The Open Door Web Site
HOME PAGE BIOLOGY CHEMISTRY PHYSICS ELECTRONICS HISTORY HISTORY of SCI & TECH MATH STUDIES LEARN FRENCH STUDY GUIDE PHOTO GALLERY
ATOMIC and NUCLEAR ELECTRICITY and MAGNETISM MEASUREMENTS MECHANICS OPTICS PRACTICAL WORK QUESTIONS RELATIVITY THERMAL PHYSICS WAVES
RELATIVITY
Google
Custom Search
Length Contraction
Our friends, A and B, of time dilation and relativity of simultaneity fame, have put their heads together again and dreamed up another very-difficult-but-theoretically-not-impossible-to-perform experiment (a thought experiment).  
This time, the intention is to find out how the measured length of a rigid rod depends on the state of its motion relative to the person making the measurement.
   
   
A carries a "light beam clock" (as seen in time dilation).
 
A and B have a relative velocity v directed at 90 to the path of the light pulses.
 
A is going to use the clock to measure the length of a rigid rod carried by B.
 
The time, measured by A, for the light pulse to go from torch to mirror is to.
 
 
 
 
 
A and B carefully arrange things such that the light pulse leaves the torch (not shown in the diagrams below) at the instant when it is just next to the end 1 of the rigid rod and returns to the torch when it is just next to the other end.  
   
 
This means that in a time (measured by A) equal to 2to , B moved a distance equal to the length of the rod, L.  
Therefore, the length of the rod, as measured by A is  
 
   
Now, how does this length compare with a measurement made by B of the same rigid rod, using the same light beam clock?  
The diagrams above are all drawn from A's point of view. As in the time dilation situation, B considers that the light took a different path, as shown below.  
   
If the time taken for the light to go to the mirror from B's point of view is t, then B will conclude that the length of the rod is given by  
 
Dividing the first equation above by the second gives  
 
and, as has been shown, the relation between t and to is    
   
This means that the relation between L (length of rod in A's frame of reference) and Lo (length of rod in B's frame of reference) is   
   
usually written as   
 
and, using the abbreviation mentioned on the time dilation page, this becomes  
 
   
Points to note:  
The length of a rod as measured by an observer who is at rest relative to it is called the proper length, Lo . In this example, B gives the "proper length".  
   
Other inertial observers moving relative to the rod will find lengths less than the proper length, hence the term length contraction in the title.  
This effect is often called the Fitzgerald contraction after the Irish mathematician who predicted it using a different theory.  
   
This contraction only affects the dimension of the rod which is parallel to the direction of the relative motion.  
   
   
One way of explaining why two current carrying conductors attract or repel each other is based on this effect (see here).   
SITE MAP
WHAT'S NEW
ABOUT
PRIVACY
COPYRIGHT
SPONSORSHIP
DONATIONS
ADVERTISING
 

© The Open Door Team
2016
Any questions or
problems regarding
this site should be
addressed to
the webmaster

David Hoult 2017

Hosted By
Web Hosting by HostCentric

 
SiteLock
 
 
Relativity Index Page