Phsics 2130: General Phsics 3 Lecture 8 Length contraction and Lorent Transformations. Reading for Monda: Sec. 1.13, start Chap. 2 Homework: HWK3 due Wednesda at 5PM.
Last Time: Time Dilation Who measures the Proper Time, Dt 0? One obserer with one clock who sees local eents at their location. That s Proper Time Obserers in other frames DON T measure Proper Time. More obserers are required. The see more time Dt pass: Dt Dt 0 1 1 c 2 2
Length of an object... -3-2 -1 0 1 2 3... This length, measured in the stick s rest frame, is its proper length. This stick is 3m long. I measure both ends at the same time in m frame of reference. Or not. It doesn t matter, because the stick isn t going anwhere. But as we know, at the same time is relatie it depends on how ou re moing.
Length of an object S... -3-2 -1 0 1 2 3... S 0 Another obserer comes whiing b at uniform elocit,. This obserer can measure the length of the stick, b measuring time. Eent 1 Origin of S passes left end of stick.
Length of an object S... -3-2 -1 0 1 2 3... S 0 Eent 1 Origin of S passes left end of stick. Eent 2 Origin of S passes right end of stick.
Length of an object S... -3-2 -1 0 1 2 3... S 0 Eent 1 Origin of S passes left end of stick. Eent 2 Origin of S passes right end of stick. How man obserers are needed in S to measure the time between eents? A) 0 B) 1 C) 2 D) 57
Length of an object S... -3-2 -1 0 1 2 3... S 0 Eent 1 Origin of S passes left end of stick. Eent 2 Origin of S passes right end of stick. Which frame measures the Proper Time between the eents? A) S B) S C) neither
The frames agree on relatie speed In frame S: length of stick = L (this is the proper length) time between measurments = Dt Relatie speed of frame S is = L/Dt In frame S : length of stick = L (this is what we re looking for) time between measurements = Dt Relatie speed of frame S is = L /Dt Q: a) Dt = Dt or b) Dt = Dt Follow the proper time!
A little math Speeds are the same (both refer to the relatie speed). And so L L L Dt Dt Dt L L Length measured in frame moing rel. stick Length in stick s rest frame (proper length) Length contraction is a consequence of time dilation (and ice-ersa).
Curl measures L C Larr measures L L Moe measures L M Curl runs b real fast with a stick he knows to be of length L C. Larr and Moe are both standing on the ground and each measures the stick as it goes b. How are the three measurements related? a) L C < L L < L M b) L C > L L > L M c) L C = L L = L M d) L C < L L = L M e) L C > L L = L M
Space-time ct A useful wa to isualie things in relatiit. Think of eents as four (x,,, ct) coordinates. x Suppose something is moing to the right in frame S. It starts at x=0 at t=0. It moes to positie x at positie time. Connect the dots this is the world line.
The Lorent transformation S S 0 A stick is at rest in S. Its endpoints are the eents (position, c*time) = (0,0) and (x,0) in S. S is moing to the right with respect to frame S. x Eent 1 left of stick passes origin of S. Its coordinates are (0,0) in S and (0,0) in S.
The Lorent transformation S x As iewed from S, the stick s length is x /. Then, time t passes. According to S, where is the right end of the stick? a) x = t b) x = -t c) x = t + x / d) x = -t + x / e) x = t x /
The Lorent transformation S x = t + x /. This relates the coordinates of an eent in one frame to its coordinates in the other. Algebra x = (x-t)
Transformations If S is moing with speed in the positie x direction relatie to S, then the coordinates of the same eent in the two frames is related b: In Galilean relatiit x t x t t In Special relatiit x t ( x ( t t) x 2 c ) In a minute Remark: this assumes (0,0) is the same eent in both frames.
The Lorent transformation 2 S S 0 A stick is at rest in S. Its endpoints are the eents (position, c*time) = (0,0) and (x,0) in S. S is moing to the left with respect to frame S. x Eent 1 left of stick passes origin of S. Its coordinates are (0,0) in S and (0,0) in S.
The Lorent transformation 2 0 As iewed from S, the stick s length is x/. Time t passes. According to S, where is the right end of the stick? a) x = t b) x = -t c) x = t + x/ d) x = -t + x/ e) x = t x/
The Lorent transformation 2 t x x x x t 0 x = -t + x/. This relates the coordinates of an eent in one frame to its coordinates in the other. Algebra t t x 2 c
Transformations t t t x x If S is moing with speed in the positie x direction relatie to S, then the coordinates of the same eent in the two frames is related b: In Galilean relatiit In Special relatiit ) ( ) ( 2 x c t t t x x Remark: this assumes (0,0) is the same eent in both frames. 2 ( ) ( ) x x t t t x c
Transformations We now hae the tools to compare positions and times in different inertial reference frames. NOW we can talk about how elocities, etc. compare.: In Galilean relatiit x t t x t Newton s Laws worked with these x t ( x ( t In Special relatiit t) c 2 x ) x ( xt) t ( t x) 2 c Newton s Laws need attention. Momentum and Energ definitions!