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3063_Exam1_solutions_sp06

3063_Exam1_solutions_sp06 - PHY3063 Spring 2006 R D Field...

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PHY3063 Spring 2006 R. D. Field Exam 1 Solutions Page 1 of 7 January 31, 2006 PHY 3063 Exam 1 Solutions Problem 1 (25 points): A rod with is at rest in the O-frame ( parallel to the x-axis ) with the left end of the rod ( L ) at x = 0 and right end ( R ) at x = R and a cart with rest length L 0 = 300 m is at rest in the O'-frame ( parallel to the x'-axis ) with the right end of the cart ( r ) at x' = 0 and left end (l) at x' = -L 0 . as shown in the figure . The O'-frame is moving to the right along the positive x-axis of the O-frame with velocity V = 0.8c and the origins of the two frames coincide at t = t' = 0 . (Note that c = 3×10 8 m/s.) Consider the following four events: Event Description A Right end of cart crosses left end of rod (Lr) B Right end of cart crosses right end of rod (Rr) C Left end of cart crosses left end of rod (Ll) D Left end of cart crosses right end of rod (Rl) In the O-frame event B and event C occur simultaneously ( i.e. 0 = = B C t t t ). Part A (10 points): Plot the four events A , B , C , and D on the space-time plots below and label the (ct, x) coordinates of each event in the O-frame and the (ct',x') coordinates of each event in the O'-frame . (Express the coordinates in terms of c, β , γ , and L 0 ). . O-frame x ct Path of r Path of l A = (0,0) (L 0 / βγ ,0) = C B = (L 0 / βγ ,L 0 / γ ) D = (2L 0 / βγ ,L 0 / γ ) θ O'-frame x' ct' Path of L Path of R θ (0,0) = A B = (L 0 / βγ 2 ,0) (L 0 / β ,- L 0 ) = C D = (L 0 (2- β 2 )/ β ,-L 0 ) Solution: O-frame: The key to plotting the events is to first plot the paths of the right (r) and left (l) end of the cart in the O-frame. Both r and l follow straight (parallel) lines with β θ / 1 tan = . Events A and C lie at x = 0 and events B and D lie at x = R. Also, / R ct B = and ) /( / 0 βγ L L ct C = = , where I used γ / 0 L L = . Thus, / 0 L R = and ) /( 2 0 L ct D = . O'-frame: We can use the Lorentz transformations to compute the coordinates of the four events in the O'-frame as follows: y x z y' z' x' V O O' Cart at rest in the O'-frame. Rod at rest in the O-frame. L R l r
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PHY3063 Spring 2006 R. D. Field Exam 1 Solutions Page 2 of 7 January 31, 2006 0 ) ( ' 0 ) ( ' = = = = A A A A A A ct x x x ct ct β γ 0 0 0 0 ) ( ' ) ( ' L L ct x x L L x ct ct C C C C C C = = = = = = βγ 0 ) ( ' ) 1 ( ) ( ' 0 0 2 0 2 0 0 0 = = = = = = = L L ct x x L L L L x ct ct B B B B B B 0 0 0 2 0 0 0 0 0 2 ) ( ' ) 2 ( 2 2 ) ( ' L L L ct x x L L L L L x ct ct D D D D D D = = = = = = = Both R and L follow straight (parallel) lines with θ / 1 tan = . Events A and B lie at x' = 0 and events C and D lie at x' = -L 0 . Also, / ' 0 L ct C = , ) /( ' 2 0 L ct B = , and / ) 2 ( ) /( / ' 2 0 2 0 0 = = L L L ct D . Part B (5 points): How long ( in microseconds ) does it take for the cart to cross over the rod according to an observer in the O-frame ( i.e. what is A D t t t = )?
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3063_Exam1_solutions_sp06 - PHY3063 Spring 2006 R D Field...

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