Relativity rod Question

In summary, the "Relativity rod question" is a thought experiment proposed by Albert Einstein to demonstrate the effects of relativity on the measurement of length. It is directly related to his theory of special relativity and demonstrates the concept of length contraction. Factors such as precision of instruments, orientation, and gravity can also affect the measurement of length in this scenario. While primarily a thought experiment, it has been tested in experiments and confirmed by numerous observations as an accurate description of the behavior of objects at high speeds.
  • #1
frankR
91
0
An observer, moving at a speed of 0.995c relative to a rod as shown measures its length to be 2.00m and sees its length to be oriented at 30.0 degrees with respect to the direction of motion.

a) What is the proper length of the rod? (Length as measured at rest)

b) What is the orientation angle in a reference frame moving with the rod? (Again, in rest frame)


I've found the correct values, however the math seems ambiguous. I was wondering if there was another way to solve it, with cleaner mathematics.

Do = the length of the rod as measure when the rod is moving a .995c
Lo = the adjacent side of the triangle formed by the rod moving at .995

Lo = DoCos(30.0)

D = the actual length of the rod as observed from the rest frame
L = the actual length of the adjacent side of the triangle formed by the rod in the rest frame
@ = the angle formed between L and D.
h = height of the rest frame triangle

L = Lo/sqrt(1-v^2/c^2) = DoCos(30)/sqrt{(1-(.995c)^2/c^2)}

L = 17.34m

To find D you can use the following equations.

Tan(@) = h/L
Sin(@) = h/D
Cos(@)=L/D
D^2 = L^2 + h^2

I ended up getting this to solve for D.

D = sqrt{L^2 + L^2*Sin^2{Cos^-1(L/D)}^2}

My calculator solved it and got D = 17.26m or 17.42m.

The answers are 17.3m and 3.30 degree.

Is there a cleaner way to do it?

Am I missing something?
 
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  • #2
I would only clean up the notation. Instead of using different letters for each quantity, I would use subscripts and primes.

L=Length of rod in frame S
Lx=Lcos(30)=component of length in direction of motion in frame S
Ly=Lsin(30)=component of length perpendicular to motion in S

and use L', Lx' and Ly' for the corresponding quantities in frame S'.
 
  • #3
My math is okay then?

My notation is a lot cleaner on paper, a lot cleaner the second and third time I worked the problem out. How do you make subscripts?

Thanks for your help!
 
  • #4
Originally posted by frankR
My math is okay then?

Yes, there's no other way to do it. But, using a more systematic notation will help whoever reads it (like the grader).

How do you make subscripts?

Subscripts:
If you type this: v[ sub ]0[ /sub ]
without the spaces you will get: v0.

Superscripts:
If you type this: v[ sup ]2[ /sup ]
without the spaces you will get: v2.

You can also combine the two as follows: v[ sub ]0[ /sub ][ sup ]2[ /sup ]
to get v02.
 
  • #5
CoolThanks!
 
  • #6
Something else: Regarding Greek letters. The ones in the "Smiley" menu (the ones you've been using) are really ugly, and what's worse they don't respond well to being superscripted or subscripted.

e[mu]
e[mu]

See?

Instead, use the forum code for this, like so:

& mu ;

without the spaces gives

μ

and for capitals, simply capitalize the name of the Greek letter, like so:

& Mu ;

to get

Μ

These will super- and subscript nicely, like so:

eμ
eμ
 

1. What is the "Relativity rod question"?

The "Relativity rod question" is a thought experiment that was proposed by physicist Albert Einstein to illustrate the effects of relativity on the measurement of length. It involves a rod moving at high speeds and the resulting differences in its measured length for observers in different reference frames.

2. How does the Relativity rod question relate to Einstein's theory of relativity?

The Relativity rod question is directly related to Einstein's theory of relativity, specifically the theory of special relativity. It is used to demonstrate the principle of relativity, which states that the laws of physics are the same for all observers in uniform motion.

3. Can you explain the concept of length contraction in relation to the Relativity rod question?

According to Einstein's theory of special relativity, the length of an object appears to contract in the direction of motion when measured by an observer in a different reference frame. This is known as length contraction, and it is demonstrated in the Relativity rod question as the rod appears shorter to the observer moving at a high speed compared to the stationary observer.

4. What other factors can affect the measurement of length in the Relativity rod question?

In addition to the effects of relativity, there are other factors that can impact the measurement of length in the Relativity rod question. These include the precision of the measuring instruments, the orientation of the rod in relation to the observer, and the effects of gravity.

5. Is the Relativity rod question purely theoretical or has it been tested in experiments?

The Relativity rod question is primarily a thought experiment used to illustrate the principles of relativity. However, experiments have been conducted to verify the predictions of relativity, including the effects of length contraction on moving objects. These experiments have shown that relativity accurately describes the behavior of objects at high speeds and has been confirmed by numerous observations and experiments.

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