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39. Relativity 39.4 Consequences of the Special Theory of Relativity 39.6 Relativistic Linear Momentum and the Relativistic Form of Newton's Laws Suggested HW problems: Chp39: 5-12,16-21,27-31 Consequences of the Special Theory of Relativity a) Simultaneity of two events b) Length Contraction c) Time Dilatation d) The Twins Paradox 1 Simultaneity of two events two events are simultaneous if t1= t2 ; then, in general the events are not simultaneous in another reference frame 2 ∆t ' = t2′ − t1′ = γ ( ∆ !t − V ∆x / c ) =0 =− γV c 2 ∆x = − γV c2 ( x2 − x1 ) ≠ 0 Example: two lightning bolts striking the ends of a moving boxcar occur simultaneously for a stationary observer at O (AO=OB) but not for a moving observer at O’ Length Contraction 1 2 proper length in the rest frame 1 2 to measure the length of the ruler in the frame in which it moves with velocity v= βc requires the simultaneous determination of the coordinates x2 and x1 (i.e., t2-t1=0) (in the rest frame you may “take your time” to measure the proper length of the ruler which is motionless) Lp = x2′ − x1′ = γ ( x2 − x1 ) = γ L ⇒ L = L p 1 − β 2 < Lp 2 Time Dilatation the proper time , i.e., the time shown by a clock in its rest frame, is τ = t2′ − t1′ x2′ = x1′ and clearly in the “lab(oratory) frame” (in which the clock moves with velocity v , from the Lorentz transformations ∆t = γ [( t2′ − t1′) + (v / c 2 )( x2′ − x1′)] = γτ > τ ! "#$ =τ =0 If v is not constant: τ dt ′ 0 1− v / c ∆t = ∫ 2 2 <τ Time Dilatation proper time interval between sending and receiving the light signal is 2d τ= c The time measured by the observer O ∆t = τ 1 − (v / c ) 2 = 2d c −v 2 2 >τ 3 The Twins Paradox S G before the cosmic trip after the cosmic trip ∆tS = 1 − (v / c) 2 ∆tG < ∆tG assuming ∆tG = 50 years, and v / c = 0.95 ⇒ ∆tS = ? The Twins Paradox: Space-Time Graph Finish Start 4 Problem Relativistic Linear Momentum p= mv 1 − (v / c ) 2 = γ mv • p is conserved if the sum (or impulse) of all forces acting upon the system is zero (e.g., in collisions) • for c→∞ (or v<<c), p assumes its classical expression 5 Relativistic Form of Newton's Laws dp d mv F= = 2 dt dt 1 − (v / c) ma mv (v ⋅ a ) = + 2 2 3/ 2 1 − (v / c ) 1 − (v / c) where a= dv dt is the acceleration Problem 6 For next Lecture: From the text READ Sec. 39.7-9! 39. Relativity 39.7 Relativistic Energy 39.8 Equivalence of Mass and Energy 39.9 Relativity and Electromagnetism Suggested HW problems: Chp39:35-42,46-50 7