Length Contraction in Relativity

In summary, an electron traveling at a speed of 0.997 c relative to Earth sees the length of an accelerator tube as 1.20 m in its own frame of reference. To find the length of the tube relative to Earth, the formula L = Lo / γ is used, where L0 is the proper length and γ is the Lorentz factor. Plugging in the values correctly, the length of the tube relative to Earth is calculated to be 15.5 m, as shown in the textbook answer.
  • #1
bjgawp
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0

Homework Statement


An electron travaels in an accelerator tube at a speed of 0.997 c relative to Earth. In the frame of reference of the electron, the length of the tube is 1.20 m. What is the length of the tube relative to earth?

Homework Equations


L = Lo / γ
Textbook answer: 15.5 m


The Attempt at a Solution


I know it is a simple plug-in problem but when I plug in the numbers, they don't work out and I'm guessing that I'm assigning the numbers wrong. For this problem, would 1.20m be L or Lo? I'd assume it to be Lo since that's the proper length in terms of the electron's reference frame. However, the numbers do not work out then - indicating that L = 1.20 m. This does not make sense to me since 1.20 m is the length measured from the electron's frame, is it not?
 
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  • #2
Yes, 1.2m is the length as measured from the electron's frame. In your formula above, L0 is the proper length of the tube as measured in the frame we are working in, and L is the length as observed by the moving observer. So, suppose we are in the frame at rest relative to the earth. Then, L0 is the proper length relative to the earth, L is the length which the electron sees, and v (in the factor gamma) is the relative velocity between the Earth and the electron.
 
  • #3
Hmm. So if 1.20m is from the electron's frame, wouldn't that mean it is assigned to L0? But, since this is the proper length we're dealing with where we have to measure the tube at the same time, it has to be from the Earth's frame .. doesn't it? Sorry, I guess I don't understand the assignation/ the nature of the variables.

L = L0 / γ

Either:
L = (1.20m)√(1 - 0.997²)
L = 0.093 m

Or:
1.20m = (L0)√(1 - 0.997²)
L0 = 15.5 m (textbook answer)
 
  • #4
bjgawp said:
Hmm. So if 1.20m is from the electron's frame, wouldn't that mean it is assigned to L0? But, since this is the proper length we're dealing with where we have to measure the tube at the same time, it has to be from the Earth's frame .. doesn't it? Sorry, I guess I don't understand the assignation/ the nature of the variables.
The 0 in L0 stands for the proper value, the value in its own restframe. Clearly from the electron's rest frame the tube is not at rest and thus it cannot be assigned L0.
 
  • #5
Ah, it just clicked. Thanks guys. Relativity is so fun :rolleyes:
 

What is length contraction?

Length contraction is a phenomenon in which an object appears to be shorter when moving at high speeds relative to an observer. This is a consequence of Einstein's theory of special relativity.

Why does length contraction occur?

According to special relativity, the speed of light is the same for all observers, regardless of their relative motion. This means that as an object moves closer to the speed of light, the distance between two points on the object will appear shorter to an observer at rest.

How is length contraction calculated?

The formula for calculating length contraction is L = L₀/γ, where L is the contracted length, L₀ is the rest length (length at rest), and γ is the Lorentz factor which depends on the velocity of the object relative to the observer.

Does length contraction only occur in one direction?

No, length contraction occurs in the direction of motion as well as in all other directions perpendicular to the direction of motion. This means that an object will appear shorter in all dimensions when moving at high speeds.

Is length contraction a real physical phenomenon or just a visual illusion?

Length contraction is a real physical phenomenon that has been confirmed through numerous experiments. It is not just a visual illusion, but a consequence of the fundamental principles of special relativity.

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