# How can length contraction of a rod be measured?

Here is a simple question.

How can an observer measure length (or length contraction) of a cylindrical Iron rod, when it is moving with a relative velocity v with respect to the observer?

The reason to ask is not to inquire the math about it, but if the rod is moving say in x direction, from the observer's point of view, he will se only a circle (cross section of the rod in yz plane)!, assuming the observer is at the origin of the stationary frame, or at least on x axis.

(I think, If observer is not on x axis, and the rod is moving in x direction, the rod is not moving with uniform relative velocity with respect to the observer)

Edit: The length of the road is parallel to (infect coinciding with) x axis.

Last edited:

jtbell
Mentor
You need two or more observers or measuring devices, spaced along the x-axis, at rest with respect to each other, and with synchronized clocks. The observers or devices record the times when the ends of the rod pass them. The times of observation and the positions of the observers allow you to calculate where each end of the rod is at any time, and how fast the rod is moving.

Hi AntigenX,

Say the rod is 2m long when at rest. Now if you measure the velocity of the rod to be 10m/s then it should take L/v = 2/10 = 0.2 seconds for the rod to pass you. (i.e start the stopwatch when the front is level with you and stop the stopwatch when rear of the rod is level with you.) If you measure a time of significantly less than 0.2 seconds then you know the rod has length contracted. Measuring the speed could be done with two observers a fixed distance apart and with synchronised clocks. It could also be done by a single observer with a radar gun in the same way a police officer can check the speed of your vehicle but using the correct relativistic doppler equations. In relativity we generally allow as many observers as required to make the measurements easier with the requirement that all the observers in a single reference frame are at rest with respect to each other and all have their clocks synchronised with respect to each other.

You need two or more observers or measuring devices, spaced along the x-axis, at rest with respect to each other, and with synchronized clocks. The observers or devices record the times when the ends of the rod pass them. The times of observation and the positions of the observers allow you to calculate where each end of the rod is at any time, and how fast the rod is moving.

Does that mean, a single observer can not accomplish the measurement? The suggested approach came to me, but I though, what if the rod has passed the observer?

Hi AntigenX,

Say the rod is 2m long when at rest. Now if you measure the velocity of the rod to be 10m/s then it should take L/v = 2/10 = 0.2 seconds for the rod to pass you. (i.e start the stopwatch when the front is level with you and stop the stopwatch when rear of the rod is level with you.) If you measure a time of significantly less than 0.2 seconds then you know the rod has length contracted. Measuring the speed could be done with two observers a fixed distance apart and with synchronised clocks. It could also be done by a single observer with a radar gun in the same way a police officer can check the speed of your vehicle but using the correct relativistic doppler equations. In relativity we generally allow as many observers as required to make the measurements easier with the requirement that all the observers in a single reference frame are at rest with respect to each other and all have their clocks synchronised with respect to each other.

Greetings kev!
Yes, but that is understood, however, I was wondering how one observer would accomplish the task at hand. Also, If we introduce more observers in same frame, they are supposed to be along x axis, otherwise, the relative velocity will no longer be uniform. And if they are all on x axis, the situation will be same as the rod passing the observer, which is precisely the case I wish to avoid, because, almost in no real case this will be the situation.
The radar gun will observe the velocity, and needs to be used from a point perpendicular from the path of the rod, the motion will no longer be with uniform velocity with respect to the radar gun.

To AntigenX;
Yes the rod confined to the x axis, is an idealistic example, and not verifiable.
If the rod is offset from the x axis, and the rod is in constant uniform motion, the observer could measure the time of reflected signal as it passes.
The velocity will be constant for the device. The perceived length of the rod will not.