3063_Exam1_solutions_sp06

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

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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 = )?
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 05/29/2011 for the course PHY 3063 taught by Professor Field during the Spring '07 term at University of Florida.

Page1 / 7

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online