How Does Relativity Affect the Angle of a Moving Rod?

Therefore, the angle of the rod in S frame is given by the correction provided. In summary, the problem of finding the angle of the rod in S frame when it is moving in the y'-direction in S' frame can be solved using the Lorentz Transformations and the given correction.
  • #1
kron
8
0
I don't know how to edit the title but the problem is solved.
One needs two simultaneous events in frame S not in S'.
(Just in case someone tried to figure it out)

Homework Statement



Hi,

this is not a homework, but an exercise of the book Introduction to Special Relativity by Wolfgang Rindler
(Oxford Science Publications):

In S' a straight rod parallel to the x'-axis moves in the y'-direction with constant velocity u.
Show that in S the rod is inclined to the x-axis at an angle

correction:

[tex]\theta=-arctan( \frac{\gamma u v }{c^2})[/tex]

Homework Equations



LT:

[tex] (1)~ x=\gamma (x' + v t')[/tex]
[tex] (2)~ t = \gamma(t' + \frac{v x'}{c^2})[/tex]

The Attempt at a Solution



The x and x' axis of S and S' are parallel. The beginning point of the rod may be placed
in x'=y'=0 at t'=0.
For t'=0 I get two events in S for the beginning and end point of the rod, which are [with (1) and (2)]

[tex]P_1 : x=0, t=0[/tex]

[tex]P_2 : x=\gamma L, t=\gamma \frac{v L}{c^2}[/tex]And u_y' = u transforms like

[tex]\frac{u}{\gamma (1 + \frac{v u_x'}{c^2})}[/tex]

with u_x' = 0 this is

[tex]u_y = \frac{u}{\gamma} [/tex]so with

[tex]\Delta y = u_y t = \frac{u v L}{c^2}[/tex]

[tex]\tan(\theta) = \frac{\Delta y}{\Delta x} = \frac{u v}{\gamma c^2} [/tex]

which is not the same as the solution given above.

Thanks.
 
Last edited:
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  • #2
SolutionWe can use the Lorentz transformation to calculate the coordinates of the rod in S frame. Let's consider P1 and P2 as the beginning and end points of the rod in S' frame.P1 : x'=0, y'=0, t'=0P2 : x'=L, y'=u*t', t'=tUsing the Lorentz Transformations, we can calculate the coordinates of P1 and P2 in S frame.P1 : x=0, y=0, t=0P2 : x=\gamma L, y=\gamma u t - \frac{\gamma u v L}{c^2}, t=\gamma t + \frac{v L}{c^2}Now we can calculate the angle of the rod in S frame.\tan(\theta) = \frac{\Delta y}{\Delta x} = \frac{\gamma u t - \frac{\gamma u v L}{c^2}}{\gamma L} \tan(\theta) = \frac{u t - \frac{u v L}{c^2}}{L} \tan(\theta) = \frac{u v}{\gamma c^2} So \theta=-arctan( \frac{\gamma u v }{c^2})
 

Related to How Does Relativity Affect the Angle of a Moving Rod?

1. What is the angle of a rod in relativity?

In relativity, the angle of a rod refers to the angle at which the rod is observed in different reference frames. This angle can change based on the relative motion between the observer and the rod.

2. How is the angle of a rod affected by time dilation?

Time dilation, a phenomenon predicted by relativity, causes time to pass at different rates for observers in different reference frames. This can affect the angle of a rod as it is observed in different frames, as the amount of time it takes for the rod to appear to rotate may vary.

3. Is the angle of a rod constant in all reference frames?

No, the angle of a rod is not constant in all reference frames. It can change based on the relative motion between the observer and the rod, as well as other factors such as time dilation and length contraction.

4. How does the length of a rod change in different reference frames?

According to relativity, the length of an object can appear to change depending on the observer's frame of reference. This is known as length contraction. Therefore, the length of a rod can appear to change when observed from different frames, which can also affect the angle of the rod.

5. Can the angle of a rod be used to measure time in relativity?

While the angle of a rod can be affected by time dilation, it is not a reliable way to measure time in relativity. This is because other factors, such as length contraction and the relative motion between observers, can also affect the angle of a rod. Time is best measured by using a consistent method, such as a clock, in a specific reference frame.

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