Problems are listed in approximate order of difficulty. A single dot (•) indicates straightforward problems involving just one main concept and sometimes requiring no more than substitution of numbers in the appropriate formula. Two dots (••) identify problems that are slightly more challenging and usually involve more than one concept. Three dots (•••) indicate problems that are distinctly more challenging, either because they are intrinsically difficult or involve lengthy calculations. Needless to say, these distinctions are hard to draw and are only approximate.
•• Two perfectly synchronized clocks A and B are at rest in S, a distance d apart. If we wanted to verify that they really are synchronized, we might try using a third clock, C. We could bring C close to A and check that A and C agree, then move C over to B and check the agreement of B and C. Unfortunately, this procedure is suspect since clock C will run differently while it is being moved. (a) Suppose that A and C are found to be in perfect agreement and that C is then moved at constant speed v to B. Derive an expression for the disagreement τ between B and C, in terms of v and d. What is τ if v = 300 m/s and d = 1000 km? (b) Show that the method can nevertheless be made satisfactory to any desired accuracy by moving clock C slowly enough; that is, we can make τ as small as we please by choosing v sufficiently small.
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