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

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

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

Physics 215 – Fall 2014Lecture 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 N. m 2 /kg 2

Physics 215 – Fall 2014Lecture 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 2014Lecture 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 2014Lecture Apple and Moon calculation But: a M = (2  r M /T) 2 /r M = 4  2 r M /T 2 = 2.7 x m/s 2 a M /a apple = 2.7 x 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

Physics 215 – Fall 2014Lecture 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)

Physics 215 – Fall 2014Lecture 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 2014Lecture 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 2014Lecture 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 !!

Physics 215 – Fall 2014Lecture 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

Physics 215 – Fall 2014Lecture 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 

Physics 215 – Fall 2014Lecture 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!

Physics 215 – Fall 2014Lecture 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

Physics 215 – Fall 2014Lecture 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

Physics 215 – Fall 2014Lecture 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

Physics 215 – Fall 2014Lecture 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

Physics 215 – Fall 2014Lecture 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 ?

Physics 215 – Fall 2014Lecture 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 2014Lecture 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 = kg/m 3 for object with R = 1m approx Need very high densities – possible for collapsed stars

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