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Inertial Reference Frames and the Galilean Transform

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Inertial Reference Frames and the Galilean Transform Modern Physics Unit 12: Special Relativity - Introduction Lecture 12.1: Inertial Reference Frames and the Galilean Transform Ron Reifenberger Professor of Physics Purdue University 1 Relativity This word is usually associated with Einstein, but……………really it is related to a 2,500 year long effort to understand our place in the universe: • ~ 500 BC - Earth was at rest at the center of a flat universe that was constructed with a mathematical plan • ~ 1500 – suggestion that Earth actually moves around the sun • ~ 1900 - no center to the universe, there is no place in the universe that is at rest, the universe is not flat. Q: What is Relativity? A: Relativity is the idea that the laws of the universe are the same no matter what direction you are facing, no matter where you are standing, no matter how fast you are moving, so long as the speed you are moving is constant. 2 Scientific origin of Relativity “Does the earth move?” In Galileo’s time (early 1600’s) this was THE problem of the century. Arguments against earth motion: • There is NO speed sensation when standing on the surface of the earth. • If the earth were moving (caused by say the Earth’s rotation), anything dropped from a tall height would rapidly “fall behind” and appear to drift backward. Galileo’s counter argument – a simple thought experiment: Consider a man below deck on a ship. Is the ship docked or is it moving smoothly through the water? Modern equivalent: Do some vintage 1600 experiments!! • stretch springs, • fish swim in a fish bowl, • insects fly about, • pendulums swing back and forth, • and so on. You could play a good game of pool if either the ship is moving at a constant speed or standing still. 3 Galileo’s Conclusions Can’t determine if the ship is moving so long as it moves smoothly, at a constant velocity. There is no mechanical experiment you can perform on board the ship to determine if the ship is at rest or is moving at say velocity v, or 2v, or 5v, or 10v. (Of course, you could look outside through a window, but that doesn’t count). 4 The Inertial Reference Frame Galileo’s conclusions leads to the definition of an Inertial frame of reference and the notion that the Laws of Physics are the same in ALL inertial reference frames. No inertial frame is different in any way from another. Galileo further concludes it is impossible to tell if the earth is moving as long as it moves at a constant speed. One example of many possible reference frames 5 Frame of Reference A convenient way to identify an inertial frame of reference is when ALL the laws of physics take their simplest form. Pick a reference frame, then apply Newtonian mechanics to explain the results of an experiment. What form do Newton's Laws of Motion take to explain the experimental observations? A Newtonian inertial frame is a reference frame where a free particle travels in a straight line at constant speed, or is at rest - Newton’s 1st Law (often referred to as the Principle of Inertia). If you pick a different reference frame AND Newton’s laws hold true, how is the 2nd frame related to the first frame? These two frames are related by a Galilean transformation. (In special relativity these frames are related by a Lorentz transformation) 6 Non-Inertial Frame of Reference? If you choose a non-inertial frame of reference, fictitious forces are required to explain observations. Fictitious forces depend on the motion of the observer and therefore vary with the motion of the observer. These forces vanish for some observers, namely those in an inertial frame of reference. An example of a non-inertial frame of reference? One that is rotating about a point on the Earth's surface. The consequences? Projectile motion must include: • gravity • a fictitious force known as the Coriolis force • etc. 7 Example of a Non-Inertial Frame on Earth Case 1: Reference frame Case 2: Reference frame attached to rotating platform – attached to earth – inertial frame non inertial frame. ball Time for one revolution 2 comparable to duration of experiment Ω Scientist 2 appears Ω stationary 2 1 1 1 1 Scientist 1 appears stationary http://www.phys.unsw.edu.au/einsteinlight/jw/module1_Inertial.htm 8 Animation of a Non-Inertial Frame on Earth Case 2: Reference frame Case 1 : Reference frame attached to earth attached to rotating platform. Two clicks 9 Slow Motion: Non-Inertial Frame on Earth Case 1: Non- inertial frame Case 2: Inertial frame 10 In an inertial frame (Case 2), Newton's second law for a particle takes the form: Newton's second law in a rotating frame of reference (Case 1), rotating at angular rate Ω about an axis, takes the form: Coriolis Force Centrifugal Euler Force (mass has velocity in Force (angular velocity rotating reference frame) changes with time) Fictitious forces must be invoked to recover Newton’s 2nd Law of Motion. The angular rotation of the frame is expressed by the vector Ω pointing in the direction of the axis of rotation, vector xB locates the mass m and vector vB is the velocity of the mass according to a rotating observer (which is different from the velocity seen by the inertial observer). 11 Technically, the earth is not strictly an inertial reference frame! a) b) c) The earth a) has an orbital motion around the sun (the earth constantly accelerates radially toward the sun); b) it rotates and c) a particle placed at rest will not stay at rest – it will fall due to gravity. However, if an experiment happens so quickly that we can ignore the rotational, orbital, and gravitational acceleration, then we can approximate the earth’s reference frame as inertial. In what follows, we focus only on Inertial Frames. 12 Measuring the position of a baseball in an inertial frame Baseball is at x=2 meters At t=0 x -4 -3 -2 -1 0 +1 +2 +3 +4 Baseball is at x’=2 meters x’ vf=1 m/s -4 -3 -2 -1 0 +1 +2 +3 +4 At t=2s Baseball is at x=2 meters x -4 -3 -2 -1 0 +1 +2 +3 +4 xf Baseball is at x’=0 meters x = xf + x’ =v t + x’ f x’ vf=1 m/s -4 -3 -2 -1 0 +1 +2 +3 +4 Location of an object depends on when you measure it AND 13 on the motion of your reference frame Galilean Transformation and Inertial Reference Frames m Fixed with Moving frame respect to what?? drf v f = = constant Fixed dt frame The statement “You were Time is tt= ' travelling at 65 immutable mph” really means rr=f + r' “You were Classical travelling at 65 drdr dr ' mph wrt earth.” velocity ≡=f + =+ addition u vuf ' Quoting a speed dt dt dt always implies wrt 0 some reference dudv f du ' a≡= + =+aaf ' frame. dt dt dt 14 The Standard Form of the Galilean Transformation – align the x and x’ axis - ˆˆ let rff= x xˆˆ +=00 y x f i + j S y’ S’ y vf xo u’ x x’ -2 -1 0 +1 +2 +3 +4 -4 -3 -2 -1 0 +1 +2 +3 +4 x f xo’ Given x,y,z,t and u Given x’,y’,z’,t’ and u’ x’= x-xf = x-vft (I) x= x’+xf = x’+vft’ y’=y y=y’ take z’=z z=z’ derivative t’=t t=t’ u’=u-vf u=u’+vf 1. Often useful to focus on one object at some fixed position xo’, a “point event”. 2. The variable t measures the “time interval” since the two frames were aligned at t=0. 15 3. To obtain the inverse transform: switch the primes and let vf -> -vf Example of the Galilean Transform The experiment: Kyle the observer bridge Cartman’s firecracker vf=68 mph 10 m flat bed truck on I65 16 Thinking it through Position of truck when t’ = t = 0 S at t’ = 10 s, the firecracker explodes S’ bridge S’ vf=68 mph x’=10 m = 30 m/s x’=10 m Position and time in frame S Position and time in when fire-cracker explodes: frame S’ when fire- x=x’+vft’= 10m + 30 m/s × 10 s cracker explodes: x’=10 m; t’=10 s = 310 m; t=t’=10 s 17 Up Next – Measuring the speed of light APPENDIX: Useful web site on relativity (including simulations): http://www.phys.unsw.edu.au/einsteinlight/index.html 18 .
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