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  • Describe proper length.
  • Calculate length contraction.
  • Explain why we don’t notice these effects at everyday scales.
A long isolated double-lane road banked by barren land on both sides.
People might describe distances differently, but at relativistic speeds, the distances really are different. (credit: Corey Leopold, Flickr)

Have you ever driven on a road that seems like it goes on forever? If you look ahead, you might say you have about 10 km left to go. Another traveler might say the road ahead looks like it’s about 15 km long. If you both measured the road, however, you would agree. Traveling at everyday speeds, the distance you both measure would be the same. You will read in this section, however, that this is not true at relativistic speeds. Close to the speed of light, distances measured are not the same when measured by different observers.

Proper length

One thing all observers agree upon is relative speed. Even though clocks measure different elapsed times for the same process, they still agree that relative speed, which is distance divided by elapsed time, is the same. This implies that distance, too, depends on the observer’s relative motion. If two observers see different times, then they must also see different distances for relative speed to be the same to each of them.

The muon discussed in [link] illustrates this concept. To an observer on the Earth, the muon travels at 0.950 c size 12{c} {} for 7.05 μ s size 12{c} {} from the time it is produced until it decays. Thus it travels a distance

L 0 = v Δ t = ( 0.950 ) ( 3.00 × 10 8 m/s ) ( 7.05 × 10 6 s ) = 2.01 km

relative to the Earth. In the muon’s frame of reference, its lifetime is only 2.20 μ s . It has enough time to travel only

L = v Δ t 0 = ( 0 . 950 ) ( 3 . 00 × 10 8 m/s ) ( 2 . 20 × 10 6 s ) = 0 .627 km .

The distance between the same two events (production and decay of a muon) depends on who measures it and how they are moving relative to it.

Proper length

Proper length L 0 size 12{L rSub { size 8{0} } } {} is the distance between two points measured by an observer who is at rest relative to both of the points.

The Earth-bound observer measures the proper length L 0 size 12{L rSub { size 8{0} } } {} , because the points at which the muon is produced and decays are stationary relative to the Earth. To the muon, the Earth, air, and clouds are moving, and so the distance L size 12{L} {} it sees is not the proper length.

In part a observer observes from ground frame of reference a muon above earth with speed v in the rightward direction. The distance between the muon and the place where it disintegrates is two point zero one. In part b the system is shown in motion having velocity v in the leftward direction. So, the cloud and ground are displaced zero point six two seven kilo meter in the opposite direction.
(a) The Earth-bound observer sees the muon travel 2.01 km between clouds. (b) The muon sees itself travel the same path, but only a distance of 0.627 km. The Earth, air, and clouds are moving relative to the muon in its frame, and all appear to have smaller lengths along the direction of travel.

Length contraction

To develop an equation relating distances measured by different observers, we note that the velocity relative to the Earth-bound observer in our muon example is given by

v = L 0 Δ t . size 12{v= { {L rSub { size 8{0} } } over {Δt} } } {}

The time relative to the Earth-bound observer is Δ t size 12{Δt} {} , since the object being timed is moving relative to this observer. The velocity relative to the moving observer is given by

v = L Δ t 0 . size 12{v= { {L rSub { size 8{0} } } over {Δt} } } {}

The moving observer travels with the muon and therefore observes the proper time Δ t 0 size 12{Δt rSub { size 8{0} } } {} . The two velocities are identical; thus,

L 0 Δ t = L Δ t 0 . size 12{ { {L rSub { size 8{0} } } over {Δt} } = { {L} over {Δt rSub { size 8{0} } } } } {}

We know that Δ t = γ Δ t 0 size 12{Δt=γΔt rSub { size 8{0} } } {} . Substituting this equation into the relationship above gives

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
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Adin
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Joe
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Damian Reply
research.net
kanaga
sciencedirect big data base
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Introduction about quantum dots in nanotechnology
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what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
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it is a goid question and i want to know the answer as well
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characteristics of micro business
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for teaching engĺish at school how nano technology help us
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Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
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Cied
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
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Source:  OpenStax, General physics i phy2201ca. OpenStax CNX. Jul 03, 2013 Download for free at http://legacy.cnx.org/content/col11523/1.4
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