# Measuring contracted length

Consider an observer equipped with a good ruler and a rod. This observer measures the length of the rod and finds it 30 cm. Now this observer with his ruler and rod rides a space ship and moves with a high speed. He then tries to measure the length of the rod using his ruler. According the SR, the length of the rod is expected to be contracted, but he will measure 30 cm because the ruler will also undergo length contraction with the same proportion. Therefore he will not measure any length contraction!!

How is this possible?

Or length contraction is a "theoretical" consequence of SR which could never be "measured" - since the tools of measuring will undergo length change with same ratio??

Thanks to any ideas.

Doc Al
Mentor
According the SR, the length of the rod is expected to be contracted, but he will measure 30 cm because the ruler will also undergo length contraction with the same proportion. Therefore he will not measure any length contraction!!
No, according to SR the rod will not contract as it is not moving with respect to the spaceship observer. (But other observers, who see the rod moving with respect to them, will claim that it contracts along the direction of motion.)

1 person
ghwellsjr
Gold Member
Consider an observer equipped with a good ruler and a rod. This observer measures the length of the rod and finds it 30 cm. Now this observer with his ruler and rod rides a space ship and moves with a high speed. He then tries to measure the length of the rod using his ruler. According the SR, the length of the rod is expected to be contracted, but he will measure 30 cm because the ruler will also undergo length contraction with the same proportion. Therefore he will not measure any length contraction!!

How is this possible?

Or length contraction is a "theoretical" consequence of SR which could never be "measured" - since the tools of measuring will undergo length change with same ratio??

Thanks to any ideas.
You pretty much have the right idea.

When a rod and ruler have no relative velocity, you don't have to worry about when you look at both ends of the rod to see where they are in relation to the ruler. You could look at one end today and the other end tomorrow and you get the same answer that you would get if you looked at both ends today.

But when the rod is moving with respect to the ruler, you will get different answers if you look at one end today and the other end some other day. So how do you know what is the right answer? Of course you could cheat and say you need to get the same answer you got when you measured the rod with a ruler that had no relative velocity but that leads to inconsistencies.

Wouldn't it be nice to have a means to measure the length of the rod with a single process that doesn't care about its motion and yet gets the same answer as the ruler gets when there is no relative motion?

That's what Special Relativity does by means of a process using radar signals. This is essentially what a laser rangefinder does. It works by measuring how long it takes for a light signal to make a round trip to some object. If you then assign the time of the measurement as the average between when the signal was sent out and when it was received, you are half way there. Then all you have to do is make measurements to both ends of the rod making sure that the assigned time is the same for both and merely take the difference.

1 person
Doc Al
Mentor
Or length contraction is a "theoretical" consequence of SR which could never be "measured" - since the tools of measuring will undergo length change with same ratio??
Realize that to see length contraction you would not measure the length of a moving rod with a co-moving rod, but with a "stationary" rod--or the equivalent.

Nugatory
Mentor
Or length contraction is a "theoretical" consequence of SR which could never be "measured" - since the tools of measuring will undergo length change with same ratio??

As others have pointed out, the length contraction is between a measured object and a measuring stick that are not at rest with respect to one another. If they're both moving at the same speed (which is equivalent to saying that they're both at rest relative to one another) then there is no length contraction.

All the cool relativistic phenomena (time dilation, length contraction, mass increase) work this way - they're observed between observers moving relative to one another, and as far as you're concerned you're always at rest and it's the other guy who is moving so he's the one is dilated/contracted/mass-increased. This will be a bit less surprising if you remember that right now you are moving at 99% of the speed of light relative to some observer somewhere - you are, but you don't expect it to affect you.

It is very unlikely that you'll ever see length contraction measured with macroscopic objects like meter sticks; consider the logistical difficulties of getting close to a macroscopic object moving at an appreciable fraction of the speed of light relative to you and you'll see why. However, length contraction has been observed: From the FAQ at the top of this page we have http://www.edu-observatory.org/physics-faq/Relativity/SR/experiments.html#Length_Contraction; and if you search this forum for some of the threads on the muon lifetime experiments you'll see how they can be explained in terms of length contraction as well as time dilation.