2 /kg 2

7. Physics 215 – Fall 2014Lecture 13-17 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

8. Physics 215 – Fall 2014Lecture 13-18 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

9. Physics 215 – Fall 2014Lecture 13-19 Apple and Moon calculation But: a M = (2  r M /T) 2 /r M = 4  2 r M /T 2 = 2.7 x 10 -3 m/s 2 a M /a apple = 2.7 x 10 -4 r M /r E = 3.8 x10 8 /6.4 x 10 6 = 59.0  p = 2! a M = kr M -p a apple = kr E -p a M /a apple = (r M /r E ) -p

10. Physics 215 – Fall 2014Lecture 13-110 What is g ? Force on body close to r E = GM E m/r E 2 = mg  g = GM E /r E 2 = 9.81 m/s 2 Constant for bodies near surface Assumed gravitational effect of Earth can be thought of as acting at center (ultimately justified for p = 2)

11. Physics 215 – Fall 2014Lecture 13-111 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).

12. Physics 215 – Fall 2014Lecture 13-112 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

13. Physics 215 – Fall 2014Lecture 13-113 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  GM S M/r 2 = M(2  r/T) 2 /r T 2 = (4   /GM S )r 3 !!

14. Physics 215 – Fall 2014Lecture 13-114 Orbits of Satellites Following similar reasoning to Kepler’s 3rd law  GM E M sat /r 2 = M sat v 2 /r v = (GM E /r) 1/2

15. Physics 215 – Fall 2014Lecture 13-115 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  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 = m test 

16. Physics 215 – Fall 2014Lecture 13-116 Gravitational Potential Energy M m P Work done moving small mass along path P W =  F.  x But F acts along line of action! x x Therefore, only component of F to do work is along r W = - F(r)dr Independent of P!

17. Physics 215 – Fall 2014Lecture 13-117 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  U = - F(r)dr = -GMm/r

18. Physics 215 – Fall 2014Lecture 13-118 A football is dropped from a height of 2 m. Does the football’s gravitational potential energy increase or decrease ? 1.decreases 2.increases 3.stays the same 4.depends on the mass of football

19. Physics 215 – Fall 2014Lecture 13-119 Gravitational Potential Energy near Earth’s surface U = -GM E m/(R E +h) = -(GM E m/R E ) 1/(1+h/ R E ) For small h/R E  (GM E m/R E 2 )h = mgh!! as we expect

20. Physics 215 – Fall 2014Lecture 13-120 Energy conservation Consider mass moving in gravitational field of much larger mass M Since W = -  U =  K we have:  = 0 where E = K+U =  mv 2 - GmM/r Notice E < 0 if object bound

21. Physics 215 – Fall 2014Lecture 13-121 Escape speed Object can just escape to infinite r if E=0  (1/2)mv esc 2 = GM E m/R E  v esc 2 = 2GM E /R E Magnitude ? 1.1x10 4 m/s on Earth What about on the moon ? Sun ?

22. Physics 215 – Fall 2014Lecture 13-122 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

23. Physics 215 – Fall 2014Lecture 13-123 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 = c 2 /2G  density = 10 27 kg/m 3 for object with R = 1m approx Need very high densities – possible for collapsed stars

24. Physics 215 – Fall 2014Lecture 13-124 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

25. Physics 215 – Fall 2014Lecture 13-125 Reading assignment Prepare for Exam 3 ! Chapter 15 in textbook (fluids)