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Chapter1_13 - i.e L 1< L 2 Measurement C is the same as...

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PHY3063 R. D. Field Department of Physics Chapter1_13.doc University of Florida Moving Lengths Parallel Moving Lengths: The “proper length”, L 0 , of a rod is defined to be its length at rest. Suppose the rod is at rest in the O'-frame so that x' = L 0 . The length of the moving rod as observed in the O-frame , L, is defined be marking both ends of the rod simultaneously in S ( i.e. t=0). Thus, ) ( t c x x = β γ implies that L x L x = = = 0 and / 0 L L = where 2 1 / 1 / = = c V . Perpendicular Moving Lengths: Perpendicular moving distances are invariant under uniform velocities since 0 L y y L = = = Proof: Suppose two rods both have length L 0 at rest. Now suppose that Rod 2 is at rest and Rod 1 is moving to the right at speed V. As the two rods cross they mark each other to see which one is longer (measurement A) and suppose that Rod 1 is shorter L 1 < L 2 ( i.e. moving rods are shorter). Measurement B is equivalent to measurement A by rotational invariance (
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Unformatted text preview: i.e. L 1 < L 2 ). Measurement C is the same as measurement A except Rods 1 and 2 are interchanged and we deduce that L 2 < L 1 . But measurement C must be equivalent to measurement B by the 1 st postulate of special relativity ( i.e. “Relativity Principle”). Measurement A and measurement C are not compatible unless L 1 = L 2 . Hence, L 1 = L 2 .! y x z y’ z’ x’ V Rod at rest in the O' frame O' O Mark ends simultaneously in the O frame L Parallel moving lengths are shorter! Perpendicular moving lengths are invariant! Length Contraction! Measurement A Rod 2 Rod 1 V Rod 2 at rest. L 2 =L L 1 <L Rod 1 moving to the right at speed V. Measurement B Rod 2 Rod 1 V Rod 2 at rest. L 2 =L L 1 <L Rod 1 moving to the left at speed V. Measurement C Rod 2 Rod 1 V Rod 1 at rest. L 2 <L L 1 =L Rod 2 moving to the right at speed V....
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